Download x - carpath - Alexandru Ioan Cuza

ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
QUANTIZATION OF GAUGE FIELDS ON DE-SITTER GROUP
BY FUNCTIONAL INTEGRAL METHOD
Viorel Chiriţoiu*, Gheorghe Zet**, Simona Babeţi*
*Technical Physics Department, "Politehnica” University Timişoara, Romania
**Department of Physics, ”Gh. Asachi” Technical University, Iaşi, Romania
Abstract. A formulation of the de-Sitter symmetry as purely inner symmetry defined on a fixed Minkowski
space-time is presented. We define the generators of the de-Sitter group and write the equations of structure
using a constant deformation parameter λ. The conserved gauge currents are calculated and their physical
meaning is given. Local gauge transformations and corresponding covariant derivative depending on gauge
fields are also obtained. We study the behaviour of gauge fields, the torsion and curvature tensors and give a
regularization technique in terms of ζ - function.
1. Introduction
The gauge theory of gravitation allows us to describe the gravity in a
similar way with other interactions (electromagnetic, weak, strong). As gauge
group of gravitation we use the de-Sitter group endowed with a constant
deformation parameter λ .
In Section 2 we define the generators of de-Sitter group and the
commutation relations between generators are obtained. We present de-Sitter
symmetry as a pure inner symmetry in Section 3.We also calculate the
conserved current and functions δf γ (x) which ensure us the invariance of the
action.
Sections 4 and 5 are reserved to introduce the covariant derivative, the
gauge fields and to study their local infinitesimal transformations. We also
introduce the torsion and curvature tensors and the strength field tensor.
We write the invariant matter actions for a scalar, spinor and vector fields in
Section 6 and present a renormalization technique of gauge fields in Section 7.
2. The generators of de-Sitter group
We consider a gauge theory of gravitation having de-Sitter (DS) group with
generators
 Jα 5 ≡ Πα = pα + λ2 Kα + λΣα 5


1
 J αβ ≡ mαβ = i (xα ∂ β − xβ ∂α ) + Σαβ
2

where
(2.1)
2
V. CHIRITOIU ET AL..
γ
Kα = tα pγ ; tα γ = 2ηαβ x β xγ − σ 2δ α γ ; σ 2 = η µν x µ xν
(2.2)
Because tα β depends of coordinates only we have
[t
β
α
, tγ
δ
]= 0
(2.3)
We can find the next commutation relations
 pα , pβ = 0

 Kα , K β = 0
 p , K + K , p = 4L
α
β
αβ
 α β
 pα , Lβγ = ηαβ pγ − ηαγ pβ

 Kα , Lβγ = ηαβ Kγ − ηαγ K β

 pα , Σ βγ = 0; Kα , Σ βγ = 0
 p , Σ = 0; K , Σ = 0
α
β5
 α β5
 Σα 5 , Σ β 5 = −2iΣαβ

 Σα 5 , Σ βγ = 2i (ηαβ Σγ 5 − ηαγ Σ β 5 )
where Lαβ = xα ∂ β − xβ ∂α
[
[
[
[
[
[
[
[
[
]
]
] [
]
]
]
] [
]
] [
]
]
]
(2.4)
Using equations (2.4) the commutation relations between de-Sitter generators
become
[Π α , Π β ] = −4iλ2 mαβ

[Π α , mβγ ] = i (ηαβ Π γ − ηαγ Π β )
(2.5)

[ mαβ , mγδ ] = i(η βγ mαδ − η βδ mαγ + ηαγ mδβ −ηαδ mγβ )
[Σ , Σ ] = 2i (η Σ − η Σ + η Σ − η Σ )
αδ γβ
βγ αδ
βδ αγ
αγ δβ
 αβ γδ
In the limit λ → 0 process named contraction de-Sitter algebra is
transformed into the Poincaré algebra and
Πα = pα
(2.6)
3. Pure de-Sitter internal symmetry
From Noether theorem and considering a set of fields ϕ j , with j = 1,..., n ,
∫
their dynamics will be specified by the action S M = d 4 x LM ( x,ϕ j , ∂α ϕ j ) .
3
QUANTIZATION OF GAUGE FIELDS
δS M = 0 yields then the equations of motion. If there are functions δf γ (x) for
which
(
)
d 4 x′LM ( x′, ϕ ′j ( x′), ∂′α ϕ ′j ( x′)) = d 4 x LM ( x, ϕ ′j ( x), ∂′α ϕ ′j ( x)) + ∂ γ δf γ ( x) (3.1)
and imposing δS M = 0 , where
 ∂LM
 
∂LM
⋅ δTϕ j + 
⋅ ∂αϕ j − η γ α LM δxα  +
 ∂ (∂ ϕ )
  (3.2)
 ∂ (∂ γ ϕ j )
γ j

 

δS M = − ∫ d 4 x∂ γ −
+ ∫ d 4 x∂ γ δf γ
then, there is a conserved current J γ found to be
Jγ = −
∂LM
⋅ δ T ϕ j + Θγ α ⋅ δxα + δf γ
∂ (∂ γ ϕ j )
(3.3)
where Θγ α is the energy – momentum tensor.
The concept of global de-Sitter symmetry is given by the relation
 xα → x′α = xα + ε α + λ2ε ξ tξ α + ω α β x β


i αβ
α
ϕ j ( x) → ϕ ′j ( x′) = ϕ j ( x) − ω Σαβ ϕ j ( x) + iλε Σα 5ϕ j ( x)
4

(3.4)
In this case the variation of coordinates and fields are
δxα = ε α + λ2ε ξ tξ α + ω α β x β
(3.5)
i
4
δϕ j ( x) = − ω αβ Σαβ ϕ j ( x) + iλε α Σα 5ϕ j ( x)
(3.6)
and
δf γ = 0
(3.7)
If we consider the de-Sitter symmetry as a pure inner symmetry the
infinitesimal transformations are

 xα → x′α = xα

2 ξ α
α
α
β
ϕ j ( x) → ϕ ′j ( x′) = ϕ j ( x) − ε + λ ε tξ + ω β x ⋅ ∂α ϕ j ( x) − (3.8)

− i ω αβ Σαβ ϕ j ( x) + iλε α Σα 5ϕ j ( x)
 4
(
)
where the parameters of de-Sitter group ε α and ω αβ depend of x. In this case
the variation of action is
4
V. CHIRITOIU ET AL..
(
)
δS M = − ∫ d 4 x ε α + λ2ε ξ tξ α + ω α β x β δ γ α ⋅ ∂ γ LM −
(3.9)
− λ2ε ξ ∫ d 4 xLM δ γ α ∂ γ tξ + ∫ d 4 x∂ γ δf γ
α
and from δS M = 0 results
δf γ = −δ γ α (ε α + λ2ε ξ tξ α + ω α β x β )LM
(3.10)
and the conserved current is
J γ = Θγ α ⋅ ε α +
1 γ
∂LM
Σα 5ϕ j (3.11)
M αβ ⋅ ω αβ + λ2Θγ α ⋅ ε ξ tξ α − iλε α
2
∂ (∂ γ ϕ j )
where the angular momentum tensor is
M γ αβ = Θγ α xβ − Θγ β xα +
i ∂LM
Σαβ ϕ j
2 ∂ (∂ γ ϕ j )
(3.12)
We can rewrite the relation as
 xα → x′α = xα

ϕ j ( x) → ϕ ′j ( x′) = ((1 + Θ )ϕ j )(x )
with
{
(3.13)
}
Θ( x) = − ε γ ( x) + λ2ε ξ ( x)tξ γ ( x) + ω γδ ( x) xδ ⋅ ∂ γ −
(3.14)
i
i
− ω γδ ( x)Σγδ + iλε γ ( x)Σγ 5 ≡ iε γ ⋅ Π γ − ω γδ ⋅ mγδ
4
2
In this case
d 4 xLM (ϕ ′j ( x), ∂αϕ ′j ( x) ) = d 4 xLM (ϕ j ( x), ∂αϕ j ( x) ) −
(
)
− d 4 x ε γ (x ) + λ2ε ξ tξ ( x ) + ω γ β (x )x β ⋅ ∂ γ LM (ϕ j ( x), ∂α ϕ j ( x) )
γ
(3.15)
So that equation (3.15) does not lead the invariance of action because the
second term in the right hand side is no longer a pure divergence.
~
4. Local DS gauge invariance. The covariant derivative ∇α and its
decomposition in function of Πα and mγδ
As Θ(x) can be written as a function of DS generators we introduce the
~
covariant derivative ∇α
~
∇α = ∂α + Bα
together with the decomposition of Bα function of Πα and mγδ
(4.1)
5
QUANTIZATION OF GAUGE FIELDS
Bα = −iBα γ ⋅ Π γ +
γ
introducing 16 fields Bα
Bα
γδ
i
Bα
2
γδ
⋅ mγδ
(4.2)
for local translations and 24 antisymmetric fields
for the local Lorentz rotations.
We are ready now to discuss the behaviour of Bα under the local gauge
transformations. In fact
[
]
δBα = Bα′ − Bα = Θ, ∇α = [Θ, ∂α + Bα ] = Θ∂α − ∂α Θ + [Θ, Bα ] (4.3)
~
It can be easily find that
β
Θ∂α = ωα ∂ β
(4.4)
and
i
γ
β
∂α Θ = i∂α ε γ ⋅ Π γ + iλ2ε ξ ⋅ ∂α tξ ⋅ pγ − ∂α ω γδ ⋅ mγδ + ωα ∂ β (4.5)
2
After evaluation of the commutation relation
[Θ, Bα ] = iε γ ⋅ Π γ − i ω γδ ⋅ mγδ ,−iBα ε ⋅ Π ε + i Bα εξ ⋅ mεξ 

2
(4.6)

2
and writing the variation of gauge fields as a function of generators
i
2
δBα = −iδBα γ ⋅ Π γ + δBα γδ ⋅ mγδ
we obtain
δBα
γ
ξ
= ∂α ε γ + Bα ⋅ ∂ξ ε γ − Bα
γ
+ ω εξ xε ⋅ ∂ ξ Bα − ε δ Bα
γδ
εξ
xε ⋅ ∂ ξ ε γ − ε ξ ⋅ ∂ ξ Bα γ +
β
+ ωα Bβ γ + ω γ δ Bα
⋅ ∂ ε tξ + λ2 Bα tε ⋅ ∂ ξ ε γ +
+ λ2 Bα ε ξ ⋅ ∂ ε tξ − λ2ε δ Bα
εξ
xε ⋅ ∂ξ tδ γ + λ2 Bα ω εξ xε ⋅ ∂ ξ tδ
γ
ε
ε
+ λ2ε ξ ⋅ ∂α tξ
ξ
γ
γ
δ
− λ2ε ε Bα
δ
− λ2ε ξ tξ ⋅ ∂ δ Bα
(4.7)
γ
−
ξ
δ
(4.8)
γ
and
δBα
γδ
ξ
= ∂α ω γδ + Bα ⋅ ∂ξ ω γδ − Bα
+ ω εξ xε ⋅ ∂ξ Bα
ξ
γδ
− λ2ε ε tε ⋅ ∂ξ Bα
β
+ ωα Bβ
γδ
εξ
γδ
ε
xε ⋅ ∂ξ ω γδ − ε ξ ⋅ ∂ξ Bα
+ ω γ ξ Bα
ξ
ξδ
+ λ2 Bα tε ⋅ ∂ξ ω γδ
+ ω δ ξ Bα
γξ
γδ
−
+
(4.9)
6
V. CHIRITOIU ET AL..
~
5. Local DS gauge invariance. The covariant derivative ∇α and its
decomposition in function of ∂α , Σγδ and Σγ 5
~
In this section we try to decompose the covariant derivative ∇α in terms of
~
γ
∂α , Σγδ and Σγ 5 , introducing an extra gauge field eα . We recast ∇α in form
i
~
γ
∇α = eα ∂ γ + Bα
4
γδ
γ
Σγδ − iλBα Σγ 5
(5.1)
where
eα
γ
γ
= δα
γ
+ Bα
β
γ
+ λ2 Bα t β
+ Bα
γδ
xδ
(5.2)
Abbreviating
i
Bα
4
γ
dα = eα ∂ γ ; Bα =
we write
γδ
γ
Σγδ ; Cα = −iλBα Σγ 5
(5.3)
~
∇α = dα + Bα + Cα
(5.4)
γ
The variation of eα become
δeα
γ
= eα
ξ
{
{
⋅ ∂ ξ ε γ + λ2ε ε tε
− ε ξ + λ2ε ε tε
γ
γ
}
+ ω γδ xδ −
}
+ ω ξη xη ⋅ ∂ ξ eα
γ
ξ
+ ωα eξ
γ
and is expressed in terms of eα only. For the variation of Bα
δBα
γδ
{
ξ
= eα ⋅ ∂ξ ω γδ − ε ξ + λ2ε ε tε
+ ω γ ξ Bα
ξδ
+ ω δ ξ Bα
ξ
}
+ ω ξη xη ⋅ ∂ξ Bα
γδ
(5.5)
γ
γδ
ξ
we obtain
+ ωα Bξ
γξ
γδ
+
(5.6)
As a determinant det e −1 will enter the locally DS invariant actions and we
give its transformation behaviour here
{
}
δ det e −1 = − det e −1 ⋅ ∂ξ ε ξ + λ2ε ε tε ξ + ω ξη xη −
{
− ε ξ + λ2ε ε tε
ξ
}
+ ω ξη xη ⋅ ∂ξ det e −1
(5.7)
Before presenting the field strength operator we introduce the commutation
relation between the translational derivatives
[d
H αβ
γ
, d β ] = H αβ dγ
γ
α
(5.8)
γ
is expressed in terms of eα as
γ
(
ξ
ξ
H αβ = e −1γ ε eα ⋅ ∂ ξ eβ ε − eβ ⋅ ∂ξ eα
ε
)
(5.9)
7
QUANTIZATION OF GAUGE FIELDS
γ
ε
γ
where e −1γ ε is the matrix inverse of eα , i.e. eα ⋅ e −1γ ε = δ α .
In order to obtain the field strength operator we calculate the commutation
relation
[
(
]
)
~ ~
γ
γ
γ
Sαβ ≡ ∇α , ∇ β = H αβ dγ − Bαβ − Bβα dγ + dα Bβ − d β Bα + dα Cβ −
− d β Cα + [Bα , Bβ ] + [Bα , Cβ ] + [Cα , Bβ ] + [Cα , Cβ ]
(5.10)
The expressions for commutation relations in eq. (5.10) are
i
i δε γ

γ
δε
[Bα , Bβ ] = − 4 Bαε Bβ Σγδ + 4 Bα Bβε Σγδ

[Bα , Cβ ] = −iλBαε γ Bβ ε Σγ 5

[Cα , Bβ ] = iλBα δ Bβδ γ Σγ 5

[Cα , Cβ ] = 2iλ2 Bα γ Bβ δ Σγδ

(5.11)
Introducing the tensor
Tαβ
γ
γ
γ
γ
= Bαβ − Bβα − H αβ
(5.12)
we can rewrite Sαβ as
[
]
i ~
i ~
~ ~
γ~
Sαβ ≡ ∇α , ∇ β = −Tαβ ∇γ + R γδ αβ Σγδ + R γ 5 αβ Σγ 5
4
4
~ γδ
~γ 5
where R αβ and R αβ are found to be
~
γ
γ
R γδ αβ = dα Bβ γδ − d β Bα γδ + Bα δε Bβε − Bβ δε Bαε −
ε
− H αβ Bε
and
(
(
γδ
(
γ
δ
δ
+ 4λ2 Bα Bβ − Bα Bβ
)
γ
(5.13)
)
(5.14)
(
)
~
γ
γ
γ
ε
γ
δ
ε
γ
R γ 5 αβ = −4λ dα Bβ − d β Bα − 4λ Bαε Bβ − Bβδ Bα − H αβ Bε =
(5.15)
~
~
γ
γ
ε
γ
= −4λ ∇α Bβ − ∇ β Bα + 4λH αβ Bε
~
γ
γ
We give the local infinitesimal transformations of H αβ , Tαβ and R γδ αβ .
)
{
}
δH αβ γ = − ε ρ + λ2ε µ t µ ρ + ω ρη xη ⋅ ∂ ρ H αβ γ + ωα ρ H ρβ γ +
ρ
γ
ξ
(
+ ωβ H αρ + ω γ ξ H αβ + eα
ξ
⋅ ∂ξ ωβ
γ
− eβ
ξ
⋅ ∂ ξ ωα
γ
)
(5.16)
8
V. CHIRITOIU ET AL..
δTαβ γ = −{ε ξ + λ2ε ε tε ξ + ω ξη xη }⋅ ∂ξ Tαβ γ +
ξ
γ
ξ
γ
and
{
~
}
(5.17)
ξ
+ ωα H ξβ + ωβ H αξ + ω γ ξ Tαβ
~
~
δR γδ αβ = − ε ξ + λ2ε ε tε ξ + ω ξη xη ⋅ ∂ξ R γδ αβ + ωα ξ R γδ ξβ +
~
~
ξ ~
+ ωβ R γδ αξ + ω γ ξ R ξδ αβ + ω δ ξ R γξ αβ
(
(5.18)
)
In all calculations we have worked on Minkowski space-time R 4 ,η . If we
consider an indefinite metric tensor
µ
(
g µν = eα eαν
4
(5.19)
)
and a Riemannian manifold R , g for an arbitrary base êα we find the
commutation coefficients (structure functions) from
[eˆ , eˆ ] = c
α
γ
β
γ
αβ
eˆγ
(5.20)
γ
In this case eˆα = dα and cαβ = H αβ . The connection coefficients are then
δ
γδ
to be identified as Γγ α = − Bα . In this situation Tα
~
torsion and curvature tensors and R γ 5 αβ = 4λTαβ
γ
γδ
~
and R γδ αβ are the
.
6. DS gauge invariant matter actions. Scalar, spinor and vector fields
As we already mentioned in Section 3 the relation (3.15) is not yet sufficient
for the original action to be locally DS gauge invariant. We have to complete
the Lagrangian density with another term in order to obtain a pure divergence.
Using the transformation law for det e −1 we consider the combination
(
)
(
~
~
LM ϕ j , ∇αϕ j → det e −1 ⋅ LM ϕ j , ∇α ϕ j
and under local DS gauge transformations
(
)
(
)
~
~
det e′−1 ⋅ LM ϕ ′j , ∇′α ϕ ′j = det e −1 ⋅ LM ϕ j , ∇α ϕ j −
{(
)
)
(6.1)
(
~
γ
− ∂ γ ε γ (x ) + λ2ε ξ (x )tξ (x ) + ω γδ (x )xδ det e −1 LM ϕ j , ∇α ϕ j
)}
(6.2)
Therefore the minimally extended DS gauge invariant matter action become
~
(6.3)
S M = d 4 x det e −1 ( x )LM ϕ j ( x ), ∇α ϕ j (x )
∫
(
)
The actions for a massive scalar field is extended to a locally DS gauge
invariant action
9
QUANTIZATION OF GAUGE FIELDS
1
1

S M = ∫ d 4 x det e −1  dα ϕ ⋅ d α ϕ − m 2ϕ 2 
2
2

(6.4)
For Dirac spinor field and for a massive vector field the actions have the
forms
(
)
( )
i ~
~

i
S M = ∫ d 4 x det e −1  ψ γ α ∇αψ − ∇αψ γ αψ − mψ ψ 
2
2

(6.5)
and
with Fαβ
1
 1

S M = ∫ d 4 x det e −1 − Fαβ F αβ + m 2 Aα Aα 
2
 4

~
~
= ∇α Aβ − ∇ β Aα .
(6.6)
7. Regularization technique with ζ - function
In this section we determine the heat kernel coefficients c1 and c2 belonging
to a general hermitean second order differential operator M defined on the
Minkowski space-time R 4 ,η .
We introduce the operator M
(7.1)
M = − Dα Dα + E ,
Dα = ∇α + Aα
(
)
The heat kernel K (is; x, y ), s > 0 , belonging to M x fulfils
 ∂


+ M x  K (is; x, y ) = 0
 ∂ (is )

(7.2)
together with the initial condition
lim K (is; x, y ) =
s →0
1
δ 4 (x − y )
−1
det e
(7.3)
For y → x and s → 0 we are interested in the small s–expansion of
K (is; x, y ) as
K (is; x, y ) ~
i
(4πis ) 2
d
e
−
r 2 ( x, y ) ∞
4 is
∑ (is ) c (x, y )
k =0
k
k
(7.4)
The our main task is to evaluate the derivatives of different orders for
function r 2 ( x ) and the coefficient functions ck ( x, y ) . After some
calculations we get
10
V. CHIRITOIU ET AL..
∇α r 2 ( x) = 0

2
∇ βα r ( x) = 2η βα

∇γβα r 2 ( x) = 0

2

2
∇δγβα r ( x) = 3 (Rαδβγ + Rαγβδ )
and the coefficients c1 and c2
1
c1 ( x) = − Rαβ αβ − E
6
(7.5)
(7.6)
1
1
1
∇γ γ Rαβ αβ +
R αβ αβ ⋅ R γδ γδ +
R αβγδ ⋅ Rαβγδ
30
72
180
1
1
1
αβ
−
R αγ α δ ⋅ Rβ γβδ + F αβ ⋅ F αβ + R αβ ⋅ E −
180
12
6
1
1
− Dα , Dα , E + E 2
6
2
where, in the relation (7.7) Fαβ = ∇α Aβ − ∇ β Aα + Aα , Aβ
c2 ( x ) = −
[ [
]]
[
(7.7)
]
It is known that
d
ζ (u; µ ; M )
u → 0 du
ln det M = − lim
(7.8)
If we consider the behaviour of the functional determinant under a change of
~ = κµ
scale µ
(7.9)
ζ ′(0; µ~; M ) = ζ ′(0; µ ; M ) + 2 ln κ ⋅ ζ (0; µ ; M )
and the change of the functional determinant under rescaling is fully determined
by ζ (0; µ ; M ) . For a d-dimensional Minkowski space-time the function
ζ (0; µ ; M ) has the form
ζ (0; µ ; M ) =
i
d
(4π ) ∫
d
d
2
x det e −1trcd ( x )
(7.10)
2
If we turn back at the actions written in the previous section we can construct
the functional integrals for different types of fields. The simplest case is for
scalar field when the functional integral is
Z ϕ [e] = ∫ Dϕ e
iS M (ϕ ; e )
After a partial integration we can rewrite the action as a scalar product
(7.11)
11
QUANTIZATION OF GAUGE FIELDS
S M (ϕ ; e ) =
1
(ϕ , M ϕ (e)ϕ )
2
(7.12)
Introducing the second order operator M
M ϕ (e) = −∇α ∇α − m 2
(7.13)
Performing a Gaussian integration we can write
Zϕ [e] = e
1
− ln det M ϕ ( e )
2
(7.14)
We are interested in the behaviour of Zϕ [e] under rescaling using ζ – function.
At the scale µ we we can write
1
ζ ′ (0; µ ; M ϕ (e ))
2
Zϕ [µ , e] = e
~
If we consider a new scale µ = κµ we get
ln κ ⋅ζ (0; µ ; M ϕ (e ))
Z [µ~, e] = Z [µ , e]e
ϕ
ϕ
(7.15)
(7.16)
In the same way we can construct the second order operator M and the
functional integral for spinor and vector fields.
8. Concluding remarks
Based on the complementary conception of de-Sitter symmetry as a pure
inner symmetry we have developed a DS gauge theory of gravitation. The
gravitational interaction is mediated by gauge fields defined on a fixed
Minkowski space-time. Their dynamics can be determined imposing
consistency requirements with renormalization properties of matter fields in
gravitational backgrounds.
We have also presented a technique of renormalization using ζ – function.
To use this technique we have to find out the second order differential operator
M, and then we must to calculate the coefficients c1 and c2 belonging to this
operator. Using these coefficients we can write ζ – function ζ (0; µ ; M ) . Its
insertion into the functional integral gives us the anomalous term in scalar,
spinor or vector fields. At this point we remark that this work was made on a
space with null torsion.
Finally, we will be able to construct a minimal action for the gauge fields
defined on a fixed Minkowski space-time R 4 ,η . This action is invariant on
one hand under local DS gauge transformations, on the other hand under global
DS transformations.
(
)
12
V. CHIRITOIU ET AL..
References
[1] M. Blagojević, Gravitation and gauge symmetries, IOP Publishing, London,
2002;
[2] C. Wiesendanger, Poincaré gauge invariance and gravitation in Minkowski
spacetime, arXiv:gr-qc/9505049;
[3] R. Aldrovandi, J.G. Pereira, A second Poincaré group, arXiv:gr-qc/9809061;
[4] R. Banerjee, Gauge theories on de-Sitter space and Killing vectors,
arXiv:hep-th/0608045;
[5] R. Aldrovandi, J.G. Pereira, The case for a gravitational de-Sitter gauge
theory, arXiv:gr-qc/9610068;
[6] D. Bailin, A. Love, Introduction to gauge field theory, IOP Publishing,
Bristol, 1993;
[7] E. Fradkin, Course of general field theory, University of Illinois Urbana,
2005;
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
FRACTAL PATTERNS IN DISCHARGE PLASMAS
GABRIELA CIOBANU1, CORINA MARIN2,3, M. AGOP2
1
“Al.I.Cuza” University, Faculty of Physics,
Department of Theoretical Physics,
Blvd. Carol No.1, 700506, Iasi, Romania
2
“Gh. Asachi” Technical University, Department of Physics,
Blvd. Mangeron, 700029, Iasi, Romania
3
Ministery of Education and Research, Bucharest, Romania
Abstact
Using a set of coupled equations for concentrations, the dynamic of two discharge plasmas
interface is analyzed. The numerical solutions of the concentration breather, concentration breather pairs and
the concentration clusters type are obtained. In our opinion, these numerical solutions may be assimilated to a
double layer or multiple layers. Moreover, the self-organization process of the interface as a double layer or
multiple layer is a fractal process - the equal ambipolar diffusion curves are of Koch type.
The validity of the model is given by means of some applications. First of all, the improved
confinement of neon plasma inside a high - TC superconducting tube, is given. Secondly, using the scale
relativity theory, the multiple double layers problem is analyzed: the current in the interface is given by the
variation in time of the concentration difference between plasmas, the potential is also given by the time
variation of the phase difference induced by the plasmas carriers, the peaks of the continuous current result for
“quantified” values of the potential and from here, the generation of a negative resistance. In such context the
double layer or multiple layer plasma structures are projections of a higher dimensional fractal.
PACS: 52.20.-j, 52.25.-b, 52.40.-w
Keywords: discharge plasma, double layer, multiple layers, fractal patterns
1. Introduction.
Kolmogorov’s works [1] on the concept of self-similarity in
fluctuations has helped people understand some of the basic features of fluid
turbulence. This concept was extended to the hydrological data mainly by
Mandelbrot [2] and latter applied to a variety of natural phenomena. It is now
widely accepted that physical systems exhibiting chaotic behaviour are generic
in Nature.
14
G. CIOBANU et. al.
Turbulence is not only active on a wide range of length scales, but it is
filled with localized coherent structures which makes quantities such as energy,
anisotropy, pressure highly intermittent. The plasma turbulence [3] is usually
characterized by assuming that the fluctuations are randomly distributed in
space and time. It was shown that the turbulence in plasma discharge have the
same origin as the periodical ones, but in this case, stochastic causes decide
about the moment when the unstable state of one or more double layers (DLs),
coupled by current, starts its proper evolution process [4]. The experimental
data evidence that the global evolution of the plasma discharge containing a
system of DLs is the result of the mutual influence of two or more individual
DLs coupled by the current flowing through them [4-6]. The chaotic regime
observed as non-coherent variations of the current appears when the
correlations between the proper dynamics of each of DLs do not exist and each
of their dynamics starts aleatory, independently one of the other. The observed
behaviour is similar to the chaotic dynamics of coupled non-linear oscillators
[7,8]. Two general properties of turbulence in plasma discharge resulted: a)
non-linearity, a basic property of all numerical models of chaos; b)
intermittency which causes the characteristics of turbulence change on a
relatively short time and/or space scale, necessitates the use of suitable
statistical description to study the long term dynamical behaviour. On such a
description is the fractal formalism.
In the present paper, using a set of coupled equations for the
concentrations, one shows that the self-organization of the two discharge
plasma interface by means of DLs or multiple layers is a fractal process.
2. Mathematical model.
For discharge plasmas of relatively high pressure, the variation of the
parameters is imposed by the ambipolar diffusion [9,10]. Thus, imposing the
electronic and ionic current densities,
je = −eDe∇ne − ene µe E
ji = −eDi∇ni + eni µi E
,
(1a,b)
by the plasma quasineutrality conditions, ne = ni = n and je = ji = j , and by
eliminating E, we obtain
(2a,b)
j = −eDa ∇n , Da = (µ i De + µ e Di ) (µ i + µ e )
with Da, the ambipolar diffusion coefficient. The parameters which appear in
the relations (1a,b) and (2a,b) have the usual significances from Refs. [9,10], i.e.
n the concentration, D the diffusion coefficient, µ the mobility, j the current
density, E the electric field, the subscript e and i denoting the electronic and
FRACTAL PATTERNS IN DISCHARGED
15
ionic component respectively. This means that by E, the two sets of charged
particles are diffusing together.
On the other side, electronic and ionic current densities satisfy the
continuity equations [9,10],
(3)
∂ t en + ∇ ⋅ j = 0
Assuming the electronic temperature, Te, to be constant along the r direction,
and the ambipolar diffusion coefficient, Da, to be independent on r, from
relations (2a,b) and (3)
one gets the equation:
(4)
∂ t n = Da ∆n
Hence, through separation of the variables n(r , t ) = T (t )N (r ) we obtain
−
1
τ
=
1 dT
,
T dt
∆N
1
1
=−
=− 2
N
τDa
ξ
(5a,b)
respectively the decay time (or the relaxation time of the plasmas) [9, 10]
τ = (ξ 2 Da )
(6)
with ξ the characteristic length of diffusion which depends on the geometry of
the discharge tube [9,10].
Now let us consider two plasmas P1, P2, separated by an interface,
which is assimilated with two different phases of the same system [9, 10] (in
experiments [11,12], the difference between the two plasmas was accomplished
through the different polarization of a plasma, according to the other one). To
explain such situation, let us observe that the plasma P1 – plasma P2 interaction
induces in the interface, for local quasi-neutrality of the plasmas, n1e = n1i = n1 ,
j1e = j1i = j1 , n2 e = n2i = n2 , j 2 e = j 2i = j 2 the current densities,
j1 = −eD1 ∇n1
j 2 = −eD2 ∇n2
(7a,b)
which satisfy the continuity equations with sources [9,10]
∂ t en1 + ∇ ⋅ j1 = −eν 2 n2
∂ t en2 + ∇ ⋅ j 2 = eν 1 n1
(8a,b)
where D1, D2 are the ambipolar diffusion coefficients and ν1, ν2, the source
coefficients. By substituting (7a,b) in (8a,b) and considering D1, D2 not
depending on r, one obtains the coupled equations set [9,10]
16
G. CIOBANU et. al.
∂ t n1 = D1 ∆n1 − ν 2 n2
∂ t n2 = D2 ∆n2 + ν 1 n1
(9a,b)
The source terms are obtained out of the analysis of the trajectories of
the particles in the phase space [9, 10]. Thus, if P1 plasma is polarized at V1 < 0
potential, and P2 at V2 > 0 potential, then the electrons that come from P2
plasma and the ions that come from P1 plasma are reflected in the interface,
while the electrons that come from P1 plasma and the ions that come from P2
plasma are accelerated by the electric field of the interface, generating, when
getting out, fascicles of energetic particles. Now, neglecting the particles
trapped to interface, it results that the source terms are induced by the
accelerated particles (the electrons coming from P1 plasma, respectively the
ions coming from P2 plasma).
17
FRACTAL PATTERNS IN DISCHARGED
Figures 1a-j: The numerical solutions of the Eqs. (11a,b) for α1=α2=1/3,
β1=0.2, β2=2, tr=0÷2.25, ∆tr=0.25 and the initial condition
Φ 0 ( xr , y r ,0) ≈ exp[−( xr2 + y r2 )]
For a plane symmetry and through an adequate normalization of the
parameters from eqs. (9a,b), i.e.
t r = ωt , x r = kx, y r = ky, α G = D1 k 2 / ω 2 , α 2 = D2 k 2 / ω 2
β1 = ν 1 / ω , β 2 = ν 2 / ω , Φ 1 = n1 / n01 , Φ 2 = n2 / n02
the previous set of equations becomes:
(
= α (∂
)
)Φ
∂ tr Φ 1 = α 1 ∂ 2xr + ∂ 2yr Φ 1 − β 2Φ 2
∂ tr Φ 2
2
2
xr
+ ∂ 2yr
2
+ β1Φ 1
(10a-i)
(11a,b)
Figures 2a-j: The numerical solutions of the Eqs. (11a,b) for α1=α2=1/3,
β1=1.8, β2=1.9, tr=0÷2.25, ∆tr=0.25 and the initial condition
Φ 0 ( xr , y r ,0) ≈ exp[−( xr2 + y r2 )]
18
G. CIOBANU et. al.
In the general case the β1 , β 2 coefficients are functions of φ1 , φ 2 . In
other words, the (11a,b) system is nonlinear [13] and admits nonlinear solutions
[13]. To these non-linear solutions we can associate certain physical structures
with non-linear behaviour (non-linear structures). For example, the compressive
solitons are associate with the compressive double layer [14, 15] the breathers
(two-dimensional (2D) dark solitons) with magnetic domains [13], the kink with
the magnetic flux quanta [16] etc.
In the present context we shall solve the (11a,b) system using a finite
differences method [19]. For α1=α2=1/3, β1=0.2, β2=2, tr=0÷2.25 and the initial
condition Φ 0 ( xr , y r ,0) ~ exp[−( xr2 + y r2 )] , the numerical solutions are
presented in Figs. 1a-j. It results concentration breather. For α1=α2=1/3, β1=1.8,
β2=1.9, tr=0÷2.25 and the initial condition Φ 0 ( xr , y r ,0) ~ exp[−( xr2 + y r2 )] ,
the numerical solutions are presented in Figs. 2a-j. It results concentration
breather pairs for decreased time sequences and concentration clusters for
increased time sequences, respectively.
In our opinion, and according to [14-16], the numerical solutions from
Figures 1a-j, 2a-j are sequences of the DLs generating and multiplication.
Through self-organization (correlation in amplitude and phase of the
concentration breathers – for details see Ref. [11,13], one gets first the
concentration breather pairs, i.e. DL, and then concentration clusters, i.e.
multiple layers. As a matter of fact, numerical simulations on type (11a,b)
system, with type gauss initial conditions, leads with good approximation, to
breathers or clusters of breathers [13,19].
The self-organization is a fractal process since the equal ambipolar
diffusion curves are of Koch type (the fractal dimension (for details see Ref.
[20]) of the physical objects from Figure 2f is D ≈ 1.26 )
3. Applications.
3.1. Improved confinement of neon plasma inside a high-TC
superconducting tube.
In a recent note [21], a neon pulsed plasma was produced along the bore
of a superconducting tube, to observe its influence on the plasma characteristics,
or the compression and confinement of the plasma. After observing spatial and
temporal behaviours of the plasma through a high-speed streak camera, the
decay time of the plasma for both normal and superconducting tube was
evaluated and superconducting tube proved to be superior to the conventional
one.
FRACTAL PATTERNS IN DISCHARGED
19
To explain such a situation, let us firstly observe that, in the absence of
the superconducting tube, for a discharge plasma of characteristics [21]: the
electronic temperature, Te ≈ 5eV , the ionic temperature, Ti ≈ 2.5 ⋅10 −2 eV , the
electronic-ionic mass ratio (me / mi )1Ne/ 2 ≈ 1 / 192 , the electronicneutral collision frequency, ν en ≈ 2 ⋅109 s −1 , the length of the discharge
tube, d = 60mm , the radius of the discharge tube, r = 3.25mm , the electronic
diffusion coefficient, De = (k BTe / meν en ) ≈ 400 m 2 / s , the ambipolar diffusion
coefficient Da ≈ De (me / mi )1/ 2 (1 + Ti / Te ) ≈ 2 m 2 / s , and the characteristic
length of diffusion, ξ = [(π / d 2 ) + (2.405 / r ) 2 ] −1 / 2 ≈ 1.34 ⋅ 10 −3 m , from Eq.
(6) it results the plasma relaxation time τ ≈ 10 −6 s , a value close to the
experimental one [21].
The presence of the superconducting tube generates through the
interaction plasma-wall, a double electric layer, as described by the (11a,b) set
of equations. Then, by an analysis of the numerical solutions (see paragraph 2),
it results that β=β1=β2=2 is the optimal value of the self-organization process.
Since from relations (6) and (10f,g), the parameter β defines the ratio between
the decay time of plasma in the presence of the superconducting tube, τS, and in
its absence, τ, one gets δ = τ S τ ≈ 2.0 , a value close to the experimental one
[21].
3.2. Multiple DLs.
The experimental results regarding the multiple DLs show that its
generation is achieved by a ‘negative conduction’ mechanism (i.e. negative
resistance) [4-6], and its multiplication by the ‘quantification’ of the potential
[4,5,6,11,12]. To explain such a situation, let us observe firstly that, according
to the results from paragraph 2, the self organization of the plasma – plasma
interface as electric DL is a fractal process and consequently the scale relativity
theory (SRT) [22] can be applied – see Appendix A. Then, the interface
dynamics is described by the coupled equations set,
2imD∂ tΨ 1 = T1Ψ 1 + KΨ 2
2imD∂ tΨ 2 = T2Ψ 2 + KΨ 1
,
result obtained from (9a,b) with the restrictions
D1=D2=D, ν1=ν2=ν, K=2mDν
(12a,b)
(13a-c)
20
G. CIOBANU et. al.
by the analytical extension t→it, i.e. by means of the correspondences (for
details see Appendix A)
n1 → Ψ 1 , n2 → Ψ 2
− 2mD ∆Ψ 1 → T1Ψ 1 , − 2mD 2 ∆Ψ 2 → T2Ψ 2
2
,
The analytical extension gives to the parameters,Ψ 1 =
Ψ 2 = ρ 2 e iθ
(14a-d)
ρ1 e iθ , and
1
a probabilistic meanings, the diffusion coefficient D
characterizing the fractal behaviour of trajectories (namely, it defines the fractal
– non-fractal transition in scale space) [17,18,22,23].
Replacing the expressions of Ψ 1 , Ψ 2 in (12a,b) and equating the real
and the imaginary parts one obtains
2
K
K
ρ1 ρ 2 sin(θ 2 − θ1 ), ∂ t ρ 2 = −
ρ1 ρ 2 sin(θ 2 − θ1 )
mD
mD
T
T
ρ2
ρ2
K
K
∂ tθ1 = − 1 −
cos(θ 2 − θ1 ), ∂ tθ 2 = − 2 −
cos(θ 2 − θ1 )
2mD 2mD ρ1
2mD 2mD ρ1
∂ t ρ1 =
(15a-d)
or, with the substitutions
T1-T2=qV, θ = θ2-θ1
it results
∂ t ρ1 = −∂ t ρ 2 =
(16a,b)
K
qV
ρ1 ρ 2 sin θ , ∂ tθ =
2mD
mD
(17a,b)
Under these circumstances, the expression of the current in the interface
becomes
I = q[∂ t ( ρ1 − ρ 2 )] = I M sin θ
IM =
2qK
mD
or explicitly, for
ρ1 ρ 2 , θ = θ 0 +
q
V (t )dt
2mD ∫
V = V0 + v cos(ωt + ϕ )
(18a-c)
(19)
21
FRACTAL PATTERNS IN DISCHARGED
qv
 qV t

sin(ωt + ϕ ) + θ 0  =
I (t ) = I M sin  0 +
 2mD 2mDω

∞

qV0 
 qv  
= IM ∑ Jn
t + nϕ + θ 0 
 sin  nω +
2mD 
 2mDω  
−∞

(20)
where Jn is the n-order Bessel function. Then, for ω = (nqV0 / 2mD) , the
temporal average of I(t) differs from zero, i.e. it exists a continuous component
of the current of the form
 qV0 
I C = (−1) n I M J n 
 sin(θ 0 − nϕ )
 2mDω 
From here, it results peaks of the continuous current for the voltages
Vn=(2nmDω/q)
and consequently the negative resistance (for details see [16,24]).
For ρ1 = ρ 2 = ρ , θ satisfies the nonlinear differential equation
∂ ttθ + ω 2 sin θ = 0, ω 2 =
K
mD 2
(21)
(22)
(23a,b)
which for the one-dimensional (1D) case admits the solution
θ = sn[ s −1ω (t − t 0 ); s ], t 0 = const.
(24)
where sn is the elliptic Jacobi function of s modulus,
−1
 θ  2
 θ 
2
s =  0  + sin 2  0  ,
 2 
 2ω 
θ 0 = const.
(25)
22
G. CIOBANU et. al.
Figure 3: The fractal
Now, by the iterated map of the current, (I/IM)=sin(sn), one obtains the fractal
from Figure 3. Its fractal dimension (or Hausdorff-Besicovitch dimension [20])
D~2.4 is greater than the topological one, D=2. Then, the DL or multiple layers
may be considered as projections of the same higher-dimensional fractal – for
details see [20,25]. On the structure and properties of such objects one may
consult [20,25].
The previous results show the fact that, by applying the SRT to the
study of P1 plasma – P2 plasma interface, the current in the interface is given by
the variation in time of the concentration difference between plasmas (see
(18a)), and the potential by the time variation of the phase difference induced
by the plasmas carriers (see (17b)). Moreover, peaks of the continuous current
result for “quantified” values of the potential (see (22)) and from here, the
generation of a negative resistance.
FRACTAL PATTERNS IN DISCHARGED
23
Figure 4: One, two and three experimental plasma formations (DLs)
Recently, selforganization in plasma discharge was connected with the
scenario imposed through negative resistance (for details see [24]). DLs
structures or multiple DLs resulted. In the Ref. [26] some experimental results
concerning the formation of ordered plasma structures in the interspace of two
independently working discharge are given. The structured plasma has been
studied by means of electrical and spectral methods. The experimental research
of the plasma formations shows that the DL is the final stage of a more complex
structure which is formed first. The first plasma structure has a potential profile
across the plasma formation of more complexity, as a multiple DLs. The profile
of the space potential, electric field and total charge density along the axis of the
24
G. CIOBANU et. al.
discharge tube, on the spatial domain of existence of the two and three plasma
formations – see Figs. 4 confirm that in each plasma formation a DL is formed.
The fact that the potential on the spatial domain of the plasma
formations has been approximated with Vn ≈ nV , where V is the potential on
the spatial domain of a DL, proves the appearance of multiple DLs. Let us
verify this aspect for a Argon discharge plasma having the parameters [26]:
d=4⋅10-2m, r = 4 ⋅10 −3 m , ne ≈ 3⋅1018m-3, λDe ≈ 8⋅10-6m (the Debye length),
νen ≈ 6⋅108s-1, (me/mi)1/2 ≈ 1/270, Te ≈ 3.5eV, Ti ≈ 2.5⋅10-2eV, De=(kBTe/meνe)
Da ≈ De(me/mi)[1+(Ti/Te)]
≈ 933m2/s,
≈ 3.4m2/s,
ξ=[(π/d)2+(2.405/r2)] −1 2 ≈ 1.65 ⋅10 −3 m . The significance of the previously
defined measures is the same with the one from paragraph 3.1. It results the
plasma relaxation time, τ ≈ 0.8⋅10-6s. Since m=me, q=2e, D=d (the diameter of
the spherical DL) × c (the vacuum light speed) = 1.1⋅10-3m × 3⋅108m/s ≈
3.3⋅105m2/s, ω=2π/τ ≈ 7.85⋅106Hz, the relation (22) becomes Vn ≈ n×16V, which
corresponds to the experimental multiplication law of the DLs [26].
4. Conclusions.
The main conclusions of the present paper are as it follows:
i) Using a set of coupled equations, the dynamics of the plasma – plasma
interface is analysed;
ii) The numerical solutions of the concentration breather pairs and the
concentration clusters type are associated to a DL and multiple layers,
respectively;
iii) The self-organization process in discharge plasma in the form of a DL or
multiple layers is a fractal process – the equal ambipolar diffusion curves are of
Koch type;
iv) It results the confinement of the neon plasma inside a high-TC
superconducting tube by means of a DL and, in the formalism of SRT, the
negative conduction mechanism and the potential ‘quantification’ for multiple
layers. The theoretical model was applied either for Neon on plasma, in the case
of its confinemed by means of a superconductor tube, or for Argon plasmas, in
the case of multiple layers engendering. The difference between the two
“plasmas” (Neon and Argon) is obvious both at the level of elementary
processes (different ionization potentials, different ionization cross-sections,
etc. [9, 10]) and at the level of “constraints” (the Neon plasma is confined in the
magnetic field induced by the superconductor tube; the multiple layers are
generated through the different polarization of the two Argon plasmas).
v) From the fact that the process of generating and multiplication of DLs is a
fractal process, it results that such structures have the same origin, as for
FRACTAL PATTERNS IN DISCHARGED
25
example, a fractal. Then, known structures (double layer, multiple layers)
appear as projections of this fractal. A similar situation appears in
superconductivity. In this case, the 1D projection of a fractal string leads to
Cooper pairs, and the 2D projection leads to anions (for details see [16,23,25]).
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27
FRACTAL PATTERNS IN DISCHARGED
Appendix A
The theory of fractal space-time is the scale relativity (SR) theory
[22,23]. This theory extends Einstein’s principle of relativity to scale
transformation of resolution. A non-differentiable continuum is necessarily
fractal and trajectories in such a space (or space-time) own (at least) the
following three properties: i) The test particle can follow an infinity of potential
trajectories: this leads one to use a fluid-like description, v=v(x(t),t); ii) The
geometry of each trajectory is fractal (of dimension 2). Each elementary
displacement is then described in terms of the sum, dX = dx + dξ , of a mean
classical displacement dx = vdt and of a fractal fluctuation dξ , whose
behavior satisfies the principle of SR (in its simplest Galilean version). It is
such that dξ = 0 and dξ 2 = 2 Ddt . The existence of this fluctuation
implies introducing new second order terms in the differential equation of
motion; iii) Time reversibility is broken at the infinitesimal level: this can be
described in terms of a two-valuedness of the velocity vector, for which we use
a complex representation, V = (v+ + v− ) 2 − i (v+ + v− ) 2 .
These three effects can be combined to construct a complex time-derivative
operator,
(A.1)
∂ t = ∂ t + V ⋅ ∇ − iD∆
with V the complex speed
V = −i 2 D∇ lnψ
(A.2)
Applying the ∂ t covariant derivative to the V complex speed, we obtain
∂ t V = −2 Di[∂ t ∇ lnψ − 2 Di(∇ lnψ ⋅ ∇ )(∇ lnψ ) + iD∆(∇ lnψ )]
(A.3)
or still, considering the identity
∇(∆ψ ψ ) = ∆(∇ lnψ ) + 2(∇ lnψ ⋅ ∇ )(∇ lnψ )
∂ t V = −2 D∇[i (∂ tψ ψ ) + D(∆ψ ψ )]
(A.4)
(A.5)
Newton’s second principle takes the form
m0 ∂ t V = −∇φ
or explicitly,
(A.6)
[
]
− m0−1∇φ = 2im0 D(∂ tψ ψ ) + 2m0 D 2 (∆ψ ψ )
(A.7)
Integrating this equation yields
2m0 D 2 ∆ψ + 2im0 D∂ tψ − (m0 ) φψ = 0
−1
(A.8)
28
G. CIOBANU et. al.
up to an arbitrary phase factor ϕ (t ) which may be set to zero by a suitable
choice of the phase of ψ . If there is no external field, φ = 0 , the covariance is
explicit, since equation (A.6) takes the form of the equation of inertial motion
(A.9)
∂t V = 0
As a consequence, we are now able to write the equation of geodesics of the
fractal space-time under its covariant form
(A.10)
∂ t ∂ t xi = 0
The statistical meaning of the wave function (Born postulate) can now be
deduced from the very construction of the theory. Even in the case of only one
particle, the virtual geodesic family is infinite (this remains true even in the zero
particle case, i.e. for the vacuum field). The particle properties are assimilated to
those of a random subset of the geodesics in the family, and its probability to be
found at a given position must be proportional to the density of the geodesic
fluid.
For D =  2m0 , equation (A.8) is reduced to Schrödinger’s equation
2
φ
∆ψ + i∂ tψ − ψ = 0
m0
2 m0
(A.11)
In such a context, the study of P1 plasma – P2 plasma interface comes
back to the integration of the set of equations
2im0 D∂ tψ 1 = T1ψ 1 + Kψ 2
2im0 D∂ tψ 2 = T2ψ 2 + Kψ 1
(A.12)
where T1, T2 define the energies induced by means of the potentials of the two
plasmas, and K is a characteristics of the interface.
To deduced this set of equations, we can use the method from [22,23].
The system of equations (9a,b) is defined on a continuous and
differentiable space-time. Nevertheless, the analysis of the numerical solutions
imposes a fractal space-time.
The transition from a continuous and differentiable space-time to a
continuous and non-differentiable one (i.e. a fractalic one) is performed by
substitutions [22,23]
t → it ,
n →ψ ,
Then, the wave function ψ
−2m0 D 2 ∆ → T
takes probabilistic properties and
− 2m0 D ∆ operator becomes a energy (for details see [22,23]).
2
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
GAUGE GRAVITATION AND SPONTANEOUS
SYMMETRY BREAKING
1
Camelia Popa1, I. Gottlieb1, M. Agop2
“Al.I.Cuza” University, Faculty of Physics, Department of Theoretical
Physics,
Blvd. Carol No.1, 700506, Iasi, Romania
2
“Gh. Asachi” Technical University, Department of Physics,
Blvd. Mangeron, 700029, Iasi, Romania
Abstract:
A gravitational gauge theory to study the spontaneous symmetry breaking is used. The gravitational gauge
group and its corresponding gauge covariant derivative are introduced. The strength tensor of the gravitational
gauge field is obtained and a gauge invariant Lagrangian is constructed. The field equation of the gauge
potential are written. A model whose gravitational gauge potential have axial symmetry, depending only on
the radial coordinate is considered and analytical solution of these equations, which induce a spontaneous
symmetry breaking, is determined. A quantification of the gravitational Hall resistance is obtained.
1.
Introduction
N. Wu proposed a gauge theory of General Relativity (GR) based on
the gravitational gauge group (G) [1-3]: i) the gravitational interaction is
considered as a fundamental interaction in a flat Minkowski space-time; ii) the
gravitation gauge group G consists of generalized space-time translations, and
the gravity is described by gauge potentials. So, the space-time is always flat,
the gravitational field is represented by gauge potential, and gravitational
interactions are always treated as physical interactions.
In the present paper using the gauge theory of the gravity [1-3] the
spontaneous symmetry breaking is analyzed. In Section 2 we define the
gravitational gauge group G and then we introduce the gauge covariant
derivative Dµ . The strength tensor of the gravitational gauge field is obtained
and a gauge invariant Lagrangian is constructed. The field equations of the
gauge potential are written with a gravitational energy-momentum tensor
Tg µν on the right-hand side. This tensor has the same expression as in [1-4].
( )
In section 3 the spontaneous symmetry breaking is analyzed.
2.
Gravitational gauge group and field equations
30
C. POPA et al..
The infinitesimal transformations of the group G are, as usually, of the
form [1-3,4]:
(1)
U (ε ) ≅ 1 − ε α Pα ,
α = 1,2,3,0
where ε α are the infinitesimal parameters of the group and Pα = −i∂α are the
generators of the gauge group. It is known that these generators commute each
other
(2)
Pα , Pβ = 0
[
]
However, according with Wu’s model, this does not means that the group G is
Abelian, that is its elements do not commute [1-4]:
(3)
[U (ε1 ),U (ε 2 )] ≠ 0
It is emphasized that there is a difference between the group T of space-time
translations and the gravitational gauge group G. Space-time translations of T
are coordinate (passive) transformations, that is, the objects or fields (physical
system) are fixed in space-time, while the coordinates themselves undergo
transformations. Contrarly, under the transformations of G, the space-time
system of coordinates is fixed and the physical system undergoes (active)
transformations.
As usually, one introduces the gauge gravitational field with values into
the Lie algebra of the group G:
(4)
Aµ ( x ) = Aµα ( x )Pα ,(µ ,α = 1,2,3,0)
where Aµα ( x ) are the gravitational gauge potentials, and then a gauge covariant
derivative is defined as
Dµ = ∂ µ −
ig
Aµ ( x )

(5)
Here g denotes the gauge coupling constant of the gravitational interactions and
 is the Planck reduced constant. The corresponding strength tensor
Fµν (x ) = Fµνα (x )Pα , with values in the Lie algebra of G, has the components
[1-4]:
where
Fµνα ( x ) = Gµβ ∂ β Aνα − Gνβ ∂ β Aµα
Gµα = δ µα − gAµα ( x )
(6)
(7)
are new gauge potentials. We suppose that these new potentials admit the
inverses Gαµ ( x ) with the usual properties:
GAUGE GRAVITATION AND SPONTANEOUS SYMMETRY
Gαµ Gµβ = δ αβ ,
Gαµ Gνα = δνµ
31
(8)
Following the method given in [1-4], we define a metric tensor on the
gravitational gauge group space by:
(29a,b)
gαβ = η µν Gαµ Gβν ,
g αβ = η µν Gµα Gνβ
where η µν = diag (1,1,1,−1) is the metric tensor of the Minkowski space-time
M, and η µν denotes its inverse.
Now, we consider the integral of action for the gravitational gauge
potential under the form:
(10)
S=
− det gαβ Ld 4 x
∫
( )
( ) is the determinant of the metric tensor
where det gαβ
gαβ , and L is the
Lagrangian density of the gravitational field
1 µρ νσ
1
η η g αβ Fµνα Fρσβ − η µρ Gβν Gασ Fµνα Fρσβ
16
8
(11)
1 µρ ν σ α β
+ η Gα Gβ Fµν Fρσ
4
Taking δS = 0 with respect to gravitational gauge potentials Aµα ( x ) [1-4], we
L0 = −
obtain the following field equations:
1
1
1
σ
ν
∂ µ ( η µρ ηνσ gαβ Fρσ
− ηνρ Fραµ + η µρ Fρα
−
4
4
4
1
1
ν
− η µρδ αν Fρββ + ηνρ δ αµ Fρββ ) = − g (Tg )α
2
2
( )
where Tg
ν
α
(12)
is the gravitational energy-momentum tensor considered as the
source of the gravitational field [1-3]. These tensor has the same expression as
in [1-4] works.
3.
Spontaneous symmetry breaking of the gravitational field
We consider the Lagrangean L locally invariant under the gauge group of
transformations [5,6]
L=
1
(∇ µΦ )* (∇ µΦ ) + 1 m 2Φ *Φ − 1 Fµν F µν ,
2
2
4
where ∇ µ is given by the relation (5), the symbol “
conjugate and
*
(13)
” define the complex
32
C. POPA et al..
Fµν = ∂ µ Aν − ∂ν Aµ ,
(14)
Using the cylindrical coordinates (r , θ , z ) , where
∂
1 ∂
∂
, Uθ =
, Uz = ,
r ∂θ
∂r
∂z
(15)
1 ∂ ig
∂
∂
− Aθ , ∇ z = ,
∇ r = , ∇θ =
r ∂θ 
∂z
∂r
A = Aθ U θ (for a gravitomagnetic field B z ),
∂A
A
(16)
F12 = − F21 = θ + θ ,
∂r
r
the others components of Fµν vanish, the explicit form for L given by (13)
Ur =
reads
L=
1  ∂Φ * ∂Φ ig
1 ∂Φ * ∂Φ
* ∂Φ
A
+ m 2Φ *Φ +
Φ
+
+

θ
2  ∂r ∂r r
∂θ r 2 ∂θ ∂θ
2
A ∂A
A 2 
g2 2 *
ig
∂Φ *
 ∂Aθ 
Φ −
+ 2 Aθ Φ Φ − Aθ
 − 2 θ θ − θ2 .
r
r ∂r
∂θ
r 

 ∂r 
(17)
We may use the Euler-Lagrange equations
∂  ∂L
∂r  ∂Aθ ,r
 ∂L
=
 ∂A ,
θ

∂  ∂L  ∂  ∂L  ∂L
+

=
,
∂r  ∂Φ ,*r  ∂θ  ∂Φ ,*θ  ∂Φ *
∂  ∂L  ∂  ∂L  ∂L

+

=
,
∂r  ∂Φ ,r  ∂θ  ∂Φ ,θ  ∂Φ
(18)
(19)
(20)
where
L = g ′L .
(21)
We consider the statical and z-independent distribution for the fields.
In this case
g ′ = r,
with L given by (21), (22) and (17); the Eqs. (18) and (19) become
(22)
GAUGE GRAVITATION AND SPONTANEOUS SYMMETRY
33
∂ 2 Aθ 1 ∂Aθ  g 2 *
1
ig  * ∂Φ ∂Φ * 
Φ
Φ  , (23)
−
+
+  Φ Φ − 2  Aθ = −
2r 
∂θ
∂θ
r 
∂r 2 r ∂r   2

2
2
1 ∂  ∂Φ  1 ∂ Φ 2ig
∂Φ  2 g 2 
(24)
Aθ
−
−  m + 2 Aθ Φ = 0.
r
+ 2
2
r ∂r  ∂r  r ∂θ
r
∂θ 


We choose for (24) a solution of form
Φ = R(r )e inθ ,
(25)
which satisfy the angular part of Eq. (24); R (r ) is a finite function for
r → ∞.
Including (25) in (23) we get A : ( A = Aθ ).
2
d 1 d
ng 
  g
rA
+
A
−  R 2 = 0.
(
)

2


dr  r dr
r 
  
(26)
We notice that
A=
n
,
gr
(27)
is a solution for (26).
With Φ given by (25) and A given by (27), we try to find R (r ) from (24):
1 d  dR 
2
 − m R = 0.
r
r dr  dr 
(28)
One may see that (28) is a Bessel modified equation in the caseν = 0 , so the
origin smooth solution is
(29)
R(r ) = I 0 (mr ).
But, for r → ∞ , this function grows up exponentially. In order to assure a
finite limit for Φ (in case r → ∞ ) we must write (28) as
1 d  dR 
2
2
r
 − ( m − fR )R = 0.
r dr  dr 
(30)
This equation can be obtained from a Lagrangean
L′ =
1
(∇ µΦ )* (∇ µΦ ) − 1
2
4
2
 1
 m2
f 
− Φ *Φ  − Fµν Fµν , (31)
 4
 f
with f is a new constant of coupling.
But (31) is quite the Nielsen-Olesen Lagrangean for the spontaneous
symmetry breaking [5].
34
C. POPA et al..
So, in order to satisfy the natural boundary conditions at infinite for the
fields solutions Aθ and Φ , it was necessary a spontaneous symmetry breaking.
4.
The quantized of the gravitational Hall resistance
With the solution A given by (27) written as
n
,
mg r
A=
(32)
where mg is the gravitational mass of a test particle, we can calculate the
gravitomagnetic flux through the plane (xy), namely
Φ g = ∫ B ⋅ dS = ∫ Adl = ∫
S
Φ g0
2π
0
Γ
nh
n
;
rdθ =
mg
mg r
h
=
,
mg
(33)
(34)
is the gravitational flux quanta.
It is known that the gravitational Hall potential difference U gH is
U gH =
I gΦ g
Nm g
,
(35)
where Ig is the mass current, N-the number of carriers, Φ g -the gravitomagnetic
flux through the sample.
We name “gravitational Hall resistance” the expression
RgH =
U gH
Ig
=
Φ
Nm g
.
(36)
Each flux line has m charge carriers attached on it, and
m=
Φg
,
Φ g0
(37)
is the number of states per Landau level.
So the number N of the gravitational charge carriers (according to Pauli`s
principle) is
(38)
N = nm.
Considering (38), (36) can be written as
RgH =
h
,
mm g2
(39)
GAUGE GRAVITATION AND SPONTANEOUS SYMMETRY
35
where RgH is the gravitational quantized Hall resistance and h / m g2 is the
gravitational Hall resistance quanta.
Conclusions
The main conclusions of the present paper are the followings:
i) The gravitational gauge group and its corresponding gauge covariant
derivative are introduced; ii) The strength tensor of the gravitational gauge field
is obtained and a gauge invariant Lagrangian is constructed; iii) The field
equation of the gauge potential are written; iv) A model whose gravitational
gauge potential have axial symmetry, depending only on the radial coordinate is
considered and analytical solution of these equations, which induce a
spontaneous symmetry breaking, is determined; v) A quantification of the
gravitational Hall resistance is obtained.
It will be interesting to develop a unified geometric theory on a
principal fiber bundle as base manifold and with internal symmetry group G as
a structural group. The general structure of the theory is that of a combined
geometry of the space-time with the internal symmetry space.
References
[1]
N. Wu, Commun. Theor. Phys 38 (2002) 151 and arXiv:hepth/0207254
[2]
N. Wu, Commun. Theor. Phys 42 (2004) 543
[3]
N. Wu, Commun. Theor. Phys 39 (2003) 671
[4]
Gh. Zet, C. Popa, D. Partenie, Romanian J. Phys, 51,5-6,487-501
[5]
M. Chaichian, N.F. Nelipa, Introduction to gauge field theories,
Springer Verlag, Berlin-Heidelberg-New York-Tokio, 1984
[6] Marina Dariescu, M. Agop, C. Dariescu, Buletinul Institutului Politehnic
din Iasi, Tomul XXXVII (XL), Fasc. 1-4, 143, 1991
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
THRESHOLD ENERGY IN DIRECT TRANSITION
Mariana LATU*
*Technical University, Department of Physics, B-dul D.Mangeron, Corp T, nr.
67, 700050 Iasi, Romania
Abstract. The direct transition probability of an electron, which is in a repulsive interaction with another
electron via phonons, and the complex dielectric constant is derived. To determine the transition probability a
perturbation calculation of second order was used. We find the total numbers of transitions per unit volume
and time w(ω) and the imaginary part of the complex dielectric constant ε 2 , which is connected to the matrix
element for the electron-electron interaction via phonon. The threshold energy in the case of the indirect
electron-electron interaction via phonons contains an additional term.
1. INTRODUCTION
The photon absorption in a semiconductor leads to the electron
transition from the valence band to the conduction band (the fundamental
absorbtion), the electron transition within the same band, the formation of
excitons, the electron transitions from the donor impurity levels to the
conduction band or from the valence band to the acceptor impurity levels, etc.
In the fundamental electron absorbtion, the electron transition can be
direct or indirect.
The direct transitions there are in the semiconductors in which the
minimum of the conduction band and the maximum of the valence band are to
the same wave vector k . Apart from the electron and the photon, no other
elementary excitations are involved in the direct transitions.
The conservation law of energy and of the wave vector require that the
excitation energy E( k f ) − E( k f ) be equal to the energy of the absorbed

photon hν, and that k f − k i be equal to the photon wave vector K . Since for
the energies involved (of order eV) the photon wave numbers are several order

of magnitude smaller than the dimensions of the Brillouin zone, K can be
neglected.
37
THRESHOLD ENERGY IN DIRECT TRANSITION
In the indirect transitions the minimum of the conduction band and the
maximum of the valence band are not at the same k The momentum can only
be conserved if an additional phonon is absorbed or emitted.
2. THE TRANSITION PROBABILITY
The basic electron-phonon interaction process is the annihilation or
creation of a phonon with the wave vector q , with the simultaneous change of
the electron state, from k to k ± q .
The total hamiltonian of electrons, phonons and their
interaction, in units with  = 1 , is [1-6]
:
(
H t = ∑ ωq a q+ a q + ∑
ε k C k+ C k + iD ∑
C k + q C k a q − a −+q



q

k ,q
k
)
(1)
where a q+ , a q are the creation and annihilation operators for phonons and
C k+ , C k for electrons .The electron –electron interaction may be written as [1,
2, 5-12]:
H 1' = D 2
∑

q,k ,k '
ωq
(ε k − ε k − q ) 2 − ωq2
C k+' + q C k' C k+-q C k
(2)
The electron-electron interaction is repulsive for :
ε k − q − ε k > ωq
(3)
and attractive otherwise [1-4, 8-12]. From (3) result the condition:

q 2 + 2k q > 18 ⋅ 10 3 ωq
(4)
for the electrons near the maximum of the valence band in semiconductors .
Using the dispersion curves for phonons result that the relation (4) is correct for
the acoustic phonons in CdS, ZnO, ZnSe, ZnTe [8].
We have proposed that the direct transition probability of an electron in
a repulsive interaction with another electron, via phonon, be established. To
determine the transition probability we start from the Schrödinger equation:
38
M. LATU
 2 p2
 1 

 
+
+
+ ∑
p
e
A
Vk,q C k+ C k" C −+k C − k" + V(r1 ) + V(r2 )Ψ = iΨ (5)



2m q,k,− k
 2m

(
)


where A is the vector potential vector of the incident light and V( r ) is the
potential field of the lattice ions.
We have used a perturbation calculation of second order. The
hamiltonian is made up of the unperturbed part H 0 and the perturbation H ' :
H = H0 + H '
H0 =
H ' = H 1' + H 2' =
(6)


p2
+ V(r1 ) + V(r2 )
m
∑V
 
q,k ,− k

k ,q
C k+ C k" C −+k C -k" +
(7)
e 
pA
m
(8)
We have omitted the A2-terms as small. The solutions of the unperturbed
problem are the Bloch functions. We expend Ψ in terms of the solutions of the
unperturbed problem:


i




− [En(k ")+ En( − k ") ] t

Ψ = ∑
a
(
k
",
k
"
,t)e
n,n,
k
"
,
k
"
−
−
nn
 
(9)
n,k ",− k "
Introducing this relation in the Schrödinger equation and multiplying it by
e


i
Em(k ')+ En( − k ) t

[
]


m,n,k ' , − k result
i
 


[Em(k')− En(k")] t
1

⋅
a mn(k ', − k ,t) =
∑ ann(k ", − k ",t)e
i n,k",− k"
e


i
E n( − k )− E n( − k ") t

[
]
 


< m,n,k ', − k H ' n.,n,k ", − k " >
(10)
For time t we have


(2)
a jj (k ", − k ",t) = 1 + a (1)
jj + a jj + .....
(11)


(1)
(2)
+ a nn
+ ........ n ≠ j
a nn(k ", − k ",t) = a nn
(12)
39
THRESHOLD ENERGY IN DIRECT TRANSITION
If we insert (11), (12) in (3) and compare the terms of the same order
on the left and on the right member, neglecting the A 2 –terms as small
(1)
and a (2)
compared with the others, result the expressions for a nn
j 'j . By
(1)
integration we obtained a nn
, a (2)
j 'j and the transition probability W which is
equal to a (2)
j 'j .
'
2
2
< f H 2" m > < m H 2' i >
'
W(j,j ,k ,ωω,t=



 2 2πt ⋅
E j (k ) − E j (k ") + E j ( − k ) − E j ( − k ")
(
)




δ E j' (k ') + E j ( − k ) − E j (k ") − E j ( − k ") − ω
[
]
(13)
3. THE IMAGINARY PART OF THE COMPLEX DIELECTRIC
CONSTANT
The complex dielectric constant has a real part and an imaginary
part:.[6]
ε (ω ) = ε 1 (ω ) + iε 2 (ω )
(14)
(15)
ε1 = n 2 − α 2
(16)
ε 2 = 2nα
α is the extinction coefficient and n the real index of refraction .
We can find the total number of transition per unit volume and time,
dividing (13) by t, summing over all j (occupied in BV), j’ (unoccupied in BC)
and integrating throughout the Brillouin zone.

1 2
W(j,j',
k
,ωω,t)dk
3 ∫
jj' t (2π2π
The relation between ε 2 and W(ω) is:
2W(ω(ω
ε2 =
ε0 ω 2 A02
W(ω(ω= ∑
From (13), (17), (18) and the law of conservation energy result:
(17)
(18)
40
M. LATU
2
2

'
f
e
m
m
H
i
<
∇
>
<
>
1
4ππ 
2




ε2 =
∫
⋅
∑
2
3
2
ε0 ω
(E j (k ) − E j (k ') + E j ( − k ) − E j (k ")) 2
j,j' (2π2πm j



(19)
δ E j' (k ') + E j ( − k ) − E j (k ") − E j ( − k ") − ω dιk
2
4
[
]
With the ofen used assumption that the matrix element does not change
much according to k, one can place it in front of the integral. From the energy
conservation law result:
ω = E g +
mc =
2k 2
2mc
m j m j'
 2 k ''
−
mj
2
(20)
(21)
m j' − m j
Calculating the integral, for ε 2 we have obtained the relation:
ε2 = ∑
C if
ω2
(ω − E g −
 2 k" 2 1/2
)
mj
(22)
where:
2e 2 
1
(2mc ) 2


2
ε0 πm j
E j (k ) − E j' (k ) + ω
3
C if =
(
)
2
2
2

< f ∇e m > < m H 1' i > (23)
4. CONCLUSIONS
In the direct transition of an electron which is not in the interaction with
other electron via phonon, the relation of ε 2 is:
−2
ε 2 ~ω (ω − E g )
1
2
(24)
In the direct transition of an electron which is in a repulsive interaction
with another electron via phonon, in the relation of ε 2 appears the matrix
element for the electron-electron interaction. In this case the threshold energy
is:
THRESHOLD ENERGY IN DIRECT TRANSITION
ω = E g +
 2 k" 2
mj
41
(25)
Thus, if we take into account a repulsive interaction via phonon, the threshold
energy contains an additional term.
REFERENCES
[1] J. Bardeen, L. N. Cooper, J. R. Schrieffer, Phys.Rev., 106, 162, (1957)
[2] J. Bardeen, L. N. Cooper, J. R. Schrieffer, Phys.Rev., 108, 1175, (1957)
[3] M. L. Cohen, Phys.Rev.,134, 511, (1964)
[4] M. L. Cohen, Rev.Mod.Phys., 36, 240, (1964)
[5] C. Kittel, Quantum Theory of Solids, ed. J. Wiley & Sons, (1964)
[6] C. Madelung, Introduction to Solid State Theory, Springer-Verlag Berlin
Heidelberg New York, (1978)
[7] G. Nelin, Phys.Rev. B, 10, 10, 4331, (1974).
[8] R. R. Alfano, S. L. Shapiro, Phonons (International Conference, Rennes,
France), (1971)
[9] M. Cardona, Physica C, vol. 317-318, 30, (1999)
[10] N. Andrenacci, H. Beck, Physica C, vol.408-410, 275, (2004)
[11] V.C. Aguilera-Navarro, M. Fortes, M. de Llano, Solid State
Comunications, vol 129, 9, 577, (2004)
[12] M.Kitamura, A. Iric, G.I.Oya, Physica C, vol.423, 3-4, 190, (2005)
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
APPLIED GEOMETRY OF GAUGE-INVARIANCE TO
SOME PHYSICALLY IMPORTANT METRICS
Ana Camelia Lohan and Ciprian Dariescu
University “Al. I.Cuza”, Faculty of Physics, 11Blvd. Carol I, 6600, Iaşi,
România
Using the Cartan formalism for the free of coordinate differential geometry formulation, we derive
for the class of statically spherically symmetric metrics, the 1-forms connection
the Ricci tensor components
Rab
and the scalar curvature
R.
Γ ab , the curvature 2-forms,
In this way we obtain the essential
components of Einstein tensor in the pseudo-orthonormal tetradic frame. Application of these to Einstein
equations in the case of the vacuum spacetime structure and respectively, to the one sustained by a spherically
symmetric electrostatic field, leads in a unitary manner to the well-known exact solutions, of physical
interest, Schwarzschild and Reissner-Nordström, respectively. The
SO ( 3,1) -gauge invariant character of
the external group allows the generalization and application of this method in the case of more intricate
metrics for astrophysics such as the Starobinsky’s one.
1. Introduction
The theories of modern physics generally involve a mathematical
model, defined by a certain set of differential equations and supplemented by a
set of rules for translating the mathematical results into meaningful statements
about the physical world. In the case of gravitation, it is generally accepted that
the most successful is Einstein’s theory of general relativity.
Soon after Einstein gave the first form of the field equations bearing his
name, in 1915, namely Rik = κ 0Tik , he realised that this one is incorrect, because
R ik ;k ≠ 0 leads to Tik ;k ≠ 0 and therefore, it runs counter to the matter motion
equations. Two years later, taking into account the second contracted Bianchi
1


identities, Einstein gets the vanishing 4-divergence  Rik − gik R  ≡ 0 , so that,
2

 ;k
the proper form of his equations reads Gik = κ 0Tik . Nevertheless, for vacuumtype solutions, Tik = 0 implies
=
T g=
0 and consequently (from G = − R )
ik Tik
it yields R =
−κ 0T ≡ 0 , or Rik = 0 . That allowed Karl Schwarzschild to derive,
as early as 1915, the first exact solution to the relativistic gravitational field
equations, which (also) describes a previously unknown astrophysical object,
namely, the static spherically symmetric black hole. Later on, based on the
proper form of the Einstein’s field equations, it has been derived the so-called
APPLIED GEOMETRY ON GAUGE INVARIANCE
43
"Inner Schwarzschild solution" (translated from the original German text:
"Innere Schwarzschild Lösung"), valid within a sphere of radius R of
homogeneous and isotropic distributed molecules. It is applicable to solids,
incompressible fluids, the sun and other similar stars viewed as a quasi-isotropic
heated gas.
In this paper, we discuss the Schwarzschild and Reissner-Nordström
solutions for the field equations, using the Cartan formalism.
2. The Cartan formalism
In 1921-1923 Élie Cartan derived the famous equations, based on the
free of coordinate formulation in terms of congruence, which also has been
suggested by the Romanian mathematician Gheorghe Vrânceanu.
As it is known, the differential geometry is the modern language of
physics and mathematics and includes studies about space curvature and
differential equations. Differential geometry is based on p-forms which belong
to the class of skew-symmetric covariant tensors. A covariant vector is just
described by a given 1-form (co-vector), a 2-form can be the Maxwell tensor,
while the 3- and 4-forms represent pseudo-vectors (magnetic field) and pseudoscalars (mesons), respectively.
The first E. Cartan structure equation has the form [1]
(1)
=
d ω a Γ .[abc]ω b ∧ ω c , 1 ≤ b 〈 c ≤ 4
where ω a is the rigid frame 1-forms, Γ .[abc] - connection coefficients and
ω b ∧ ω c is the exterior product of ω b and ω c . The exterior product (or wedge
product) is an associative and bilinear operation. In this paper we use some of
its properties [2] like, α ∧ α =
0 and α ∧ β =− β ∧ α for all α , β ∈V where V is
a vector space. In calculus we made, the 1-forms Γ .[abc] with respect to a rigid
frame are skew-symmetric in their first two indices.
The second Cartan equation, for the curvature 2-form(s), reads [1]
d Γ ab + Γ ac ∧ Γ .bc ,1 ≤ b 〈 c ≤ 4 (2)
=
ab
where
=
 ab
1
Rabcd ω c ∧ ω d , and Rabcd designate the Riemann tensor
2
components.
The Einstein equations do generally represent a highly non-linear 2-nd
order differential system, which describes the relativistic gravitational field by
44
A.C. LOHAN , C. DARIESCU
the essential components of the space-time curvature, caused by the matterenergy-momentum contributions, and therefore, they read
1
κ 0Tab
g ab R =
2
with the Ricci tensor (measure of volume distortion) [1], [4]
Rab = R c .acb
and the scalar curvature (Ricci scalar) [1], [5]
R = g ab Rab
Rab −
(3)
(4)
(5)
The Einstein tensor is defined by [5], [6]
Rab −
where κ 0 =
1
g ab R =
Gab
2
(6)
8π G
is the Einstein (universal) constant and, Tab stands for
c4
the conservative energy-momentum tensor.
3. Statically spherically symmetric metrics
As it is very well known, the two metrics, Schwarzschild and ReissnerNordström, belong to the same class, being described by
2
2
2
2
−2 f r
2f r
(7)
ds 2= e ( ) ( dr ) + r 2 ( dθ ) + sin 2 θ ( dϕ )  − e ( ) ( dt )


where r is the usual radial coordinate and f : R+ → R is a real function of
class C 2 at least. The pseudo-orthonormal tetradic frame has the 1-forms
ω 1 = e − f dr
ω 2 = rdθ
ω 3 = r sin θ dϕ
ω 4 = e f dt
whose exterior derivatives are
(8)
45
APPLIED GEOMETRY ON GAUGE INVARIANCE
dω 1 = 0
ef 1
ω ∧ ω2
r
ef 1
ctgθ 2
3
d ω=
ω ∧ ω3 +
ω ∧ ω3
r
r
4
' f 1
4
dω
f e ω ∧ω
=
=
dω 2
(9)
Using the first Cartan equation, (1), we find the connection coefficients
ef
Γ=
Γ
=
212
313
r
ctgθ
(10)
Γ 323 =
r
Γ 414 = − f ' e f
which allow us to define the connection 1-forms as [6]
(11)
Γ ba = Γ bca ω c
namely
ef
Γ 12 = − ω 2
r
ef
Γ 13 = − ω 3
r
' f 4
(12)
Γ 14 = f e ω
ctgθ 3
ω
r
Γ=
Γ=
0
24
34
From the 2-nd Cartan equation, (2), the curvature 2-forms can be written as
Γ 23 = −
46
A.C. LOHAN , C. DARIESCU
1
r
1 2f 1
=
− e ω ∧ ω3
r
 12 =
− f ' e2 f ω 1 ∧ ω 2
 13
(
)
 14 =
f '' + 2 f ' e 2 f ω 1 ∧ ω 4
(13)
f' 2f 2
e ω ∧ ω4
r
f' 2f 3
=
 34
e ω ∧ ω4
r
1
 23 =2 1 − e 2 f ω 2 ∧ ω 3
r
leading to the essential Riemann tensor components
f'
R1212 = − e 2 f
r
1
R1313 = − f ' e 2 f
r
=
 24
(
R1414
=
)
(f
''
)
+ 2 f ' 2 e2 f
f' 2f
e
r
f'
R3434 = e 2 f
r
1
R=
1 − e2 f
2323
r2
The Ricci components defined by (4) do concretely read
2 f' 2f
R11 =
e − f '' + 2 f ' 2 e 2 f
−
r
2 f' 2f
1
R22 =
R33 =
e + 2 1 − e2 f
−
r
r
'
2 f 2f
R=
e + f '' + 2 f ' 2 e 2 f
44
r
and the scalar curvature takes the form
(14)
R2424 =
(
)
(
)
(
(
)
)
(15)
APPLIED GEOMETRY ON GAUGE INVARIANCE
47
8 f' 2f
2
(16)
−
R=
e − 2 ( f '' + 2 f ' 2 ) e 2 f + 2 ( 1 − e 2 f )
r
r
Putting everything together, one gets the Einstein tensor components, (6), as it
follows,
2 f' 2f 1
G11=
e − 2 (1 − e2 f )
r
r
'
2 f 2f
G22 = G33 =
e + ( f '' + 2 f ' 2 ) e 2 f (17)
r
'
2 f 2f 1
−
G44 =
e + 2 (1 − e2 f )
r
r
For the Schwarzschild metric [3], [4]
2m
(18)
e2 f ( r ) = 1 −
r
the expression (17) becomes
(19)
G=
G=
G=
G=
0
11
22
33
44
and does clearly show the vacuum spacetime nature.
Let us switch now to the Reissner-Nordström metric [4]
2m q 2
(20)
+ 2
e 2 f ( r ) =−
1
r
r
whose expression is sustained by the condition R = 0 for the electromagnetic
field. From R = 0 , it follows that
d2
4 d
2
(21)
1 − e2 f +
1 − e2 f + 2 1 − e2 f =
0
2
r dr
dr
r
and, using the notation h= 1 − e 2 f , one gets
d 2h
dh
(22)
r 2 2 + 4r
+ 2h =
0
dr
dr
With h = r α it yields
(23)
α+ =
−1 , α − =
−2
which leads to form
C1 C2
(24)
e 2 f =−
1
− 2
r
r
where the constants C1 and C2 are given by C1 = 2m, C2 = −q 2 .
Taking into account the expression for the conservative energymomentum tensor [5]
(
)
(
)
(
)
48
A.C. LOHAN , C. DARIESCU
1
(25)
=
Tab Fac Fb.c − ηab F cd Fcd
4
it gets the following components
1
T11 = − F142
2
(26)
1 2
T=
T=
T=
F14
22
33
44
2
To show the relation between the geometrical electric charge ( q ) and the
rationalized charge ( Q ) , we use the relations (17), (6), (3) and the equation
Q
. For instance, the component G11 = κ 0T11 takes the form
r2
κ Q2
G11 = − 0 4
2r
On the other hand, from the equations (17) it yields
2 f' 2f 1
q2
(28)
− 4
e − 2 (1 − e2 f ) =
r
r
r
If we match the last two equations it follows that
F14
= E=
1
q=
(27)
κ0
(29)
Q
2
The same result it yields using any other component of the Einstein tensor.
4. Conclusions
The Schwarzschild solution represents the spherically symmetric empty
spacetime outside a spherically symmetric non-rotating massive body. In fact,
the difference between the Newtonian theory and General Theory of Relativity
are based on this solution. The metric is singular at r = 0 and r = 2m , the latter
been called the “gravitational radius”, or the “Schwarzschild radius”. It is also
known as the “Schwarzschild horizon” since, beyond it, going into the interior
region, t and r exchange their roles as time and space coordinates. The
vanishing of gtt suggests that the surface r = 2m , which appears to be threedimensional in the Schwarzschild coordinate system, has zero volume and thus
is actually only two-dimensional, or else is null.
Thence, the Schwarzschild black hole, located at r = 0, has a
surrounding area, called the event horizon, which is situated at the
Schwarzschild radius. Any in-coming particle that crosses the Schwarzschild
APPLIED GEOMETRY ON GAUGE INVARIANCE
49
horizon will never make it out again and thus, it will never be seen by the
exterior observers. The Schwarzschild radius can also be found by setting the
2Gm
escape velocity to the speed of light and yields rs = 2 , where G is the
c
gravitational constant and m is the mass of central object. For the Sun rs is
about 3 km , while for the Earth is almost 8,86 mm . In order to become a black
hole an object should collapse to a characteristic radial dimension smaller then
its corresponding Schwarzschild radius The name of black hole was invented by
John Archibald Wheeler, who also introduced the word wormhole to describe
tunnels in spacetime.
The Reissner-Nordström solution represents the spacetime outside a
spherically symmetric charged body carrying an electric charge Q, with no spin
or magnetic dipole. Apart of r = 0, where the black hole is located, the metric
has the singular “points” r± =±
m
m 2 − q 2 , where m represents the
gravitational mass and q the electric charge of the body. If |q| > m, the metric
is non-singular everywhere except at r = 0 . On the other hand, if |q| < m, the
metric gets the two singular spheres of radii r+ and r− .
Thence, the Reissner-Nordström (electrically charged) black hole has
two concentric horizons: the event horizon and the Cauchy horizon. The former,
located at r+ , is the only one that one can escape from, while the inner horizon
is the ultimate one-way space-time membrane, surrounding the cosmic
Reissner-Nordström vacuum cleaner.
Acknowledgements.
This work is supported by the Grant Type A CNCSIS Code 1160/2007.
5. References
[1]- B.A. Bassett, C. Dariescu, M.A. Dariescu, B. Gumjudpai- Proceedings
from The First Poe School on Cosmology: Introductory Cosmology, TPCosmo I
2002
[2]- H. Flanders- Differential Forms with Applications to the Physical Sciences,
Academic Press, 1963
[3]- C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation- W.H.Freeman and
company San Francisco, 1972
[4]- S.W. Hawking, G.F.R. Ellis- The large scale structure of space-time,
Cambridge university press, 1973
50
A.C. LOHAN , C. DARIESCU
[5]- D. Kramer, H. Stephani, M. MacCallum, E. Herlt- Exact Solutions of
Einstein’s Field Equations, VEB Deutscher Verlag der Wissenschaften, Berlin
1980
[6]- C. Dariescu, M.A. Dariescu- Gravitaţie şi câmpuri în Universul Einstein,
Editura Vesper, 1997
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
THE EQUATIONS OF KINEMATIC REFLECTION
Dimitrie Olenici ¹ , Stefan-Bogdan Olenici²,
¹Planetariul Suceava , Romania
²Facultatea de Fizica, Universitatea “Al. I. Cuza” Iasi, Romania
Contact person: [email protected]
Abstract
In this paper we show that the correct application of Huygen`s Principle at the surface of a
mirror which is in movement with respect to the light propagation medium has led us to modify the classical
laws of reflection, which were established for static situations. Because in practice all optical elements are in
movement together with the Earth around the Sun at 30 km/sec, in reality, we are actually always in a
kinematic situation, and for very accurate observations we must consider the laws of this new optics,
Kinematic Optics.
Introduction
The classical laws of optics were established on the assumption that
the light sources, the medium of propagation of light, and the optical elements
such as mirrors, lenses, prisms etc. are fixed. But in reality all these are in
movement along with the Earth around the Sun. Because this we are obliged
consider a new optics: “kinematic optics”.
In this article we tackle the problems of kinematic reflection.
To understand this, we will treat in parallel aspects of classical
reflection which we have named static reflection, in comparison with this new
reflection, which we have named “kinematic reflection“.
The reasoning assumes that the light waves and the optical elements
are in movement through the ether, the light transmission medium, which is
considered to be fixed.
1. Two-dimensional static reflection
The fact is well known that the following two laws of geometrical optics have
been established in accordance with experimental results:
1st Law: The incident beam, the perpendicular to the reflective
surface, and the reflected beam lie in the same plane. See Fig.1 a).
2nd Law :The angle of reflection r and the angle i made by the
incident beam and the perpendicular N to the reflective surface at the point of
incidence are equal: i = r .
On the other hand light reflection can be explained using the HuygensFresnel principle in the framework of wave optics theory. See Fig.1 b).
52
D. OLENICI ET AL
Fig.1 Classical representation of the reflection left) and the
explanation of it using Huygens- Fresnel principle right)
Observation:
The statements above are valid only in the case of a static situation
when the light source S, the reflective surface M and the medium through which
the light wave is propagating are static ( motionless) with respect to one
another (i.e. in a situation of relative rest ).
Because we have been considering a plane wave, we term this kind of
reflection static two-dimensional reflection.
2. Considerations of kinematic optics
In practice, situations are encountered when the relative speed of optical
devices with respect to a light source has to be taken into account. An example
of such a situation is the explanation of astronomical light aberration
phenomena. It is well known from astronomy that, for the image of a star S to
appear in the reticule of a telescope, the telescope needs to be inclined at an
angle of σ in order for the speed of movement of the Earth to be compensated,
as shown in Fig. 2 a)
The aberration angle is given by the formula:
v
sin σ = sin θ
c
THE EQUATIONS OF KINEMATIC REFLECTION
53
where: v = 30 Km ⋅ s −1 is the speed of movement of the Earth around
the Sun on its orbit and c = 300000 Km ⋅ s −1 is the speed of light.
Observation:
The formula above has been arrived at by taking into account just one
single beam of light.
If the Huygens-Fresnel principle is used to explain the astronomical
aberration of light, the situation gets more complicated.
Let us consider a reflecting telescope of d length and aperture l. The
wave surface Σ coming from the star S will successively arrive at points MM’
on the surface of the mirror, which is moving at the speed v together with the
Earth, in points on a virtual surface MM’’. This virtual surface MM'' makes an
angle γ with the actual surface of the mirror MM’. See Fig.2 b)
a)
b)
Fig. 2 Astronomical light aberration phenomena a)
and the explanation using Huygens- Fresnel principle b)
Observation:
If this point of view is not accepted, then one needs to return to the
antique concept of instantaneous propagation of light (action at a distance).
54
D. OLENICI ET AL
This example forces us to reconsider the phenomena of light reflection in
kinematic situations.
3. Two-dimensional kinematic reflection
Let us consider a kinematic situation in a light-transmitting medium
when a mirror MM’ is moving with respect to the medium and is receding from
the light source. See Fig. 3.
The first contact between the wave front Σ and the mirror will take
place at point A’ and the last one at point B’’. The line A’B’’, which is a virtual
reflective surface, makes the angle γ with the mirror MM .
Putting the equations B’B’’=ct; M’B’’=v·t and using the simple
relations expressed by the triangles A’B’B’’ and A’B’M’, the follow formula
for γ angle can be obtained:
tgγ =
v sin i ⋅ sin α
where:
c ± v cos i ⋅ sin α
c is the speed of light in the medium;
v is the speed of the mirror with respect to the medium;
i is the angle of incidence;
α is the angle between the surface of the mirror and the velocity v.
The sign in the above equation is negative when the mirror is receding
from the wave front and positive when the mirror is approaching the wave front.
The situation is as though the mirror MM has experienced a rotation
through the angle γ.
Fig.3 The use of the Huygens-Fresnel principle in the case of a mirror
that moves with respect to the medium through which the light travels.
THE EQUATIONS OF KINEMATIC REFLECTION
55
Bi-dimensional kinematic reflection
The envelope of the elementary waves emanating from centres of
oscillation along the line A’B’’ is given by the reflected wave front Σ'.
The reflection takes place with respect the perpendicular N’ to the
surface A’B’’ which makes an angle γ with the perpendicular N to the surface
MM. As a result, the angle of incidence i’ with respect to the perpendicular N’
will have the value i' = i ± γ . In this situation, the 2nd law of reflection in the
case of a two-dimensional kinematic situation becomes r = i ± γ .
example
:
if
i = α = 45 o ,
v = 30 Km ⋅ s −1 and
′′. This can not be
c = 300000 Km ⋅ s −1 , th e an gle γ has the value 10.33
neglected in the astronomical determinations.
Because a plane wave has been used in our reasoning, we call this kind
of reflection 2-dimensional kinematic reflection.
For
Observation:
The first law of reflection remains unchanged.
4. Three-dimension static reflection
Let us consider a bundle of parallel light rays in which the wave front
OAB has the shape of a right angled triangle parallel with the plane XO’Y of an
O’XYZ coordinate system. We consider a mirror M placed at the origin of the
system, in a position such that the projection of the wave front OAB describes
the triangle O’A’B’ on it.
From Fig.4 it can be seen that intersection of the mirror with the plane
ZO’X is given by the line O’A’ which makes the angle β with the O’X axis, and
the intersection of the mirror with the ZO’Y plane is given by the line O’B
which makes the angle θ with the O’Y axis.
We consider three perpendiculars N, NA and NB respectively at the
points O’, A’ and B’ that are mutually parallel. These three perpendiculars
determine a bundle of perpendiculars, with respect to which the rays of light
that are parallel with the O’Z axis are reflected.
After reflection we will get a bundle of parallel rays with the wave front
O’’A”B”. For each ray of light the classical laws of reflection are valid
separately.
To find the direction of the perpendicular N in space, one has to write
the direction cosine equations with respect to the X, Y and Z axis.
56
D. OLENICI ET AL
Fig.4 Illustrating the static reflection of a bundle of parallel light rays
To do this we start from the equations determining the plane O’A’B’,
and take into account the coordinates of the points O’, A’ and B’ O'(0, 0, 0, ) ;
A' ( a, 0, a∙tgβ), B' ( 0, b, b∙tg θ), where OA =a , OB =b.
From the equations: X cos α x + Y cos α y + Z cos α z = 0
cos 2 α x + cos 2 α y + cos 2 α z = 1
We obtain for the direction cosines the following formulas:
(tg β + tg θ + 1)
= tgβ (tg β + tg θ + 1)
= 1 (tg β + tg θ + 1)
cos α x = tgβ
cos α x
cos α z
2
2
2
2
2
2
THE EQUATIONS OF KINEMATIC REFLECTION
57
5. Three-dimensional kinematic reflection
Let us now suppose that the mirror M has a movement of translation
with respect to the medium along the positive direction of the OZ axis, with the
velocity v.
From Fig.5 we see that:
The first contact of the wave front with the mirror M takes place at the
point O’;
In the XO’Z plane, the wave front will catch up the mirror at the point
along the O’A” line;
In the ZO’Y plane, the wave front will meet the mirror M at the points
along the O’B” line; and
The reflection of the waves will take place from the virtual surface of
the triangle O’A”B”. The perpendicular N” to this is no longer identical with
the perpendicular N’ to the mirror M.
Fig.5 The representation of the virtual reflection surface( O′A"B") of a
mirror (O′A′B′) that moves with the constant speed v
58
D. OLENICI ET AL
The direction in space of this perpendicular N” can be found by writing
the direction cosines with respect to the X,Y and Z axes. To do this, we start
with the equation of the O’A”B” plane, and consider the coordinates of the
points O’,A” and B”:
O' ( 0,0,0) ; A''( a, 0,atg ( β+γx) ) ; B''(0, b, b tg( θ+γy) ).
This time the angles β and θ are increased by the values γx and γy
respectively, because of the movement of the mirror and because of the
occurrence of kinematic reflection.
In a similar way the direction cosines of the perpendicular O’N” have the
following values:
cos α x'' = tg (β + γ x )
tg 2 (β + γ x ) + tg 2 (θ + γ y ) + 1
cos α y'' = tg (θ + γ y )
tg 2 (β + γ x ) + tg 2 (θ + γ y ) + 1
cos α z'' = 1
tg 2 (β + γ x ) + tg 2 (θ + γ y ) + 1
The values of the angles γx and γy are determined using the following
formula
tgγ = v sin i sin α (c ± v cos i sin α )
Fig.6 Representation of the perpendicular N’ on the mirror O ′A′B′ and
of the virtual reflection surface with respect to the O ′XYZ coordinates system
59
THE EQUATIONS OF KINEMATIC REFLECTION
And, considering the two-dimensional reflection at the plane ZO’X for γx and at
the plane ZO’Y for γx respectively, as can be seen from Fig. 6.
Starting with the following equations: Xtgβ + Ytgθ − Z = 0
Xtg (β + γ x ) + Ytg (θ + γ y ) − Z = 0
and using the formula:
cos ϕ = ( A1 A2 + B1 B2 + C1C 2 )
(A
)(
+ B12 + C12 ⋅ A22 + B22 + C 22
2
1
)
where the coefficients have the following values:
A1 = tgβ ; B1 = tgθ ; C1 = −1 ; A2 = tg (β +γ x ) ; B2 = tg θ + γ y ; C 2 = −1
(
)
we get the angle γxy between the plane of the mirror O’A’B’ and the plane
O’A”B” on which the elementary oscillating sources are placed during the
reflection process.
This angle can be computed using the following formula:
cos γ xy =
(tg
(
)
tgβ ⋅ tg (β + γ x ) + tgθ ⋅ tg θ + γ y + 1
2
)(
)
β + tg 2θ + 1 ⋅ tg 2 (β + γ x ) ⋅ tg 2 (θ + γ y ) + 1
In this case the angle of reflection r” will have the following value:
r = i + γ xy . This expression represents the second law of three-dimensional
"
kinematic reflection.
In a similar way, starting with the following equations:
X cos α y − Y cos α x = 0
X cos α y' − Y cos α 1x = 0
Where the coefficients have the following values:
A1 = cos α y ; B1 = − cos α x ; C1 = 0 and
A2 = cos α y' ; B2 = − cos α x' ; C 2 = 0
We get the value for the angle γir between the plane ZON’ containing
the perpendicular N’ to the mirror M and the plane ZO’N” containing the
perpendicular N” to the surface O’A”B” using the following formula:
cos γ ir =
(
)
tgθ ⋅ tg θ + γ y + tgβ ⋅ tg (β + γ x )
(tg θ + tg β )⋅ (tg
2
2
2
(β + γ x ) + tg 2 (θ + γ y ))
60
D. OLENICI ET AL
The fact that an angle appears between the plane ZO’N′ that contains the
incident beam and the plane ZO’N” that contains the reflected beam means that,
(see Fig. 7), in the case of three-dimensional kinematical reflection, the 1st law
of reflection is no longer valid in the form in which it has been accepted up till
now. The appearance of the angle γxy proves the change of the 2nd law of
reflection in the case of kinematic reflection.
Fig.7 The representation of the angle γxy made by the perpendicular N′
on the moving mirror and the perpendicular N’’ on the virtual reflective
surface, and the representation of the angle γir made between the planes
containing the perpendiculars N′ and N".
6. Preliminary experimental results
In order to verify whether the kinematic reflection exists, we made an original
light deviation experiment with a fixed terrestrial telescope T . In front of the
telescope a mirror M was placed at half of the objective O, see Fig.8 a). So we
were able to see directly the image of a mark A , and also the image of a mark B
after reflection from the mirror.
The directions of Mark A and mark B made an angle of ±35º with the
direction N of north.
The determinations were made for two weeks, day and night. Each hour
the positions of the marks were read with the precision of 1" arcsecond. Due to
movement of the telescope with the Earth around the Sun, the deviation of the
THE EQUATIONS OF KINEMATIC REFLECTION
61
image of the mark B after reflection was twice the deviation of mark A. See
Fig. 8 b). We consider that this difference in deviation can be explained only by
kinematic reflection.
a)
b)
Fig.8 The experimental set-up used to verify the appearance of
kinematic reflection a) and deviations in time of the images of two marks
considered in the experiment b)
Acknowledgements
We thank our Professor Dr. Ioan Gottlieb for suggestion to do studies
referring to kinematic optics.
Bibliography
1.R.J.Kennedy, Simplified theory of the Michelson-Morley Experiment,
Phy.Rev.vol 47, 1935
2. D.Olenici, The kinematic reflection: a new method of standing out
the ether, Proceedings the III-rd Conference on theoretical physics, general
relativity and gravitation held May 1993, Bistriţa ,Romania
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
DROP DEFORMATIONS UNDER SURFACE
TENSION GRADIENTS
Ioan-Raducan STAN,1 Maria TOMOAIA-COTISEL,2
Aurelia STAN3
1
Babes-Bolyai University of Cluj- Napoca, Department of Mechanics and Astronomy, Kogalniceanu Str., Nr.
1, 400084 Cluj-Napoca, Romania; [email protected]
2
Babes-Bolyai University of Cluj-Napoca, Department of Physical Chemistry, Arany Janos Str., Nr. 11,
400028 Cluj-Napoca, Romania; [email protected]
3
Augustin Maior Technical College, Calea Motilor Str., Nr. 78, 400370 Cluj-Napoca, Romania.
[email protected]
Abstract. The effect of the surfactant adsorption on the surface of a free liquid drop
immersed in an unbounded liquid (the densities of the two bulk liquids are equal) is
studied. The interfacial tension gradient caused by the surfactant adsorption at the drop
surface generates surface forces exerted within the boundary region of the drop. The effect
of the variable interfacial tension gives rise to a surface flow (i.e. Marangoni flow) which
causes the motion of the neighboring liquids by viscous traction, and generates a
hydrodynamic pressure force (named Marangoni force) which acts on the drop surface.
The Marangoni force is examined on several cases using nondeformable and deformable
drops.
Keywords: interfacial tension gradient, Marangoni force, Marangoni effect, free drop
deformations, dynamic instability, hydrodynamic model
1. INTRODUCTION
The boundary between two phases of matter is known as the
interfacial zone or, the interface. It is the thin layer surrounding a geometric
surface of separation, in which the physical properties differ much from those
in either of the bulk phases. The thickness of this layer is ill-defined because
the variations of physical properties across it are continuous. We shall consider
it as infinitely thin, i.e., as a geometric surface. Since the thickness of interface
is of the order of molecular dimensions, such an approximation is justified in
treating the macroscopic movements of liquids.
Under given conditions, tangential forces may exert in the interface of
the two liquids, together with the normal pressure. If the surface tension σ, of
the liquid interface changes from point to point, a tangential force will be
exerted in addition to the pressure normal to the surface and its magnitude is
determined by the surface tension gradient, which per unit area is [1]:

pt = grad σ.
DROP DEFORMATIONS UNDER SURFACE
63
The plus sign preceding the gradient indicates that this force tends to move the
surface of the liquid in a direction from lower to higher surface tension.
Surface active compounds (surfactants), present in even small
quantities, have an important role in determining the hydrodynamic behavior of
the two phase system. Those cases of liquid motion, in which surface tension
plays an important role, belong to the interfacial hydrodynamics.
There are many examples where the presence of a surfactant has an
important role. Probably, the best known is the effect of a surfactant on a liquid
drop, immersed in a bulk liquid, initially at rest [2]. The force, acting on the
unit volume of the drop (density ρ’) immersed in a bulk liquid (density ρ), is
cancelled,

(ρ - ρ') g = 0
when either the densities of the two liquids are equal (ρ’= ρ) or in the absence

of gravity g = 0 (zero gravity). Such a drop is called “free” and is motionless.
Free drops undergo complicated motion when a surface tension gradient
is applied on the drop surface. Translational and rotational motion, oscillations,
surface waves and deformations have been experimentally evidenced. To
explain the free drop dynamics we have developed some experiments and
theoretical models [3- 6]. We have shown that the surface Marangoni flow
causes the motion of the neighboring liquids by viscous traction, which
generates the force of hydrodynamic pressure, named Marangoni force.
The aim of this paper is to show, experimentally and theoretically, that
this Marangoni force is correlated with the surface coverage degree, namely
with the extent to which the drop surface is covered by the surfactant. Our
calculations have shown that this force acts like a hammer and like an engine.
So, we divided this force in:
• “hammer” force, responsible with the deformation and break up of the
drop, oscillations and surface waves and
• “lifting”(propulsive) force, responsible with the translational motion of
the drop.
Certainly, these results can not be attribute to a single mechanism, in
our case a mechanical one, but we must consider also other mechanisms,
namely, the surface dilution and tip-stretching of the surface tension, as well as
capillary forces. In our opinion, the real flow at the drop interface and
consequently, the Marangoni force is the principal mechanism.
64
I.RADUCAN STAN ET AL..
2. EXPERIMENTAL MODEL
We shall consider a viscous liquid drop (in our experiments with a
volume between 0.4 cm3 and 7 cm3, noted L’ of density ρ’) immersed in an
immiscible bulk liquid L, (density ρ). The two liquids having the same density
the drop is motionless or at zero gravity. The two liquids inside and outside the
drop are Newtonians, incompressible and viscous, having the viscosities μ and
μ’. The surface between the two liquids is characterized by an interfacial tension
noted σ0.
A small quantity of a surfactant (e.g. a droplet of 10-3-10-2 cm3, which is
very small compared with the volume of the initial drop) is introduced on a
well-chosen point (called injection point) at the drop surface; surfactants, also
known as tensides, are wetting agents that lower the surface tension of a liquid.
Fig. 1. The spreading of a surfactant on a free drop surface.
The surfactant, because of its molecular structure, is simultaneously
adsorbed at the liquid-liquid interface and is continuously swept along the
meridians of the drop. In the injection point, the interfacial tension is
instantaneously lowered to σ1 (σ1 < σ0) value. A gradient of interfacial tension is
established over the surface of the drop. Consequently, the Marangoni
spreading of the surfactant takes place from low surface tension to high surface
tension. We mention that local changes in temperature, in electric charge or the
presence of surface chemical reactions might produce a similar effect.
The symmetry of the problem suggests a system of spherical
coordinates (r, θ, φ) with the origin placed in the drop center and with the Oz
axis passing through the sphere in the point of the minimum interfacial tension,
65
DROP DEFORMATIONS UNDER SURFACE
i. e. the injection point of the surfactant. We underline that the surfactant
injection point at the drop surface may be taken anywhere, the drop being
initially at rest. In the following, we shall take it like shown in Fig. 1, or
otherwise specified.
3. VARIATION OF THE INTERFACIAL TENSION
The variation of the interfacial tension over the drop surface must be
defined before we can proceed with the analysis of the model. Generally, the
interfacial tension, σ, is assumed to be distributed [7] within the surfactant
invaded region, (0 ≤ θ ≤ θf ), by
σ(θ) = σm - a1 cos θ
where, σm and a1 are constants.
For the variation of the interfacial tension with θ, we have proposed for
σ − σ1
σ 0 − σ1
+ σ1 and a1 =
. Thus, the above
the constant values, σm = 0
1 − cosθ f
1 − cosθ f
equation becomes:
σ − σ1
(1– cos θ) + σ1,
σ(θ)= 0
1 − cosθ f
(1)
where σ0 = σ(θf ) and σ1 = σ(0). The Eq. (1) presents the advantage that it
contains the angle θf and permits the calculation of σ for different drop coverage
with surfactant.
In Fig. 2, we give the plot of eq. (1) for a particular case θf = π/2, σ0 =
7.5 and σ1 = 3.5.
Fig. 2. The variation of the surface tension, σ (θ), versus angle θ.
66
I.RADUCAN STAN ET AL..
For the situation when the surface of the entire drop is covered with
surfactant, θf = π, the Eq. (1) is similar to that one proposed by other authors [8].
This confirms that Eq. (1) proposed by us in its general form is correct.
Further, by derivation of Eq. (1), the interfacial tension gradient in the
invaded drop region with surfactant
σ − σ1
dσ
(2)
sinθ ,
= 0
dθ 1 − cosθ f
is obtained.
The interfacial tension σ0 is constant in any point of the uncovered drop
surface, while the interfacial tension difference П = σ 0 − σ 1 arises only in the
invaded region with surfactant. Therefore, it is clear that only σ1 and σ0, i. e. the
minimum and the maximum values of the interfacial tension can be
experimentally measured.
4. THEORETICAL MODEL
The theoretical model reported here considers that the drop is initially at
rest and the real surface flow – Marangoni flow – arises on the drop surface,
with a distinct front, which advances continuously; without a surfactant transfer
inside or outside the drop. The drop surface is considered a two dimensional,
incompressible Newtonian fluid. The Reynolds number of the inside and
outside flow is less than unity.
The equations governing the flow inside and outside the drop are the
continuity and Navier-Stokes equations [9-11]. The continuity equations for an
incompressible fluid are for the outside and inside flow

(3)
∇ ⋅ v =0

(4)
∇ ⋅ v ' =0


where v is the velocity of the bulk liquid L and v ’ represents the velocity of
the liquid L’ within the drop.
The Navier - Stokes equations for a steady flow are :
1
µ 


(v ⋅ ∇) v = − grad p + ∆v ,
ρ
ρ
1
µ' 


∆v'
(v ' ⋅ ∇) v ' = − grad p ' +
ρ
ρ
(5)
(6)
where p and p’ are the pressures outside and inside the drop. All the parameters
are considered constants.
DROP DEFORMATIONS UNDER SURFACE
67
We propose here to give some account on the equations governing the
fluid motion in an interface, considered as a two dimensional, incompressible
Newtonian fluid, having surface density Г, surface dilatational κ viscosity and
surface shear ε viscosity. Even that we have considered the interface like a bidimensional geometrical surface, it has a finite thickness, about 5 x 10-10 cm.
The flow in a surface is not just a flow in a two-dimensional space
whose governing equations will be immediate analogs of the three-dimensional
ones. In contrast with the three-dimensional space, this surface is a twodimensional space that moves within a three-dimensional space surrounding it.
In our case, the interface is the region of contact of the two liquids, namely drop
liquid and bulk liquid. This is again a new feature which oblige us to take
account on the dynamical connection between the surface and its surroundings,


namely on the traction exerted by the outer T and the inner T ' liquid upon the
drop interface.
The equation of the interfacial flow [12-14] is

 

Γ ( w ⋅ ∇ S ) w = F + ∇ S σ + (κ + ε ) ∇ S ( ∇ s ⋅ w ) ,
(7)

 
  
where w = vs is the interface velocity,=
F Γ g + T - T ' is the external force
acting on the drop surface and ∇ S is the surface gradient operator. Because the
surface density is very small (Г ≈ 10 -7g cm) the inertial term can be neglected
against the remainder terms. It is significant to underline that equation (7) can
be used in two ways: as the equation which describes the surface flow or as a
dynamical boundary layer condition.
 
In order to find the distributions of the velocities v, v ' and of the
pressures p, p’, the system of equations (3)-(6) must be solved taking into
account some appropriate boundary conditions [9-11], for the interface between
the two contiguous liquids.
The velocities of the inner and outer liquid of the drop must satisfy the
following kinematical conditions:
• the outer velocity must be zero far from the drop surface,

=
v
0 for r → ∞ ;
• the normal component of the outer and the inner velocities must be zero
on the surface of the drop

'
0 , at r = a;
v= v=
n
•
n
the tangential velocity components of the two liquids at the interface
must be continuous:
 
at
r = a;
v = v'
t
t
68
I.RADUCAN STAN ET AL..

•
the velocity v' within the drop must remain finite at all points,
particularly at the centre of the drop (r = 0 the origin of the
coordinates).
• In addition to these kinematical conditions, a dynamical condition must
be fulfilled at the interface and is given by Eq. (7).
Eqs. (3-7) with these appropriate boundary conditions lead to the

distribution of the velocity v and of the pressures p, outside of the drop:
vr ( r , θ ) =
1 1 
 - θ cos ,
(1 - cosθ )  r 3 a2 r 
f
(8a)
vθ ( r , θ ) =
1
 1
 3 + θ2
(1 - cosθ f)  2r
2a r
(8b)
p(r,θ)= -
A
A
μA
cosθ
a (1 - cos
θ )f r
2
2

 sin ,

.
(8c)
where
A=
(σ 0 - σ1 )a3
3(μ + μ' + 2κ/3a)
(8d)
Similarly expressions for the inner flow are obtained.
5. MARANGONI FORCE EXERTED ON THE DROP
Further, we calculate the Marangoni force exerted on the free drop due
to Marangoni flow. The Marangoni surface flow give rise to a stream of liquid
directed to the drop along the Oz axis.
69
DROP DEFORMATIONS UNDER SURFACE
Fig. 3. The hydrodynamic force fM(r,θ) acting on a point.
This stream arises as a consequence of the continual replacement of that
liquid layer which has been displayed by the surface flow, similar with a
ventilation effect [3], as can be seen in Fig. 3. The hydrodynamic force fM(r,θ)
acting on a point may be decomposed in its normal fn(r,θ) and tangential ft(r,θ)
components.
Immediately, it is observed that the flow occurs with the movement of
the outer liquid L (Fig. 3) driven by viscosity, while the forces of hydrodynamic
pressure will act on the drop L’. The resultant of the forces exerted by the fluid
on the drop, FM, due to the symmetry of the Marangoni flow, is oriented along
the Oz axis. This force, acting on the drop, may be calculated from the general
expression of the force [9]:
FM = ∫∫ [(prrθ -)r(p
) sin θ]
=a cos
r =a
rθ ds
,
(9)
S
where, S is the surface covered (invaded) with surfactant, prr and prθ are the
normal and tangential components, respectively, of the viscous stress tensor:
prr (r , θ ) =− p + 2 µ
∂ vr
,
∂r
 1 ∂vr ∂vθ vθ 
+
− .
θ) µ
prθ (r ,=
 r ∂θ ∂ r r 
(10)
(11)
From Eqs. (8) one obtains for the normal (Eq. (10)) and tangential
components (Eq. (11)) of the stress tensor, at the drop surface (r = a), the
following equations [15]:
70
I.RADUCAN STAN ET AL..
( prr ) r = a = −
( prθ ) r = a = −
µ (σ 0 − σ 1 )
cosθ ,
a ( µ + µ '+ 2κ / 3a ) (1 − cosθ f )
(12)
µ (σ 0 − σ 1 )
sinθ
a ( µ + µ '+ 2κ / 3a )(1 − cosθ f )
(13)
The surface element in spherical coordinates on the drop (r = a) is
ds = 2π a 2 sin θ dθ . Further, after the integration of Eq. (9) by using Eqs. (12)
and (13), the force, acting on the drop surface, is given by the following
expression:
FM(θf)= C(1-2cosθf - 2cos2θf ),
(14)
where
C=
2π aΠ
3(1 + λ )
(15)
and λ = μ’/μ is the ratio of the bulk viscosities [16].
Normal forces acting on the interface will tend to deform the drop from
a spherical shape. So, the principal mechanical factor which can modify the
shape of the drop, namely deformations and break-ups of a drop, is the normal
component of the Marangoni force (see Fig. 3). To understand better the role of
this force, on the deformations of the drop, it will be useful to calculate the
normal Fn component of the resultant force FM.
The normal component Fn of the force FM is given by
Fn = ∫∫ ( prr ) r =a cosθ ds
(16)
S
and using Eqs. (12) and (15) and the surface element in spherical coordinates,
the normal component of the force, acting on the drop surface, becomes after
integration
Fn(θf) = C(−1−cosθf −cos2θf ).
(17)
It can be seen that this force Fn depends on the θf angle, namely, on the
extent to which the drop surface is covered by the surfactant, the interfacial
tension difference П and the ratio of the bulk viscosities, λ.
The normal component of Marangoni force (in short normal force) is
negative and consequently acts in the opposite direction of the normal at the
DROP DEFORMATIONS UNDER SURFACE
71
drop surface (Fig. 3) for any degree of coverage with surfactant. Also, it can be
observed from Eq. (17) that this component is at its minimum for θf = 0 (i.e. at
the point of the surfactant injection) and increases rapidly but it never reaches
positive values for any value of θf. The maximum magnitude of the normal
force is for θf=0 and is given by
2π aΠ
Fn(0)=
.
(18)
1+ λ
It is to be observed that Fn(0) = FM(0).
The normal force acts like a hammer (hammer effect), deforming or
even breaking up the drop, for any value of θf between 0 and π. As shown by
Eq. (18), the hammer effect appears directly proportional with П and with the
radius (a) of the drop and inversely proportional to λ.
The tangential component Ft of the Marangoni force, or tangential
force, is responsible for translational motion of the drop and it is given by
Ft=− ∫∫ ( prθ )r=a sinθ ds ,
(19)
S
After the integration, Ft is given by the following expression
Ft(θf) = C(2−cosθf−cos2θf).
(20)
The tangential force Ft is positive for any value of θf between 0 and π,
namely for any extent of coverage with surfactant on the drop surface. It is
found, from Eq. (20) that the tangential force Ft is practically zero for small
value of θf, having a maximum value at θf=2π/3.
The resultant force, Fm = FM /C and its two components, normal
Fn=Fn/C and tangent Ft = Ft/C, are plotted in Fig. 4.
The resultant force, Fm, vanish for θ0 ≈ 7π/18 (≈ 68. 530). It is to be
noted that for any degree of coverage, θf <θ0, with surfactant, the resultant Fm
exerted by
the external liquid upon the drop is oriented towards the negative direction of
the Oz axis.
From Fig. 4, it results that for small angles of surfactant covering of the
drop surface, the normal force dominates the tangent force. Consequently, the
normal component of the Marangoni force induces a change in the drop shape.
For the surfactant injection moment, θf = 0, it is also observed that the following
relation FM(0) = Fn(0) is obtained.
At greater surfactant coverage degree of the drop surface, at θf > θ0,,
although Fn is not zero, the tangent force Ft is much larger than the absolute
value of the normal force, so that the resultant force FM is positive. In other
72
I.RADUCAN STAN ET AL..
words, the normal force and the tangent force do not cancel out for any value of
θf and consequently the resultant force FM changes its sign.
Fig. 4. The forces acting on a drop surface.
The presence of an interface between two liquid phases may exert an
influence on the motion of the bulk liquids when the surface tension varies from
point to point in that interface. The modification of the surface tension is not a
simply reduction of the surface tension. It involves tangential forces arising on
the liquid surface, whose magnitude is determined by the surface tension
gradient. The occurrence of these tangential forces in the interface always sets
the interface into motion.
This movement, through the ventilation effect (Fig. 3), leads to
the appearance of the Marangoni force, Fm, with its normal Fn and
tangent Ft components. The normal force component causes the drop
deformation and it is the dominant force, for θf <θ0 , and it brings the
hammer effect. The tangent force component incites the displacement of
the drop (such as lifting effect) and it is the principal force for θf > θ0.
Finally, we wish to point out also an important factor of the hammer
force as the origin of other processes. Indeed, the modification of the surface
area of a drop, when the drop is deformed to a non-spherical shape, dilutes the
surfactant surface concentration and the deformation of the drop is different
from that expected one for the equilibrium σ0 case. This is called the surface
dilution effect. Also, surfactant molecules may accumulate at the tip of the
drop due to convection, especially at the break-up process of the drop. This
decreases the local interfacial tension and causes the tip to be overstretched.
This is called the tip-stretched effect. When the deformations a drop get a
concave surface, capillary forces also appear which tend to bring the drop into
DROP DEFORMATIONS UNDER SURFACE
73
the initial spherical shape. All these deformations appear, in our opinion, only
as a consequence of the hammer effect and the convective real flow of the
surfactant molecules.
Therefore, we suggest that the primary effect, due to a reduction of the
equilibrium interfacial tension (σ0) in the injection point of a free drop surface
with a surfactant, at t=0, is the appearance of a real surface flow (Marangoni
flow) of the surfactant on the drop surface. This convection flow of the
surfactant modifies the equilibrium surface tension, σ0, and a Marangoni force
FM(θf) will act on the drop. At the beginning it acts like a hammer which
changes the shape of the drop, or break-up it and consequently, several factors
might appear, namely, the surface dilution and tip-stretching, as well as
capillary forces.
Using the asymptotic expansion of the function 1/1+λ when λ<1, we
may have simply an asymptotic representation for FM(θf)
FMθ(
f
)=
2πaΠ(1 - λ)
(1 - 2cosθ - 2cos 2
θ ) .
f
f
3
6. EXPERIMENTAL SECTION
The experimental work on drop dynamics and Marangoni instability
was performed in liquid-liquid systems of equal densities presented in Tables 1
and 2. The densities of the liquids were determined picnometrically and the bulk
viscosities by using an Ubbelhode viscometer. The surface dilatation viscosity
at the liquid/liquid interface was not directly measured and only a few
indications concerning this magnitude for the liquid/gas interface have been
found previously [3, 17].
The interfacial tension of the liquid/liquid systems was determined by a
method based on capillarity and its value is given in Table 2. The measurement
of the above parameters as well as the drop dynamics and surface flow
experiments have been performed at constant temperature (20 ± 0.1 oC). All
chemicals were of analytical purity and used without further purification.
74
I.RADUCAN STAN ET AL..
Table 1. Composition and physical characteristics of the liquid/liquid systems of equal densities.
Continuous Phase (L)
System
No.
Composition
(% vol )
μ
(cP)
•
Drop Phase (L’)
σ0
Composition
(% vol)
μ’
(cP)
Surfactant Solution (S)
Composition
(% vol)
•
σ1
L/
L’
L’/
S
(dyn/cm)
(dyn/cm)
1
Methanol 78
Water
22
1.33
10.2
2
NaNO3 15.1
Water 84.9
1.10
28.2
Chlor benzene 50
Silicon oil
50
5.46
3
NaNO3 14.9
Water 85.1
1.10
22.8
Chlor benzene 92
Silicon oil
8
1.03
Paraffin oil
80
Propanol 77.3
Water
22.7
Benzylic
alcohol
CCl4
Benzylic
alcohol
CCl4
3.5
89
11
3.6
89
11
3.6
Table 2. Surface tension gradient, П, the ratio of the bulk viscosities, λ , the drop radius, a, and
the values of the force FM(0) for the three systems given in Table 1.
П
a
(dyn/cm)
λ
6.7 ± 0.3
60.15
1.19
0.82
2
24.6 ± 0.3
4. 96
0.46
11.92
3
19.2 ± 0.3
0.94
0.46
28.57
System
No.
1
(cm)
FM(0)
Remarks
(dyn)
The free drop remains
practical nondeformable and
the translational motion is
insignificant. Fig. 5.
The free drop might have
slight or big deformations, but
after 0.6-0.8 sec., it returns to
its initial form, Fig. 6. The
translational motion (Fig. 6c)
is evident from the initial
position (Fig. 6a).
The free drop, after 0.3-0.4
sec., (Figs 7a and 7b), breaks
up into two droplets (Figs 7c
and 7d) which are also
moving.
75
DROP DEFORMATIONS UNDER SURFACE
The mixtures making up the continuous L phase were placed in a
thermostated parallelepipedic vessel of 1 dm3, made of transparent glass. The
drop was made of various radii between 0.46 and 1.19 cm, using the mixtures
described in Table 1. The L’ liquid was carefully submerged by using a pipette
into the continuous L aqueous phase and the density of the latter was then
adjusted by adding small quantities of water or alcohol until the buoyancy of the
drop practically disappeared.
After the system was stabilized, a small quantity (10-3-10-2 cm3) of the
surfactant solution was injected with a micrometric syringe, in a point on the
drop surface (injection point in Fig, 1). The injection was done either in a
vertical direction (as in Fig. 5) or in different directions (Figs 6 and 7) and no
influence of the mode of injection on the Marangoni flow or on drop
movements and deformations were observed.
At the surfactant injection moment, the Fm is negative, acting on the
opposite direction of the normal to the drop surface and having its effect the
changing of the drop shape (Figs. 6a and 6b and Figs. 7a and 7b). For θf > θ0
the resultant Fm becomes positive and its effect is the propulsion of the drop or
the translational movement of the drop along the Oz axis (Figs. 6c, 7c and 7d).
(a)
(b)
(c)
Fig. 5. Filmed pictures of a nondeformable drop characterized by the system 1. FM(0)= 0.82 dyn.
The time of the surfactant front evolution: (a) t = 0 sec, (b) t= 1.02 sec, (c) t = 1.25 sec.
76
I.RADUCAN STAN ET AL..
(a)
(b)
(c)
Fig.6. Filmed pictures of drop deformations characterized by the system 2, FM(0) = 11.92
dyn. Time evolution of the drop: (a) t = 0 (surfactant injection on drop surface); (b) t = 0.14
sec (deformation along Oz axis); (c) t = 0.44 sec (transversal deformation). Significant
translation movement (c) from its initial position (a).
Under the surface tension gradients, the drop surface movement (Fig. 1)
and the ventilation effect (Fig. 3) lead to the appearance of the Marangoni force,
Fm, with its normal Fn and tangential Ft components (Fig. 4). In this
investigation it is shown that the normal force component causes the drop
deformations and it is the dominant force, for θf <θ0 , and it causes the hammer
effect. The tangential force component provokes the displacement of the drop
(such as lifting effect) and it is the major force for θf > θ0.
(a)
(b)
(c)
(d)
Fig. 7. Filmed pictures of a breakup drop, characterized by the system 3. FM(0)= 28.57 dyn.
Time evolution of the drop: (a) t = 0 sec (surfactant injection on drop surface); (b) t = 0.12
sec (drop deformation) ; (c) t = 0.36 sec (break of the drop); (d) t = 0.43 sec
The normal force acts like a hammer (hammer effect), deforming and
even breaking up the drop. As shown by Eq. (18), the hammer effect appears
DROP DEFORMATIONS UNDER SURFACE
77
directly proportional with П and with the radius (a) of the drop and inversely
proportional to λ. Therefore, at big values of λ ratio (Fig. 5, Table 2), the force
is small and a significant deformation and the displacement of the drop, after
the surfactant injection, are not observed. However, the normal force may
produce some gentle surface waves, but these waves were not detectable in
these experimental conditions.
At the surfactant injection moment the resultant Marangoni Fm force is
negative, acting in the opposite direction of the normal to the drop surface. For
intermediary (Figs. 6a and 6b) and small values of λ ratios (Figs. 7a and 7b),
and for θf <θ0, the resultant Marangoni force Fm causes the visible
modifications of the drop shape or even the break-up of the drop (Fig. 7c). For
θf > θ0 the resultant force Fm becomes positive and its effect is the propulsion of
the drops or the translational movement of the drops along the Oz axis (Figs. 7c
and 7d).
7. CONCLUSION
As shown above, the deformations of the drop under surfactants
adsorption can not be attribute to a single mechanism, in our case a mechanical
one, but we must consider also other factors, namely, the surface dilution and
tip-stretching of the surface tension, as well as capillary forces. The results of
our theoretical hydrodynamic model are in substantial agreement with the
observed experimental data.
8. REFERENCES
[1] V. G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, New
Jersey, 1962.
[2] R. S. Valentine, W. J. Heideger, Ind. Eng. Chem. Fund, 2, 242, 1963.
[3] E.Chifu, I. Stan, Z. Finta, E. Gavrila, J .Colloid Interface Sci., 93, 140, 1983.
[4] I. Stan, E. Chifu, Z. Finta, E. Gavrila, Rev. Roum. Chim., 34, 603, 1989.
[5] I. Stan, C.I. Gheorghiu, Z. Kasa, Studia Univ. Babes- Bolyai, Math., 38, 113, 1993.
[6] E. Chifu, I. Stan, M. Tomoaia-Cotisel, Rev. Roum. Chim., 50, 297, 2005.
[7] [R. S. Schechter, R. W. Farley, Canadian J. of Chem. Eng., 41,103, 1963.
[8] M. D. Levan, J. Colloid Interface Sci., 83, 11, 1981.
[9] L. Landau, E. Lifschitz, Mecanique des fluides, Ed. Mir, Moscow, 1971.
[10] G. K. Batchelor, An Introduction to Fluid Mechanics, Cambridge Univ. Press, Cambridge
1967.
[11] T. Oroveanu, “Mecanica fluidelor vâscoase”, Editura Academiei, Bucharest, 1967.
[12] L. E. Scriven, Chem. Eng. Sci., 12, 98, 1960.
[13] R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall,
Englewood Cliffs, New Jersey, 1962.
78
I.RADUCAN STAN ET AL..
[14]A. Sanfeld, in “Physical Chemistry Series”, W. Jost (Ed.), Vol. I, Acad. Press, New York,
1971.
[15] I. R. Stan, M. Tomoaia-Cotisel, A. Stan, Bull. Transilvania Univ. Brasov, 13(48), 357, 2006.
[16] Y. T. Hu, A. Lips, Phys. Rev. Lett., 91, 1, 2003.
[17] M. Tomoaia-Cotisel, E. Gavrila, I. Albu, I.-R. Stan, Studia, Univ. Babes-Bolyai, Chem., 52
(3), 7, 2007.
[18] G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [19] C. G. Callan
and E. Witten, Nucl. Phys. B 239 (1984) 161.
[20] B. M. A. Piette and D. H. Tchrakian, Phys. Rev. D 62 (2000) 025020 [arXiv:hepth/9709189].
[21] E. Radu and D. H. Tchrakian, arXiv:hep-th/0509014.
[22] F. E. Schunck and E. W. Mielke, Class. Quant. Grav. 20 (2003) R301. [23] M. S. Volkov
and E. Wohnert, Phys. Rev. D 66 (2002) 085003
[arXiv:hep-th/0205157].
[24] S. Yoshida, Y. Eriguchi, Phys.Rev.D56 (1997) 762.
[25] B. Kleihaus, J. Kunz and M. List, Phys. Rev. D 72 (2005) 064002 [arXiv:gr-qc/0505143].
[26] F. E. Schunck, E. W. Mielke Phys. Lett. A 249 (1998) 389; F. E. Schunck and E.W. Mielke,
Gen. Rel. Grav. 31 (1999), 787.
[27] M. S. Volkov and E. Wohnert, Phys. Rev. D 67 (2003) 105006 [arXiv:hep-th/0302032].
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
CRITICAL CURVE APPROACH ON SAF SWITCHING
Cristina Stefania OLARIU*, Alexandru STANCU*
“Alexandru Ioan Cuza” University, Department of Physics, Blvd. Carol I, 11,
Iasi, 700506, Romania
[email protected], [email protected]
*
Keywords: SAF, toggle MRAM, critical curves
ABSTRACT One of the biggest challenges in magnetic memories is represented by operating field margin
precision determination. Operating field region for classical Stoner – Wohlfarth model based memories is
limited by neighbor cells interactions and half selected disturbs problems. Toggle magnetic memories
eliminate many of these interactions related problems. A synthetic antiferromagnetic (SAF), formed by two
antiferromagnetically coupled magnetic layers, is used as a free layer. To determine the toggle mode operation
margin is important to take into consideration the strength of antiferromagnetic interactions between
ferromagnetic layers, the geometric characteristics of the memory cell, the relative thickness of the
ferromagnetic layers and the magnetocrystalline anisotropy energy of ferromagnetic materials of the layers.
These parameters control and determine the active domain for toggle MRAM cell.
1. Introduction
The discovery of Giant Magnetoresistance in magnetic multilayer and
the invention of the spin valve device show great promises for diverse
applications, magnetic fields sensors, magnetic read-heads and magnetic
memories. Magnetic tunnel junction devices are made from ferromagnetic
layers separated by a very thin non – magnetic insulating layer [1, 2]. The
magnetization’s relative orientation in the ferromagnetic layers determines the
resistance of the device. The multilayer system presents a high resistance when
the magnetizations of each ferromagnetic layer are anti-parallel orientated.
When the magnetizations of magnetic layers have the same orientation, the
system has a low resistance.
A conventional MRAM cell is made from two ferromagnetic layers
separated by a non magnetic thin layer [3]. One of the magnetic layers is
pinned while the other magnetic layer is free to switch from parallel to anti –
parallel orientation. The data storage is given by the change in electrical
resistance of the device in accordance with the directions of the free layer
magnetization. When the magnetization of the free layer is parallel with the
magnetization of pinned layer, the memory cell resistance is low. This state is
conventionally defined as the “0” memory state. When the magnetization of the
80
C. OLARIU ET AL
free layer is anti-parallel orientated according to the magnetization of the pinned
layer, the memory cell resistance is high. This state is conventionally defined as
“1” memory state (Figure 1.a). The relative orientation of the ferromagnetic
layer’s magnetization leads to an important resistance variation (Figure 1.b) [4].
The relative orientation of magnetization of the two magnetic layers is
switching by a magnetic field large enough to reverse magnetization of the free
layer but not large enough to switch the pinned layer. Control of the
magnetization switching is relevant for the data storage domain and mainly for
MRAM devices.
Free layer
Non – magnetic
layer
Fixed layer
„0” Bit
„1” Bit
(a)
R
„0”
„1”
H-
H+
(b)
Figure 1. (a) Classical memory cell made from two ferromagnetic layers coupled by an
insulator layer and the “0” and “1” memory state, b) Resistance hysteresis of memory cell.
Conventional magnetic memories have considerable errors because of
the problems in controlling write operation margin and activation energy. It is
important that none of the write operation causes a switching in any of the half
– selected bits. Write margin requires that the fully selected fields always lay
outside the Stoner – Wohlfarth astroid, while the half – selected fields must lie
CRITICAL CURVE APPROACH ON SAF SWITCHING….
81
within the astroid. Dispersion effects, temperature effects and imperfect form
of the memory element cell degrade the write margin. Therefore, errors may
appear in half – selected memory cells [5]. Activation energy preserves the
magnetization from reversing its orientation by thermal excitation and is
associated with magnetization reversal. A drop in activation energy can lead to
a spontaneous reversal of the magnetization because of local heating or ½
selectivity of the bit cell [6].
This two problems, write selected margins and activation errors, limits
the application’s area of Stoner – Wohlfarth classical magnetic memories cell.
In 2003, Leonid Savtchenko patents a new method that resolves these
problems [7]. This method relies on the unique behaviors of a Synthetic
Antiferromagnet (SAF) that is formed from two ferromagnetic layers separated
by a non magnetic coupling spacer layer [8]. The upper free magnetic layer and
the lower fix layer of a conventional MRAM have been replaced with two
ferromagnetic layers separated by a non – magnetic spacer layer (Figure 2.a).
The layers closest to the tunnel barrier influence the resistance across the tunnel
barrier, and they form the bit memory cell.
The SAF system has balanced magnetic moments, antiferromagnetic
coupled, that leads to a flux closure and increases the magnetic stability of the
pinned layer and reduces coupling to the free layer [9]. Thus, the density of a bit
cells is increased and the neighboring memory cells interferences are
eliminated.
The SAF system responds differently then the single ferromagnetic
layer to an applied magnetic field. The magnetizations of ferromagnetic layers
align themselves in such a way that the resulting moment of the coupled layers
lies in the direction of the external field. To rotate the free magnetic layer
moments by 180 degree, from one state to another, a current pulse sequence is
used. Because of the symmetry, the sequence toggles the bit to the opposite
state regardless of existing state. A single pulse alone cannot switch the bit,
proving an accurate selectivity over the previous approaches to MRAM
switching [10].
82
C. OLARIU ET AL
Free SAF
BIT
Tunnel barier
Fixed SAF
Pinned Layer
(a)
HB + HW
HW
WORD Line
Oy
Ox
HB
BIT Line
Easy Axis
(b)
Figure 2. (a) Toggle mode memory cell. (b) Orientation of the bit cell, at 45 degrees
with respect to the current lines.
The magnetic moments of the free and the fixed layers are oriented at
an angle of 45 degrees with respect to the current lines, along the “easy” axis.
The two layers have their orientations aligned opposite to each other at a stable
state. A current pulse sequence is used to generate a rotating magnetic field that
moves the magnetic moments through 180 degrees and switch the bit from one
83
CRITICAL CURVE APPROACH ON SAF SWITCHING….
state to the other (Figure 2.b). Write line BIT induces the magnetic field H B
and write line WORD induces magnetic field H W . The pulse current is
sequentially applied and the bit cell orientation rotates from the easy axis, to the
hard axis, and then back along the easy axis.
For a synthetic antiferromagnet having some net anisotropy H K in
each layer, there is a critical spin flop field H SF at witch the two antiparallel
magnetizations will rotate (flop) as such as to be orthogonal to the applied field
H [11]. This spin – flop field is very important for switching the bit memory
cell control. For fields lower than H SF , antiferromagnetic coupling is strong
enough to preserve the initial state of the bit. In H SF point, the moments
discontinuously jump, the net magnetic moment of the system is pointing in the
direction of the applied field. If the field is further increased, magnetic moments
smoothly rotate together until they become parallel, in a saturation state.
To rotate the magnetic moments of the memory cell, it is important to
exactly determine the value of this spin flop point and to avoid it. In the plane of
the applied magnetic field hx , h y 2 regions with spin – flop points
(
)
symmetrically orientated along the easy axis appear [12]. To obtain the
magnetization switching, the applied field sequences must go around the spin –
flop region. But the applied magnetic field must be small enough to not
overcome the saturation limits. The work region for safely switching of SAF
system is located between the saturation curve and spin – flop region [13].
2. Critical curves approach
(
)
To represent the work region in applied magnetic field plane hx , h y it
is important to accurate determine the saturation curve and interior spin – flop
region. The SAF system is assumed to have two ferromagnetic layers (layer 1
and layer 2) with the same planar shape but with the thickness t1 and t 2 ,
magnetizations M 1 and M 2 , and uniaxial anisotropy constant K 1 and K 2 . The
layers have the easy axes parallel to each other, both in Ox direction.
To exactly determine the shape of exterior saturation curve and the
shape of interior spin – flop region is important to precisely determine the
equilibrium and stability equations for the SAF system [14]. Thus, the critical
point in hx , h y plane can be obtained. The spin – flop points form an astroid
(
)
84
C. OLARIU ET AL
shape, limited by the interior critical curve. The saturation points are situated on
the exterior critical curve. The bit switching is obtained only if the applied pulse
sequence lies between the interior critical curve and exterior critical curve.
Energy of the SAF system is the sum of the anisotropy energy for each
magnetic layers, interaction energy and antiferromagnetic coupling energy [15]:
(1)
W = K 1t1 sin 2 θ1 + K 2 t 2 sin 2 θ 2 − M 1t1 (H x cos θ1 + H y sin θ1 ) −
− M 2 t 2 (H x cos θ 2 + H y sin θ 2 ) + J cos(θ1 − θ 2 )
where θ1 and θ 2 are the magnetization angle of the layer 1 and the layer 2 with
respect to the easy axes, H x and H y the applied field in the easy and hard axis
directions, and J is the antiferromagnetic coupling strength between the two
layers.
Normalizing W by 2 K 1t1 , we obtain:
(2)
w=
where t =
hJ =
1 2
1
sin θ1 + t sin 2 θ 2 − (hx cos θ1 + h y sin θ1 ) −
2
2
− t (hx cos θ 2 + h y sin θ 2 ) + hJ cos(θ 2 − θ1 )
Hy
H
t2
is the relative thickness of the layers, hx = x , h y =
,
t1
HK
HK
J
and H K is the anisotropy field.
2 K 1 t1
Equilibrium and stability conditions are given by:
∂w
∂w

w1 = ∂θ = 0, w2 = ∂θ = 0
1
2

2

∂ w
∂2w
(3)
=
0
,
w
w
>
=
> 0,
 11
22
2
2
θ
θ
∂
∂
1
2

∆ = w w − w w > 0
12 21
11 22


where w12 and w21 are the second mixture derivates of w with respect to θ1
and θ 2 respectively.
It obtains the critical fields components:
CRITICAL CURVE APPROACH ON SAF SWITCHING….
(4)
85
t cos θ 1 cos θ 2 (sin θ1 − sin θ 2 ) + hJ sin (θ 2 − θ1 )(cos θ1 + t cos θ 2 )

h x =
t sin (θ 2 − θ1 )


h = t sin θ1 sin θ 2 (cos θ1 − cos θ 2 ) + hJ sin (θ 2 − θ1 )(sin θ1 + t sin θ 2 )
 y
t sin (θ 2 − θ1 )
To accurately determine the interior critic curve and the exterior critical
curve means accurate determination of the work region.
For a good understanding of how the two magnetizations M 1 and M 2
(
)
behave in the plane of applied field hx , h y , new medium angular
coordinates (ξ ,η ) , defined as ξ =
θ1 + θ 2
2
andη =
θ 2 − θ1
2
, are used. This
new coordinates give a detailed image about relative orientations of the
magnetizations and about their relative position to the easy axis.
In new coordinates (ξ ,η ) , critical magnetic field components have the
expressions:

 1 

cos ξ  2
 1
2 
h x =
sin ξ + 1 + hJ − 1 cos η  − 1 − hJ sin ξ sin η
cosη 
 t 


  t
(5) 
 1 

sin ξ 
 1

2
2 
h y = cosη − cos ξ + 1 + t hJ + 1 cos η  + 1 − t hJ cos ξ sin η





In (hx , h y ) plane, critical and stability equations are situated on the
interior switching curve and the exterior saturation curve. In the case of a large
applied field, strong enough to parallel align the layers magnetization vectors
one can obtain the saturation curve. In this case, θ1 = θ 2 so thatη = 0 , the
exterior critical curve is describe by the following equations:
(6)

 1
3
hx = − cos ξ + 1 + t hJ cos ξ




h = sin 3 ξ + 1 + 1 h sin ξ
J
 y
 t
From these equations, one may observe that there is dependence
between the shape of the exterior saturation curve and the relative thickness of
ferromagnetic layers of the SAF system. A big difference between the thickness
of the SAF layers leads to an increased exterior critical curve area [16]. To
86
C. OLARIU ET AL
obtain an increasing of the exterior critical curve, one may increase the value of
antiferromagnetic coupling field (Figure 3).
Figure 3. Exterior critical curve delimit the different area for different values of the
relative thickness of ferromagnetic layers and for different values of antiferromagnetic coupling
fields.
Interior spin - flop regions are situated at the intersection points of the
ξ = const. curves. Critical curves that delimit this region are given by the
intersection points from two nearest ξ = const. . For symmetrically
ferromagnetic layers, with the same thickness, t = 1 the ξ = const. curves
intersect in one point in interior spin – flop region. For different thickness of the
layers, t1 ≠ t 2 , t ≠ 1 , ξ = const. curves intersect in two ore more points
(Figure 4). Interior critical curves have different form for different relative
thickness of the ferromagnetic layers of the SAF system.
Figure 4. For
points for t
≠ 1.
t = 1 , ξ = const.
curves intersect in one point and in two or more
CRITICAL CURVE APPROACH ON SAF SWITCHING….
87
For a symmetrically SAF system, with the same thickness of the
ferromagnetic layers t1 = t 2 , t = 1 , the critical field expressions are:
cos ξ

2
2
hx = cosη sin ξ + (2hJ − 1) cos η


h = sin ξ − cos 2 ξ + (2h + 1) cos 2 η
J
 y cosη
[
(7)
]
[
]
These expressions allow one to eliminate the cosη term and to find the
dependence between critical magnetic field components hx , h y and the ξ angle
coordinates. From the second equation in (7) the expression of cosη can be
obtaining, with two real solutions:
(8)
(cosη )1, 2 =
h y ± h y + 4(2hJ + 1)sin 2 ξ cos 2 ξ
2
2(2hJ + 1)sin ξ
One can introduce this expression in the first equation (7) and find the
dependence between the applied magnetic field components and the ξ angle
(
)
coordinate, hx = hx h y , ξ . This dependence helps to accurately determine the
exterior contour of spin – flop region, i.e. the interior critical curve. Graphical
representation of hx and h y components for different values of the ξ angle,
show that spin –flop region is determined by intersection points of this curves.
Interior critical curves are formed by the intersection points from two
ξ = const. closest curves. These points give the magnetic field coordinates for
(
)
which hx = hx h y , ξ curve derivative in respect to ξ is zero. Thus, the exact
shape of the interior critical curves can be obtained.
For a nonsymmetrical SAF system, t1 ≠ t 2 , t ≠ 1 , the critical magnetic
field components are expressed by (5). In this case, it is not just a
cosη dependence, and the same analytical algorithm as in the t = 1 case can
not be used. Determination of the interior critical curve is done numerically.
The obtained shape of the interior critical curves, for different values of the
relative thickness of the SAF layers t is represented in Figure 5.
88
C. OLARIU ET AL
Figure 5. Shapes of the interior critical curves have different shapes for different
relative thickness of the two ferromagnetic layers of the SAF system, t = 1 , t = 0.8 ,
t = 0.4 , antiferromagnetic coupling field is hJ = 2 .
The relative thickness of the ferromagnetic layers of the SAF system
affects the shape of the interior critical curve and limits the spin – flop region.
For a symmetric system t = 1 , two astroids inside of the exterior critical curves
are obtained. For a nonsymmetrical system, t ≠ 1 the shapes of the interior
critical curves are influenced by the relative thickness of the magnetic layers.
The exterior saturation curves are larger but the surface delimited by the interior
critical curves increases as well. For larger differences between ferromagnetic
layers thickness, the interior surface of spin – flop regions can become unified,
that leads to an instability state for the system.
3. The second term of the anisotropy energy of the system influence on
the interior and exterior critical curves shapes.
In the equation of the SAF system energy (1), for the anisotropy energy
only the first term of the expanded series is taken into consideration. If the
second term of anisotropy expanded energy series is taken into consideration,
the shapes of critical curves is modified [17]. In this case, from equation (2), the
reduce system energy is given by:
(9)
w=
1 2
1
1
1
sin θ1 + k 2 sin 4 θ1 + t sin 2 θ 2 + tk 2 sin 4 θ 2 −
2
4
2
4
− (hx cos θ1 + h y sin θ1 ) − t (hx cos θ 2 + h y sin θ 2 ) + hJ cos(θ 2 − θ1 )
89
CRITICAL CURVE APPROACH ON SAF SWITCHING….
where k 2 is the coefficient for the second term of the anisotropy energy series.
From equilibrium and stability conditions (3), expressions for the critical field
components for this case, where the second term of the series expansion of the
anisotropy energy is taken into considerations, are similar with (5):


cos ξ  2
1
2 
h x =
sin ξ (1 + K 1 ) + cos η 1 + hJ − 1 − K 1   −
cosη 
 t 



 1

− hJ 1 −  sin ξ sin η

 t
(10)


1
sin ξ 

2
2 
(
)
cos
η
1
1
cos
1
+
h
K
h
K
=
−
+
+
+
+



2
2
y
J

 +

cosη 
t





 1
+ hJ 1 −  cos ξ sin η

 t

and
Where
K 1 = k 2 cos 2 ξ + 3 cos 2 η − 4 cos 2 ξ cos 2 η
2
2
2
2
are the corrections due the
K 2 = k 2 cos ξ + cos η − 4 cos ξ cos η
(
(
)
)
second term of the anisotropy energy expansion.
The second term of the anisotropy energy changes the shape and the
area of the interior and exterior critical curves, if the second term is significant.
For meaningful values of the second terms of anisotropy energy, the spin – flop
region have different shape (Figure 6).
90
C. OLARIU ET AL
Figure 6. For significant values of the second term of series expansion of anisotropy
energy, the area delimited by exterior saturation curve increases and the shape of the interior
critical curves that delimits the spin – flop region is changed dramatically.
It is observed a shape variation of the interior and exterior critical
curves for the case when the second term of anisotropy energy expansion is
taken into consideration toward the case when only the first term of energy
anisotropy is considered. As long as the second term of anisotropy energy k 2 is
significant, the spin – flop region area is increased. The shape and the area of
the exterior critical curves are modified too. Thus, a different work area for the
toggle mode switching of the SAF system is obtained.
4. Conclusions:
Exterior saturation curves and interior switching curves are important to
exactly determine the work region of SAF systems. The applied sequenced
magnetic fields must avoid the spin – flop region and must be situated inside the
critical saturation curve.
The region between the interior and exterior critical curves is
determined by the geometrical parameters of the magnetic memory cell i.e. the
relative thickness of the ferromagnetic layers of the SAF system t = t1 t 2 , by
the antiferromagnetic coupling field between layers hJ and by the material of
the ferromagnetic layers through the anisotropy energy of the layers and the
influence of the second term of the anisotropy series expand k 2 .
CRITICAL CURVE APPROACH ON SAF SWITCHING….
91
To exactly determine the work region for toggle mode memories cells it
is important that all these parameters are taken into considerations
5. References:
[1] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, P. Etienne, G.
Creuzet, A. Friederich, J. Chazelas, “Giant magnetoresistance in
Fe(001)/Cr(001) superlattices”, Physical Review Letters Volume 61, number
21 2472, (1988)
[2] G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, "Enhanced
magnetoresistance in layered magnetic structures with antiferromagnetic
interlayer exchange", Physical Review B, vol. 39, 4282, (1989)
[3] S. Tehrani, J. M. Slaughter, M. Deherrera, B.N. Engel, N.D. Rizzo, J.
Salter, M. Durlam, R. W. Dave, J. Janesky, B. Butcher, K. Smith, G.
Grynkewich, „Magnetoresistive Random Access Memory Using Magnetic
Tunnel Junctions”, Proceedings of the IEEE, Volume 91, Issue 5, pp. 703 – 714
(2003)
[4] P. K. Naji, M. Durlam, S. Tehrani, J. Calder, and M. F. DeHerrera, „A
256 Kb 3.0 V 1T1MTJ Nonvolatile Magnetoresistive RAM”, Digest of
Technical Papers, IEEE International Solid-State Circuits Conference (IEEE
Cat. No. 01CH37177), pp. 122–438, (2001)
[5] R. Scheuerlein, W. Gallagher, S. Parkin, A. Lee, S. Ray, R. Robertazzi,
and W. Reohr, “A 10 ns Read and Write Non-Volatile Memory Array Using a
Magnetic Tunnel Junction and FET Switch in Each Cell”, Digest of Technical
Papers, IEEE International Solid-State Circuits Conference, pp. 128–129,
(2000)
[6] M. Durlam, P. Naji, A. Omair, M. DeHerrera, J. Calder, J. M. Slaughter,
B. Engel, N. Rizzo, G. Grynkewich, B. Butcher, C. Tracy, K. Smith, K. Kyler,
J. Ren, J. Molla, B. Feil, R. Williams, and S. Tehrani, “A Low Power 1 Mbit
MRAM Based on 1T1MTJ Bit Cell Integrated with Copper Interconnects”,
Digest of Technical Papers, Symposium on VLSI Circuits, pp. 158–161, (2002)
[7] L. Savtchenko, A. A. Korki, B. N. Engel, N. D. Rizzo, and J. A.
Janesky, “Method of writing to scalable magnetoresistance random access
memory element,” US. Patent No. 6 545 906 B1, (2003)
[8] M. Durlam, D. Addie, J. Akerman, B. Butcher, P. Brown, J. Chan, M.
DeHerrera, B. N. Engel, B. Feil, G. Grynkewich, J. Janesky, M. Johnson, K.
Kyler, J. Molla, J. Martin, K. Nagel, J. Ren, N. D. Rizzo, T. Rodriguez, L.
Savtchenko, J. Salter, J. M. Slaughter, K. Smith, J. J. Sun, M. Lien, K.
Papworth, P. Shah, W. Qin, R. Williams, L. Wise, and S. Tehrani, "A 0.18 µm
92
C. OLARIU ET AL
4Mb Toggling MRAM", IEEE International Electron Devices Meeting,
Technical Digest, pp. 34.6.1 - 34.6.3., (2003)
[9] B. N. Engel, J. Akerman, B. Butcher, R. W. Dave, M. DeHerrera, M.
Durlam, G. Grynkewich, J. Janesky, S. V. Pietambaram, N. D. Rizzo, J. M.
Slaughter, K. Smith, J. J. Sun, and S. Tehrani, “A 4-Mbit Toggle MRAM Based
on a Novel Bit and Switching Method”, IEEE Transaction on Magnetics, vol.
41, pp. 132–136 (2005).
[10] J. DeBrosse, C. Arndt, C. Barwin, A. Bette, D. Gogl, E. Gow, H.
Hoenigschmid, S. Lammers, M. Lamorey, Y. Lu, T. Maffitt, K. Maloney, W.
Obermeyer, A. Sturm, H. Viehmann, D. Willmott, M. Wood, W. J. Gallagher,
G. Mueller, and A. R. Sitaram, “A 16Mb MRAM Featuring Bootstrapped Write
Drivers”, Digest of Technical Papers, Symposium on VLSI Circuits, pp. 454–
457, (2005)
[11] D. C. Worledge, “Spin flop switching for magnetic random access
memory”, Applied Physics Letters, vol. 84, pp. 4559–4561, (2004)
[12] D. C. Worledge, „Single domain model for toggle MRAM”, IBM
Journal of Research and Development, Vol. 50, no.1, (2006)
[13] H. Fujiwara, S. -Y. Wang, M. Sun, “Critical field curves for switching
toggle mode MRAM devices”, Journal of Applied Physics, vol. 97, pp. 10P5071–10P507-5, (2005)
[14] H. Fujiwara, S.-Y.Wang, and M. Sun, “Magnetization behavior of
synthetic antiferromagnet and toggle- magnetoresistance random access
memory,” Transactions of the Magnetics Society of Japan, vol. 4, pp. 121–129,
(2004)
[15] S.-Y. Wang, H. Fujiwara, “Optimization of magnetic parameters for
toggle magnetoresistance random access memory”, Journal of Magnetism and
Magnetic Materials, vol. 286, 27, (2005);
[16] C. S. Olariu, L. Stoleriu, A. Stancu, “Simulation of toggle mode
switching in MRAM’s”, Journal of Optoelectronics and Advanced Materials,
vol. 9, 4, (2007)
[17]. C. S. Olariu, L. Stoleriu, A. Stancu, „Influence of material anisotropy
in the switching toggle mode in MRAM devices”, Journal of Magnetism and
Magnetic Materials, vol. 316, 2, (2007)
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
VARIOUS ELECTROSTATIC FIELD CONFIGURATIONS ON A
GLOBALLY PATHOLOGIC METRIC
Ana-Camelia Pîrghie, Ciprian Dariescu and Marina-Aura Dariescu
University “Al. I.Cuza”, Faculty of Physics, 11Blvd. Carol I, 6600, Iaşi,
România
Using the Cartan formalism, we are working out the geometric features of a special metric, which
generalizes the one for the pp gravitational waves collision. By fixing the three metric functions, we come to a
class of exact solutions with a =
G6 VII 0 ×VIII group of motion. In the spacetime endowed
with g 44 = − ch 2 ( α z ) , the null and timelike geodesics are pointing out some unusual (pathological) features.
Our analysis has led to similar results as for the so-called BTZ black holes, where the reduced onedimensional motion of the test particle evolves in a parabolic well periodically crossing the two
r = ±1 horizons. As the problem of time remains very controversal, it turns out that it is possible to get
cosmic-time traps and temporally imprisoned geodesics even when the metric contains no singular points.
Finally, we deal with the Gauss-Poisson equation and derive its mode-solutions. Such kind of studies for
physically meaningful source configurations may lead to a better understanding of the electrically charged
BTZ black holes.
1. Introduction
In the last decades, a wide interest has been focused on globally
pathologic manifolds and radical changes have occurred in understanding
gravity, matter fields and spacetime [1]. In this respect, not only intensive
studies on the cosmic strings, naked singularities, Bianchi spacetimes,
dynamical isotropization or topological domain walls have been the main
topics [2], but also the so-called BTZ black holes, [3], revealing new
intriguing features due to their causal structure singularities, such as the
closed timelike curves (CTC) and/or additional Taub-Nut pathologies at
the metric singular “point''.
A detailed analysis of a general class of Universes with VII0 x VIII
isometries have also pointed out the unexpected behavior of magnetic
static modes, such as a sort of gravitoelectromagnetic resonance, in some
spatially finite regions [4].
The existence of CTC is a very active issue ever since Gödel found,
in 1949, a solution to the Einstein field equations with nonzero
cosmological constant [5]. Soon after, such a solution was considered
94
A.C.PIRGHIE et. al.
without a physical significance, since it corresponds to a rotating,
stationary cosmology, whereas the actual universe is expanding and
apparently non-rotating. As an usual paradox, the existence of CTCs are
not allowed by physical laws because they contradict the usual notion of
causality. However, recently, there have been found solutions of
Einstein’s equations which contains CTCs that can represent the exterior
of physically admissible sources, as for example a single charged,
rotating, magnetic object, called Perjeon [6] and the possibility that a
spacetime associated to a realistic model of matter may contain CTCs has
been analyzed in [7].
The metric which is the aim of our present investigations, has
recently come into play, in the context of higher dimensional spacetime
models, and it was stated that it can be responsible for the mechanism of
particle confinement in our Minkowskian 4-dimensional world [8].
2. The geometric features of a more general metric
In order to investigate the pathological global spacetime structure,
we consider a metric belonging to the Bianchi type VIII, which is the
exact solution of the Einstein equations with unconventional matter
sources.
Starting with the metric in the compact form,
2
2
ds 2 = e 2 f ( u ,v ) ( dx ) + e 2 g( u ,v ) ( dy ) − e 2h( u ,v ) dudv (30)
we introduce the pseudo-orthonormal tetradic frame




e1 =e − f ∂ x ,e2 =e − g ∂ y ,e3 =e − h ∂ u ,e4 =e − h ∂ v (31)
having the 1-forms,
f
g
h
=
ω 1 e=
dx,ω 2 e=
dy,ω 3 e=
du,ω 4 e h dv (32)
whose exterior derivatives are
− f 3ω 1 ∧ ω 3 − f 4ω 1 ∧ ω 4
dω 1 =
− g 3ω 2 ∧ ω 3 − g 4ω 2 ∧ ω 4
dω 2 =
− h 4ω 3 ∧ ω 4
dω 3 =
ω 4 h 3ω 3 ∧ ω 4
d=
This enables us to use the first Cartan equation,
(33)
95
VARIOUS ELECTROSTATIC FIELD...
=
d ω a Γ .[abc]ω b ∧ ω c , 1 ≤ b 〈 c ≤ n ,(34)
in order to find the connection coefficients
=
Γ 131 f=
,Γ 141 f 4
3
=
Γ 232 g=
,Γ 242 g 4 (35)
3
=
Γ 343 h=
,Γ 434 h 4
3
and the connection 1-forms:
=
Γ 12 0,=
Γ 13 f 3ω 1=
,Γ 14 f 4ω 1
=
Γ 23 g=
ω 2 ,Γ 24 g 4ω 2
3
(36)
=
Γ 34 h 3ω 3 − h 4ω 4
From the 2-nd Cartan equation,
 .ba = d Γ .ba + Γ .ca ∧ Γ .bc ,(37)
the curvature 2-forms can be written as,
 12 =f 3 g 4 + f 4 g 3 ω 1 ∧ ω 2
(
)
( )
 f − f
 13 =−
3
 33

2
(
)
+ f 3 h 3  ω 1 ∧ ω 3 − f 34 + f 3 f 4 + f 3 h 4 ω 1 ∧ ω 4

(
)
( )
 14 = − f 43 − f 3 f 4 − f 4 h 3 ω 1 ∧ ω 3 +  − f 44 − f 4

2
(
( )
+ f 4 h4  ω 1 ∧ ω 4

(38)
)
2
 23=  g 3 h 3 − g 33 − g 3  ω 2 ∧ ω 3 − g 3 h 4 + g 34 + g 3 g 4 ω 2 ∧ ω 4

(
=
−(h
)
( )
− g 43 + g 3 g 4 + g 4 h 3 ω 2 ∧ ω 3 −  g 44 + g 4
 24 =

 34
43
)

2
− g 4 h4  ω 2 ∧ ω 4

+ h 34 + 2h 4 h 3 ω 3 ∧ ω 4
so that the scalar curvature reads:
(
) (
)
R = 2 f 3 g 4 + 2 f 4 g 3 + 4 f 34 + f 3 f 4 + f 3 h 4 + 4 g 34 + g 3 h 4 + g 3 g 4 .(39)
Putting everything together, one gets for the Einstein tensor
G=
Rab −
ab
1
g ab R ,
2
the following components:
96
A.C.PIRGHIE et. al.
(
)
=
−2 ( f h + f + f f )
=
− f −( f ) + f h + g h − g −(g )
=
− ( f ) + f − f h  − ( g ) + g − g h 

 

G11 =
−2 g 3 h 4 + g 3 + g43 g 4
G22
3
4
3
43
4
2
G33
33
2
3
3
3
3
3
33
2
G44
4
(40)
3
2
44
4
4
4
44
4
4
G34 = f 3 + f43 f 4 + f 3 h 4 + g 3 h 4 + g 3 + g4 3 g 4 + f 3 g 4 + f 4 g 3
The conformal map on the submanifold M 2 ( z,t ) ,
=
e h( z ) dz d=
ξ , e h( z ) ch (αξ ) (41)
helps us to find the closed expressions of the two variables z,ξ
z=
1
α
arctg  sh (αξ ) 
1
(42)
arcsh tg (α z ) 
α
which transform the metric (1) for the case f= g= 0 , into the form
ξ=
ds 2 = ( dx ) + ( dy ) + ( dz ) − ch 2 (α z )( dt ) ,(43)
2
2
2
2
where the present variable z does actually stand for the ξ .
The dual pseudo-orthoormal basis B* = {ω a }a =1,4 being correspondingly
defined by
=
ω 1 dx,
=
ω 2 dy,
=
ω 3 dz,
=
ω 4 ch (α z ) dt ,(44)
the non-trivial first Cartan equation reads
=
d ω 4 α th (α z ) ω 3 ∧ ω 4 (45)
leading to the essential connection coefficient
Γ 434 = −α th (α z ) ,(46)
and to the connection 1-form
Γ 34 = α th (α z ) ω 4 .(47)
In these circumstances, one gets for the curvature 2-forms (9) the simple
expression
=
 34 α 2ω 3 ∧ ω 4 ,(48)
all the other elements being equal to zero. Finally, the scalar curvature
reads
97
VARIOUS ELECTROSTATIC FIELD...
R = −2α 2 (49)
while the Einstein tensor has the components
G=
G=
α 2 .(50)
11
22
Now, one has to find a suitable matter-source that sustains this
kind of geometry, with G=
G=
0 and a single component for the
33
44
Riemann tensor. As the total energy density, T44 , should be zero and any
conventional source possesses a positive energy density T44( cs ) , one must
necessarily use a false vacuum state described by Tab( v ) = ληab , with λ > 0 ,
in order to get T44( cs ) − λ =
0 . Considering, for simplicity, the conventional
matter as being charactezized by a pressureless ideal fluid at rest, i.e.
Tab( d ) = ρ ua ub and ua = η a 4 , it obviously results ρ = λ . To also get G33 = 0
without violating the rest of the Gab values, we need an extra source that
can be thought of as a global cosmic string Tab( s ) = − µ X a X b , of unitary
elongation effort µ = λ along the X a = ηa 3 direction. Hence, the total
energy-momentum tensor
Tab = µ X a X b + ρ ua ub + g ab Λ (51)
describes a combined matter-source made of stuck universal dust on a zdirected global string immersed in a medium of negative energy density
and equal positive pressure that floods everything all around. Finally, the
conservation law requires a constant Λ and expresses the (false)
vacuum-type contribution as a true Λ term.
3. Timelike geodesics and Penrose diamond
In order to identify the pathological properties of the metric (14),
we compute the timelike geodesics for a test particle moving in this
universe. For x, y = const , the metric (14) becomes
dσ 2 =
− ds 2 =
ch 2 (α z )( dt ) − ( dz ) (52)
2
2
and the corresponding Lagrange function is
2
2
=
Φ ch 2 (α z ) (t ) − (z ) ≡ 1 .(53)
The Euler-Lagrange equations,
d  ∂Φ  ∂Φ
=
0 (54)

−
dσ  ∂xα  ∂xα
98
A.C.PIRGHIE et. al.
lead to the system
tch 2 (α z ) = k
 k2

z =
± 2
− 1
 ch (α z ) 
1/ 2
(55)
dz
 dz 
. For σ = 0 when t = 0 , z = z0 and 
 = 0 , we get for
dσ
 dσ σ =0
the constant k the expression k = ch (α z0 ) so that the Euler-Lagrange
where z =
equations are satisfied by the solutions,
z (σ =
)
1
α
ln  sh (α z0 ) cos (ασ ) + sh 2 (α z0 ) cos (ασ ) + 1 


 tg (ασ ) 
t (σ ) = arctg 

α
 ch (α z0 ) 
1
(56)
π
π
.
〈σ 〈
2α
2α
By inspecting the expressions (27), one can notice that the geodesics
π
and extended to the future beyond
cannot be emitted earlier than −
2α
π
, pointing out a genuine temporal imprisonment. So, the test particle
2α
is running its whole geodesic in a proper finite time, being trapped in
between these universal moments.
In order to define the Penrose coordinates, we first need the null
geodesics which can be obtained for x, y = const in ds 2 = 0 , namely
where −
1
t± =
t0 ± arctg  sh (α z )  .(57)
α
Now, the Penrose coordinates being
 1

=
u arctg t − arctg ( sh (α z ) ) 
α


.(58)
 1

=
v arctg t + arctg ( sh (α z ) ) 
 α

we can draw the corresponding “diamond”, represented in Fig. 1. We
notice that one gets, instead of spacelike and null infinities, the timelike
99
VARIOUS ELECTROSTATIC FIELD...
horizons T ± ( ±∞ ) , since the null trajectories travelling from z → −∞ to
z → ∞ do already have the above mentioned temporal imprisonment.
Fig. 1. The Penrose diagram
4. The Gauss-Poisson equation and its mode-solutions
In order to write down the Gauss-Poisson equation, we come back to
the metric (14) for which the Maxwell equations with sources have the
general expression
∂
k
− g ∂x
1
(
)
− g F ik =
j i .(59)
The electromagnetic tensor, F ik , can be written in a compact form, in
terms of the four potential as
F= Fik dx i ∧ dx k= A4 ,α dxα ∧ dt ,(60)
where A4  −Φ =
g 44V , V being the electrostatic potential, pointing out
the component
100
A.C.PIRGHIE et. al.
∂Φ
Fα 4 =
−Φ ,α =
− α .(61)
∂x
Thus, the relations (30) are given by
1
∂
k
∂
−g x
(
− g g αβ g km Fβ m =
jα
∂
k
∂
−g x
(
− g g g F4 m =
j
1
)
44
km
)
(62)
4
For m= k= 4 and − g =
ch 2 (α z ) the first equation in (33) becomes
∂ 2Φ
= jα (63)
ch 2 (α z ) ∂xα ∂t
1
while, for k= m= 1,3 , the second one is
1
ch (α z )
2
∆TΦ +
∂  1 ∂Φ

ch (α z ) ∂z  ch (α z ) ∂z
1

− j 4 .(64)
 =

Using Φ = ch 2 (α z )V , the Gauss-Poisson equation, namely
 ∂ 2V ∂ 2V
∂ 1
∂
ch 2 (α z )V )  + 2 + 2 =
− ρ e (65)
(

ch (α z ) ∂z  ch (α z ) ∂z
∂y
 ∂x
1
can be cast into the form,
ρ
∂2
∂ 2V ∂ 2V
2


ξ
+
+
+ 2 =
− e2 ,(66)
1
V
(
)
2 
2

α
∂ξ
∂ζ
∂η
where we have introduced the notations
=
ζ α=
x,η α=
y,ξ sh (α z ) . For the
above relation, with concretely means
(ξ
where =
∆⊥
2
+ 1)
ρ
∂ 2V
∂V
+ 4ξ
+ 2V + ∆ ⊥V = − e2 , (67)
2
∂ξ
∂ξ
α
∂2
∂2
+
≡ ∆  2 , the corresponding eigen-value problem
∂ζ 2 ∂η 2
reads
∂ 2ψ λ
(ξ + 1) ∂ξ 2 + 4ξ ∂∂ψξλ + 2ψ λ + ∆ ⊥ψ λ =λψ λ (68)
2
and one can apply the Laplace variable-separation method,
ψ λ (ζ ,η ,ξ ) = Fλχ (ξ )U χ (ζ ,η ) , (69)
where
101
VARIOUS ELECTROSTATIC FIELD...
{
∆ ⊥U χ (ζ ,η ) =
− χ 2U χ (ζ ,η ) ⇒ U χ (ζ ,η ) = ei ( χ1ζ + χ 2η )
}

χ ∈ 2
2
with χ=
χ12 + χ 22 .
So, one gets for the eigen-functions Fλχ (ξ ) , the liniar second-order
differential equation,
(ξ
2
+ 1)
d 2 Fλχ
dξ
2
+ 4ξ
dFλχ
dξ
+ ( 2 − χ 2 − λ ) Fλχ =
0 (70)
which, using the linear transformation
ξ= ax + b ,(71)
becomes
2
b + i 
b − i  d Fλχ 
4b  dFλχ

+
+
+  4x + 
+ ( 2 − χ 2 − λ ) Fλχ =
x
x
0 .(72)



2
a 
a  dx
a  dx


1
For b = i and a = −2i , i.e.=
x
(1 + iξ ) , the equation (41) turns into
2
d 2 Fλχ
dFλχ
+ ( 2 − 4x )
− ( 2 − χ 2 − λ ) Fλχ =
x (1 − x )
0 .(42)
2
dx
dx
By comparing the above expression with the differential equation
satisfied by the hypergeometric functions [9],
x (1 − x )
d 2F
dF
+ c3 − ( c1 + c2 + 1) x 
− c1c2 F =0 (73)
2
dx
dx
one gets
c3 =2, c1 + c2 =3, c1c2 =2 − χ 2 − λ (74)
which concretely means
3
1
3
1
c1 = − λ + χ 2 + , c2 = + λ + χ 2 + , c3 =2 .(75)
2
4
2
4
1
( c1 + c2 + 1) ≡ 2 , the two linearly independent solutions are
2
 3

1 3
1
1
Fλχ (ξ=
)  F  − λ + χ 2 + , + λ + χ 2 + , 2; (1 + iξ ) ,
4 2
4
2

  2
Since, c3=
3
 
1 3
1
1
F  − λ + χ 2 + , + λ + χ 2 + , 2; ( 1 − iξ )  
4 2
4
2
2
 
(76)
defined at least within the unit disk (in the complex plane of “x”)
102
A.C.PIRGHIE et. al.
1
1
ξ sh (α z ) ≤ 3 ,
1 + ξ 2 ≤ 1⇒ =
(1 ± iξ=)
2
2
which gives, for the physical coordinate z, the “standard” range
z ≤
1
(
)
ln 2 + 3 .(77)
α
For z out of this range, which means ξ 〉 3 ⇔ x 〉1 , each of two linearly
independent solutions are given by the analytic extensions,
=
F [ c1 ,c2 ,c3 ; x ]
Γ ( c3 ) Γ ( c2 − c1 ) 
1
−c
F c1 ,c1 + 1 − c3 ,c1 + 1 − c2 ;  ( − x ) +
x
Γ ( c2 ) Γ ( c3 − c1 ) 
1
(78)
Γ ( c3 ) Γ ( c1 − c2 ) 
1
−c
F c2 ,c2 + 1 − c3 ,c2 + 1 − c1 ;  ( − x )
x
Γ ( c1 ) Γ ( c3 − c2 ) 
Thence, formally expressed in clear, the corresponding complex-valued
eigenmodes read
2
2
2


k
k
1 + ish (α z )  i ( k1 x + k2 y )
3
1
3
1

ψ λ k ( x, y,z ) = F  − λ + 2 + , + λ + 2 + ,2;
e
α
α
2
4 2
4
2






2
2


k
k
1 − ish (α z )  − i ( k1 x + k2 y )
3
1
3
1

ψ λ k ( x, y,z ) = F  − λ + 2 + , + λ + 2 + ,2;
e
α
α
2
4 2
4
2




(79)
2
2
2
where k = δ AB k A k=
δ AB k A k B ≡ ( k1 ) + ( k2 ) , so that, in principle, the
B
most general form of the contravariant electrostatic field potential,
1
− g 44Φ ( x, y,z ) =
Φ ( x, y,z ) ,
A4 ( x, y,z )  V ( x, y,z ) =
2
ch (α z )
in the local coordinates ( x, y,z ) stressed out by Lorentzian M 4 pathologic metric(12) is given by the expression


=
V ( x µ ) ∫ d 2 k d λ µ ( λ ) C ( λ ,k )ψ λ k ( x µ ) + C ( λ ,k )ψ λ k ( x µ )  ,(80)


VARIOUS ELECTROSTATIC FIELD...
where
=
( x µ )µ =1,3
x, y,z ) , d 2 k
(=
103
dk1 dk2 and µ ( λ ) is a properly chosen
measure onto the space of the eigenvalue “ λ ”, so that,
2
µ
'µ
δ ( x − x' ) δ ( y − y' ) δ ( z − z' ) (81)
∫ d k d λ µ ( λ )ψ λk ( x )ψ λk ( x ) =
in order to ensure the completeness relation, symbolically written as,


∑ λ ,k λ ,k = 1 ,(82)
λ ,k
where

with x
=

 
ψ λ ,k ( x ) = x λ ,k ,(83)
x )
( x, y,z ) .
(=
µ
µ =1,3
Bibliography
[1] C. Brans and R. H. Dicke, Phys, Rev. 124 (1961) 925;
F. Hoyle and J. V. Narlikar, Proc. Roy. Soc. A 282 (1964) 191;
A. Guth, Phys. Rev. D 23 (1981) 347;
A. D. Linde, Phys. Lett. B 108 (1982) 389.
[2] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other
Topological Defects (Cambridge Univ. Press, Cambridge, 1994);
G. Clement and I. Zouzou, Phys. Rev. D 50 (1994) 7271;
A. Wang and P. S. Letelier, Phys. Rev. D 52 (1995) 1800.
[3] M. Banados et al., Phys. Rev. D 48 (1993) , 1506.
[4] C. Dariescu and M. A. Dariescu, Found. Phys. Lett. 13 (2000) 147;
C. Dariescu and M. A. Dariescu, Int. J. Mod. Phys. A 16 (2001) 3707.
[5] K. Gödel, Rev. Mod. Phys. 21 (1949) 447.
[6] W. B. Bonnor, Class. Quantum. Grav. 19 (2002) 5951;
W. B. Bonnor and B.R. Steadman, Gen. Rel. Grav. 37 (2005) 1833;
Z. Perjés, Phys. Rev. Letters, 27 (1971) 1668;
W. Israel and G. A. Wilson, J. Math. Phys. 13 (1972) 323.
[7] V. M. Rosa and P. S. Letelier, Phys.Lett.A, 370 (2007) 99.
[8] M. Gogberashvili, Mod. Phys. Lett. A 14 (1999) 2025;
F. Dahia and C. Romero, Phys. Lett. B 651 (2007) 232.
[9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products, 4th edn. (Academic, New York, 1965).
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
METROLOGICAL ASPECTS CONCERNING AN
INSTALLATION FOR BIOMAGNETISM STUDIES
Doina COSTANDACHE, Octavian BALTAG
“Gr.T. Popa” University of Medicine and Pharmacy Iasi
Faculty of Biomedical Engineering
[email protected], [email protected]
Abstract. The paper presents the metrological aspects concerning a complex installation for measuring the
biomagnetic signals produced by the electrophysiological activity. The intensity of the biomagnetical signals
is bigger with some magnitudes orders from the physical limits measured in present.
To detect such fields it is necessary a proper electromagnetic environment and a magnetic high resolution
investigation device. In laboratories, this is achieved usually by using shielding rooms made from
ferromagnetic materials. Because of its high price, in the last years is used combined systems composed from
a shielded room realized by nonferromagnetic materials and systems for compensation and control of the
external magnetic field.
Technical solutions regarding the operation of the installation in the natural electromagnetic environment or
made by the human activity, the detection and the elimination of the electromagnetic and magnetic noises to
obtain the desired signal are presented.
1. INTRODUCTION
Since the magnetic fields generated by the bioelectromagnetic activity
of the living organisms have very low values, to detect such fields it is
necessary a proper electromagnetic environment and a magnetic high resolution
investigation device. In laboratories, this is achieved usually by using shielding
rooms made from ferromagnetic materials [1]. Because of its high price, in the
last years it is used combined systems composed from a shielded room realized
by nonferromagnetic materials and systems for compensation and control of the
external magnetic field [2], [3], [4].
The paper presents a complex installation consisting of a shielded room
made from nonferromagnetic materials placed in a triaxial Helmholtz system
for compensation and dynamic control of the external magnetic field. This
installation is used for basic studies and researches in biomagnetism and
biomedical diagnosis. The aluminum material selection for the shielded room
has been done after measuring the shielding coefficients to audio range
frequencies (20 – 20000) Hz, for different materials (aluminum, copper, brass).
METROLOGICAL ASPECTS ...
105
A location study inside the building for the proper amplacement of the
installation has been done after determining the local magnetic field
components due to diurnal activity and electromagnetic level measurements.
The fig.1 presents the diurnal activity magnetogram on a 24 hours interval in
two different location (up – without diurnal activity, down – with diurnal
activity).
Fig.1. Diurnal activity magnetogram in two location
The bioelectromagnetism installation is equipped with a communication
system using microphones and piezoelectric headphones, video camera. Bed
horizontal plane movement is obtained using non-ferromagnetic cables
commanded by rotating step by step engines situated to a few meters distance
outside the shielded room.
Electromagnetic fields measurements have been realized for
higher frequencies (up to 3 GHz), inside and outside the shielded room.
2. DESCRIPTION OF THE INSTALATION
Shielded room
The shielded room is an enclosure from nonferromagnetic material able
to shield electromagnetic fields of low and high frequency.
The room having the dimensions (3 x 2 x 2) m is made from aluminum
sheets of 12 mm thickness. We used aluminum as shielding material for some
advantages: is cheaper and non-corrosive, easy to manufacture.
The room is mounted on a wood laminated structure in the centre of the
triaxial Helmholtz system (Fig.2).
106
D.COSTANDACHE et. al.
Fig. 2. The shielded room
All the clamping and fastening elements are realized in brass, and the
door elements are in bronze. The room air ventilation is realized by two aeration
apertures, an admission one in the upper side and an upsetting one in the
downside. Direct current power supply is realized by a network equipped (to the
input) with filters, the alternative circuits being disposed far away. The
electrical connections are made using twisted and shielded wires from
nonferromagnetic material. The shielded room lightening is assured using LEDs, and the walls are covered by absorbing material in order to attenuate the
reflections. The bed placed inside the shielded room is made from
nonferromagnetic material with the possibility to move in horizontally plane on
the two directions (X, Y). Above the bed, a three channels SQUID
magnetometer is fixed (Fig. 3), that may measure the biofield on the direction
X, Y, Z and the first and second order gradients.
Fig.3. The bed inside the shielded room and the SQUID
Triaxial Helmholtz system
The triaxial Helmholtz system which allows the compensation on the
three directions of the low frequency magnetic fields is realized from three pairs
METROLOGICAL ASPECTS ...
107
of large square Helmholtz coils, having the dimensions (4 x 4 x 4) m, (Fig. 4).
The currents from the coils are controlled by a triaxial magnetometer. Each coil
has two windings, one for manual compensation and the second for automatic
control. The ratio of the two coils constants is 1/5. The coils for the horizontal
components compensation (X and Y) are oriented to N-S, respectively E-W.
The coils constants ware calculated for ± 40 A/m compensation range [5], [6],
[7]. The first coil allows the manual compensation of the continuous component
of the magnetic field vector on the three directions (X, Y, Z). The second coil
allows the compensation of the alternative and slow variable components of the
earth magnetic field using an automatic system controlled by a triaxial
magnetometer. The magnetometer is working in a negative feedback system.
The negative feedback loop is closed by the Helmholtz coils field where is
placed the triaxial transducers block.
This structure presents the advantage of flexibility and a high dynamic
control of the magnetic field.
Fig.4. The triaxial Helmholtz system
The manual control system (Fig. 5) consists from a current amplifier
stage voltage controlled, using power operational amplifiers.
A digital-analog converter gives the command voltage. The field
programmed value is indicated (by means of an alphanumerical keyboard) on a
digital display.
108
D.COSTANDACHE et. al.
Fig. 5. The block diagram for manual compensation of the magnetic field
The three channels for manual control are identically; the differences
are the values of the coils constants, caused by different dimensions of the three
coils. The manual control system allows the compensation of the earth magnetic
field components in the range ± 40 A/m for each component.
The automatic control system (Fig. 6) is working in negative feedback
loop on each channel. For each channel the feedback loop is closing in a chain
made from: transducer – magnetometric channel – current source voltage
controlled – Helmholtz coil – transducer. The three channels are independent
calibrated. The response frequency of the channels is between (0 – 400) Hz. The
magnetometer noise level is 20 pT/Hz1/2.
Fig. 6. The block diagram for automatic compensation of the magnetic field
METROLOGICAL ASPECTS ...
109
The automatic system is controlled by a triaxial magnetometer mounted
on an external side of the shielded room. A second triaxial magnetometer is
used for the magnetic field control inside the shielded room.
The electronic block for comand and control is placed outside of the
shielded room at a few meters of distance (Fig. 7).
Fig.7. Electronic control block
Using the notations: H – the environmental magnetic field; Uout – the
output voltage of the magnetometer for the open negative feedback loop; β –
transfer function of the negative feedback circuit; Ki – the current constant of
the automatic compensation coil; r – the coil compensation resistance, S – the
transfer function of the magnetometer, the output voltage when the negative
feedback loop is not connected is [8]:
(1)
U out = S ⋅ h
where S is the transfer function of the magnetometer.
The compensation field generated by the Helmholtz coil is:
hr = S ⋅ H ⋅ β
Ki
r
(2)
The resulted field from the application of the negative feedback loop –
automatic compensation is:
H − hr = H − Shβ
Ki
 SβK i 
= H 1 −

r
r 

(3)
From (3) results:
SβK i
∆H
= 1−
r
H
(4)
110
D.COSTANDACHE et. al.
To obtain a better compensation it is necessary that
where ε is a pre-established value and results:
Sβ K i
= 1− ε ≈ 1
r
∆H
H
<< ε ,
(5)
This gives the conditions for the system: a bigger sensibility of the
magnetometer, the increased transfer coefficient of the negative feedback loop β
(0<β<1), the current constant Ki as big as possible, the resistance r of the
compensation coil as small as possible.
3. RESULTS
The compensation range of the earth magnetic field is ± 40 A/m. The
compensation level is almost (8 x 10-4) A/m residual field. The field is
homogenous (inhomogeneity 10-3) in a volume having the dimensions (0,8 x 0,8
x 0,8) m.
The coils constans for manual compensation are: KX = 0,6093x10-4T/A;
KY = 0,6292x10-4T/A; KZ = 0,4875x10-4T/A.
For automatic compensation the constants are: kX = 0,1218x10-4T/A;
kY = 0,1258x10-4T/A; kZ = 0,0975x10-4T/A.
The type of the magnetometer is HTS SQUID made by Tristan
Technology USA with three channels (Fig. 8).
Fig.8. The HTS SQUID
For frequencies higher than 1 Hz the noise (RMS) is smaller than 3x10-6
Φ0 Hz , where Φ0 represents the flux cuanta (Φ0 = 2,07x10-5 G/cm2). The
proper noise spectrum of 0,25 Hz – 800 Hz is presented in Fig. 9.
1/2
METROLOGICAL ASPECTS ...
111
Fig.9. The noise spectrum of SQUID
4. CONCLUSIONS
The accomplished installation permits to perform biomagnetic field
measurements. The environmental magnetic fields are dimished to levels which
ensures the operation of SQUID magnetometer by using the static and dynamic
compensation method. The shielded room accomplishes the shielding of
medium and high frequency electromagnetic fields. The control system for the
subject movement in horizontal level permits to set out biomagnetic field maps.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Kajiwara G., Harakawa K., Ogata H., High performance magnetically shielded room, IEEE Trans. Mag.,
vol. 32, pp.2582-2585, 1996
Holmlund C., Keipi M., Meinander T., A.Penttinen, H.Seppa, Novel concepts in magnetic shielding,
Proc. of 12th Int. Conf. on Biomagnetism Biomag 2000, Espoo/Helsinki, Finland, 2000
Stroink G., Blackford B., Brown B., Horacek M., Aluminum shielded room for biomagnetic
measurements, Rev. Sci. Instrum. Vol. 52(3), pp.463-468, 1981
Platzek D., Nowak H., et all, Active shielding to reduce low frequency disturbances in direct current
near biomagnetic measurements, Rev. Sci.Instrum., vol. 70 (5), pp.2465-2470, 1999
O’Neill D., Mathematical Modeling of the Magnetic Field of a Helmholtz Coil (unpublished paper)
Szelc A., Helmholtz field calculations for the magnetic test experiment (unpublished paper)
Koike K., Nakajima S., Shimizu Y., On the accuracy of flux-gate magnetometers, Calibration
experiment, memoirs of the Kakioka Magnetic Observatory, Vol.24, No.1, 1-13, 1990
Baltag O., Robu O., Costandache D., Ignat C., Magnetometrie, Ed Performantica, Iasi, 2003
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
MAGNETIZATION OF THE QUANTUM HALL
SYSTEM OF BOSONS
Marina–Aura Dariescu
Department of Solid State and Theoretical Physics
Faculty of Physics, Al. I. Cuza University
Bd. Carol I no. 11, 700506 Iasi, Romania
Abstract
In the present paper we perform a thermodynamic analysis of the quantum Hall-type behavior of relativistic
charged scalar particles subjected to orthogonal electric and magnetic fields. In Cartesian coordinates, using a
convenient complex variable, we obtain the analytic solution to the Euler–Lagrange equations and the
Landau-type energy levels. For an ultra-relativistic particle, the characteristic function leads to explicit
expressions of magnetization and susceptibility, in terms of magnetic induction and temperature.
Introduction.
As it known by now, the mesoscopic systems posses intermediate size, between
small enough so that they still move in a coherent way, to big enough for some
measurable consequences. In the last 20 years, they have been intensively
studied, starting with the celebrated Quantum Hall Effect, [1], to the theory of
quantum computers, encoding quantum information which is inaccessible to
local interactions and decoherence, [2]. In a series of papers, [3], it has been
shown that it is possible and very convenient to compute mesoscopic quantities,
as for example the persistent currents, especially after they have been observed
in mesoscopic rings, at very low temperatures. In this respect, the planary
dynamics of charged relativistic particles evolving in static orthogonal magnetic
and electric fields has been a main target of investigations, pointing out
significant differences from the non-relativistic case, [4].
The aim of the present work is to generalize the results gotten in [5],
where we have derived the main thermodynamic quantities and the bound for
the ratio of entropy to energy, by considering a general dependence of the
Landau-type energy levels on the temperature, external fields and relativistic
particle momentum.
Hall-type evolution of charged scalar field.
MAGNETIZATION OF THE QUATUM HALL SYSTEMS..
113
The U(1)-gauge invariant Lagrangian for the massive charged scalar field of
mass m0 is given by the expression
1 ij
(1)
F Fij ,
4
where Fij is the Maxwell tensor and ‘’;’’ is denoting the U(1)-gauge covariant
L = η ijψ ;∗iψ ;i + m02ψ ∗ψ +
derivatives
ψ ;i = ψ ,i − iqAiψ ,
ψ *;i = ψ *,i +iqAiψ * .
(2)
As previously, we employ the Cartesian coordinates
ds 2 = dx 2 + dy 2 + dz 2 − dt 2
and the gauge:
Ax = Az = 0, Ay = B0 x, A4 = E0 x ,
(3)
(4)
where E0 and B0 are the orthogonal electric and magnetic fields. By performing
the variational procedure, one come to the Euler–Lagrange equations as being:
η ijψ ,ij − 2iqB0 xψ , y + 2iqE0 xψ ,t − [m02 + q 2 x 2 ( B02 − E02 )]ψ = 0 ,
η ijψ *,ij + 2iqB0 xψ *, y − 2iqE0 xψ *,t − [m02 + q 2 x 2 ( B02 − E02 )]ψ * = 0 (5)
Inspecting the above formulas, one may notice that they look simpler once we
use the Compton recalibration:
ξ = m0 x ,
(6)
η = m0 y .
With the standard variables separation
(7)
ψ = χ (ξ )e −ipη e iωt
the first field equation in (5) turns into the following differential equation, for
the ξ -depending part,


αω 
d 2χ ω 2
ξ − Ω 2 − α 2 ξ 2  χ = 0
+  2 − p 2 − 1 − 2 Ωp +
2
m0 
dξ
 m0


where we have introduced the parameters
qB
Ω = 20 ,
m0
qE
α = 20 .
m0
With the new space variable:
(
)
(8)
(9)
114
M. A. DARIESCU et. al.
z = Ωξ +
α
(10)
+ p,
Ω
we get, within Newtonian energy approximation ε = ω − m0 , the solution
χ ( z ) = Cn e − z
2
/( 2 Ω )
H n (z / Ω ) ,
(11)
where H n ( z / Ω ) are the Hermite polynomials.
Moreover, the corresponding Landau-type energy spectrum
εn 
1
α2
α
(12)
=  n + Ω −
−p ,
2
m0 
2
Ω
2Ω
shows a nontrivial dependence on the external fields and on the particle
momentum. For an ultra-relativistic boson evolving in a strong magnetic field,
the above formula can be approximated to
1
εn 
α
(13)
=  n + Ω − p ,
2
Ω
m0 
which will be used in the the following thermodynamic analysis.
Magnetization and Susceptibility.
In this section, we are going to generalize the theory developed in [5].
Following the same approach, we start by introducing the temperaturedependence via the dimensionless parameter
Ω
Ωβ
,
(14)
=
δ=
4πT 4π
where β is the usual thermodynamic parameter β = 1 / T .
In order to deal with the whole range of temperatures, it is convenient to keep
δ as being arbitrary and apply a similar formalism as the one developed in [6],
for the analysis of the TM modes of the electromagnetic field in Einstein's
Universe. Let us write the energy spectrum (13) written as
pα  Ω
Ω
(15)
ωn = 2n + 1 − 2 2  = (2n + a ) ,
2
Ω  2
where we have introduced the notation:
a = 1− 2 p
α
.
(16)
Ω2
As usual in a thermodynamic analysis, we start with the characteristic function,
namely
MAGNETIZATION OF THE QUATUM HALL SYSTEMS..
G = ln Z =
∞
∑
ln[1 − exp(− βω n )] =
n =1
115
∞
∑ ln[1 − exp(− 2π (2n + a )δ )] , (17)
n =1
and apply the technique presented in [6]. Thus, by expanding the logarithm and
taking the derivative with respect to δ , we get
∞
∞
∂G
(18)
= 2π (2n + a ) exp[− 2π (2n + a )kδ ]
∂δ
n =1
k =1
and, with
1
e−x =
dsx − s Γ(s ) ,
2πi C
∑
∑
∫
where x = 2π (2n + a )kδ , we end up with the following relation,
∂G 1
−s
= ds(2πδ ) Γ(s )ξ (s )21− s ξ [s − 1,1 + a / 2] ,
∂δ i C
∫
(19)
in terms of the Euler, Riemann and Riemann's generalized functions. We deal
with the three poles: s = 0, s = 1 and s = 2 , by applying the Residues Theorem,
and we obtain the expression of the derivative of Gas being
 1 a a2  1 a +1
∂G
π
= 2π  + +  −
+
ξ [1,1 + a / 2] . (20)
∂δ
24δ 2
12 4 8  δ 2
In the last term of (20), one may expand ξ [1 + is,1 + a / 2] up to the third order in
(a − 1) and come, by integration, to the final result of the characteristic function
G = ln(4π ) +
11 Ω π 2 T
Ω
−
− ln
T
48 T
6 Ω
1 Ω π2 
π2 T
Ω 
2 −
 − ln 
− 2 ln(4π ) +
−
2T
3 
4  Ω
Ω 
 T 
αp 
(21)
In the ultra-relativistic case, when the energy spectrum is given by (13), in the
αp
first order of 2 , one may notice an overall dependence on the magnetic and
Ω
electric fields, particle momentum and temperature.
For the analyzed system of relativistic bosons, the free energy, defined
by
(22)
F = −T ln Z = −TG ,
allows us to compute the magnetization,
116
M. A. DARIESCU et. al.
∂F
q ∂F
(23)
.
=− 2
∂T
m0 ∂Ω
For G given by (21), we obtain the following expression, as a function of
temperature and external fields intensities,
q
M= 2
m0
(24)
 11 T T 2π 2 αp  T
T 2π 2 
T Ω 1 
π2 
 − 2 ln +  
2 −
+ 2  (1 + 2 ln 4π ) −
 − +
2
Ω T 2  
4 
Ω  Ω
Ω 2 
 48 Ω 6Ω
In Figure 1, we represent the contour plots of constant magnetization, for
different values of magnetic induction and temperature. Their intersections with
the dashing curve gives us the pairs of B and T corresponding to zero
susceptibility. This is defined by the first derivative of magnetization with
respect to induction,
∂M
.
(25)
χ=
∂B
M =−
0.3
0.25
T
0.2
0.15
0.1
0.05
0
0.5 0.75 1 1.25 1.5 1.75 2
B
MAGNETIZATION OF THE QUATUM HALL SYSTEMS..
117
Figure 1. Contour plots of constant magnetization, for different values of induction and
temperature. The dashing curve corresponds to zero susceptibility
In the particular case αp ≅ 0 , the susceptibility has the explicit expression
T  π 2 m02 T 
(26)
1 −
,
3 q B
B2 
which vanishes for the critical value of the temperature equal to 3 / π 2 times the
ratio between the Planck–Larmor quanta and the rest energy.
In figures 2 and 3, we represent the magnetization, for α × p = 10 −2 ,
plotted versus the induction, at selected temperatures (Figure 2) and versus the
temperature, for different induction values (Figure 3). In figure 2, the thickness
of the curves is increasing when T goes from 0.01 to 0.1, while in Figure 3 the
curves thickness is increasing for B ranging from 0.5 to 2, in steps of 0.25.
χ=
0.22
M
0.2
0.18
0.16
0.14
0.6 0.8 1 1.2 1.4 1.6 1.8 2
B
Figure 2. The magnetization, (24), for α ×
increasing temperatures, from 0.01 to 0.1.
p = 10 −2 , as a function of the induction, at
118
M. A. DARIESCU et. al.
0.3
M
0.25
0.2
0.15
0.1
0
0.1 0.2 0.3 0.4 0.5
T
−2
Figure 3. The magnetization, (24), for α × p = 10 , plotted versus the temperature, for
different induction values, ranging from 0.5 to 2, in steps of 0.25.
Concluding remarks.
The present paper deals with the U(1)-gauge invariant analysis of the Hall-type
evolution of charged scalar field. In Cartesian coordinates and for a convenient
gauge, the Euler–Lagrange equations (5), with the complex variable (10) have
solutions expressed in terms of the usual Hermite polynomials. The Landautype energy levels, (12), are exhibiting a general dependence on the exterior
electric and magnetic fields and on the particle momentum. For an ultrarelativistic particle in strong magnetic field, they turn into the relation (13). By
introducing the thermodynamic dimensionless parameter δ , one is able to
derive the characteristic function, (21), and to compute the magnetization and
the susceptibility. For the temperature going to zero, the corresponding
magnetization
q  11 αp 
M0 = 2  +
m0  48 2Ω 2 
is pointing out, for αp ≅ 0 , the irrational quantity M 0 = 11q / 48m02 . Finally,
we represent the magnetization as a function of the induction (Figure 2) and
temperature (Figure 3).
MAGNETIZATION OF THE QUATUM HALL SYSTEMS..
119
Acknowledgements.
This work is supported by the Grant Type A CNCSIS Code 1433/2007.
Bibliography
[1]. D. C. Tsui, H. L. Stomer and A. C. Gossard. Two-dimensional
magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559–1562
(1982);
R. B. Laughlin. Anomalous Quantum Hall Effect: An incompressible quantum
fluid with fractionallycharged excitations, Phys. Rev. Lett. 50, 1395–1404
(1983);
B. I. Halperin. Statistics of quasiparticles and the hierarchy of fractional
quantized Hall states, Phys. Rev. Lett. 52, 1583–1586 (1984);
F. D. M. Haldane. Fractional quantization of the Hall Effect: a hierarchy of
incompressible quantum fluid states, Phys. Rev. Lett. 51, 605–608 (1983);
A. M. Chang et al. Higher-order states in the multiple-series, fractional,
quantum Hall effect, Phys. Rev. Lett. 53, 997–1000 (1984).
[2] S. D. Sarma, M. Freedman and C. Nayak. Topologically-protected qubits
from a possible non-abelian fractional quantum Hall state. Phys. Rev. Lett.
94:166802 (2005).
[3] S. Datta. Electronic transport in mesoscopic systems. Cambridge:
Cambridge Univ Press; 2003.
Lachezar S. Georgiev. A universal conformal field theory approach to the chiral
persistent currents in the mesoscopic fractional quantum Hall states. Nucl. Phys.
B 707, 347–378 (2005); Chiral persistent currents and magnetic susceptibilities
in the parafermion quantum Hall states in the second Landau level with
Aharonov-Bohm flux. Phys. Rev. B 69:085305 (2004); Conformal field theory
description of mesoscopic phenomena in the fractional quantum Hall effect.
Arxiv:hep-th: 0601075.
[4] M. A. Dariescu and C. Dariescu. Quantized planary dynamics in orthogonal
magneto-electrostatic fields. Chaos, Solitons and Fractals 23, 1005–1011
(2005).
[5] M. A. Dariescu and C. Dariescu. Finite temperature analysis of quantum
Hall-type behavior of charged bosons. Chaos Solitons and Fractals 33, 776–781
(2007).
[6] I. Brevik et al. Entropy Bound for the TM Electromagnetic Field in the Half
Einstein Universe. Arxiv:hep-th/0508123 and the references therein
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
ISING-TYPE SIMULATIONS IN 1D
MAGNETOSTATICALLY INTERACTING PARTICLE
SYSTEMS
C. Rotarescu, A. Stancu
„Alexandru Ioan Cuza” University, Faculty of Physics & Carpath,
700506, Iasi, Romania
[email protected]
Numerical simulations of magnetic processes are presented for an Ising-type system. The studied system
consists of a linear array of magnetic dipoles with magnetostatic interactions between the constituting
elements. The temperature dependence of the hysteresis loops is studied using the Monte Carlo-Metropolis
algorithm with the energy barrier of each particle given by the Stoner-Wohlfarth model. The simulations were
made for a large number of particles. Also we present simulations for the first magnetization curve after
thermal demagnetization. The interactions between particles are taken as dipolar.
1. Introduction
In recent years a remarkable effort was dedicated to the study of the
magnetic properties of nanodimensional particles systems because of
their implications in technological applications like high-density
recording media or nanostructured permanent magnets.
The magnetic properties of all materials which exhibit persistent
memory are a function of temperature that is why the study of the
temperature dependence of material properties is very important [5, 8].
A considerable interest from a fundamental and technological
perspective is thermal equilibrium process in physical systems with
metastable free energy landscapes. In magnetic materials, the relaxation
of the magnetic properties that accompanies the spontaneous approach to
equilibrium are important for the practical limitations that it imposes on
the ultimate bit size of information storage media and for the stability of
permanent magnets [8].
The representation of magnetic materials as a superposition of
elementary bistable subsystems was first introduced into the literature by
Preisach [1] for the description of history-dependent phenomena [10-14].
A material exhibit hysteresis [12] if its physical response to an
imposed excitation at time t is not determined uniquely by the
ISING-TYPE SIMULATIONS ….
121
instantaneous value of the excitation, but instead depends upon its
previous history of exposure to the excitation at earlier times t ' ≤ t . The
hysteresis requires the existence of multiple local minima in the free
energy hypersurface of the material [12].
In this paper we present numerical simulations of magnetic processes
of an ensemble of thermally activated particles consisting of a collection
of two-level subsystems with double-well free energy profiles. When the
distance between particles decreases the influence of the magnetostatic
interactions becomes important.
The processes and properties of the Ising systems are studied with the
help of the Monte-Carlo method. In this method one randomly chooses a
spin from the network and tests if it switches from up to down or down to
up. If the answer is positive, the new microstate is accepted, if not one
keeps the initial state and chooses another spin. This is done until an
equilibrium is obtained.
We used the Metropolis algorithm [2] which gives the equilibrium
configuration quite fast, if the simulation temperature is high enough. For
low temperatures, the simulation becomes prohibitive and other methods
should be used.
2. The model
We developed a modified version of the Ising model in which the
elements of the systems, when they are isolated, have a rectangular
hysteresis loop [1], and the anisotropy was added to each magnetic entity
(Ising-Preisach model).
Fig. 1 Rectangular hysteresis loop
122
C. ROTARESCU ET AL
In Fig. 1 we represented the rectangular hysteresis loop of a single
particle, where Hα is the positive switching field, H β is the negative
switching field, H c is the coercive field of particle, H S is the shift of the
rectangular hysteresis loop on the field axis and mos the saturation
magnetic moment.
Fig.2 Stoner-Wohlfarth energy barrier
For an assembly of uniaxial single domain particles, with the easy
axis being aligned with the field direction, the free energy of a particle
can be written as:
(1)
− KV cos 2 θ − VPS H 0 cos θ
E=
where K is the anisotropy constant, V is the particle volume, PS is the
saturation polarization, H 0 the exterior magnetic field parallel with easy
axis and θ is the angle between the polarization vector and the easy
direction.
The height of the free energy corresponding to the magnetic moment
vector orientation out of the field direction and into the field direction is
=
∆E+
VPS
H
H k (1 + 0 ) 2
Hk
2
(2)
and
=
∆E−
VPS
H
H k (1 − 0 ) 2
2
Hk
(3)
ISING-TYPE SIMULATIONS ….
123
When one studies a system of particles, due to the magnetostatic
interactions, H 0 should be replaced by an effective field H =
H 0 + H int
eff
where H int is the interaction field.
For our simulations we have taken the distribution of the anisotropy
as having a Gaussian shape (Fig.3) and a nonlinear energy barrier given
by the Stoner-Wohlfarth model (Fig.2).
Anisotropy field distribution
Number of particles
1200
1000
800
600
400
200
0
900
950
1000
1050
1100
Hc(Oe)
Fig.3 Anisotropy field distribution
The studied system is one dimensional, N is the number of spins and
in the initially state the system is saturated i.e. all the spins are in the up
state. We used the Monte Carlo-Metropolis algorithm with the following
steps [2]:
1. Select one lattice site i at which the spin Si is considered for flipping
( Si → − Si )
2. Compute the energy change ∆E associated with that flip.
3. Calculate the transition probability P for that flip:
P =P0 ⋅ exp(−
∆E
)
k BT
4. One draws a random number Z uniformly distributed between zero
and one.
5. If Z < P the spin switches, otherwise it maintains the same orientation.
6. Analyze the resulting configuration as desired and store its properties
to calculate the magnetization M .
124
C. ROTARESCU ET AL
3. Numerical simulations
We have done numerical simulations of different magnetic processes
like hysteresis loop and first magnetization curve using an Ising-Preisach
model. The particles are spherical with radius r = 5nm . We used the
material constants of cobalt: the anisotropy constant =
K 0.7 ⋅105 J / m3 ,
magnetization and polarization at saturation M
=
1.42 ⋅106 A / m ,
S
respectively PS = 1.78T . Hysteresis loops at different temperatures and
geometric parameters are simulated for two cases: isolated particles
( H int = 0 ) and magnetostatic interactions between particles (Fig. 3).
M/MS
1.0
T=0.2TB
T=TB
T=2.07TB
0.5
0.0
-3000
-2000
-1000
0
1000
2000
3000
H(Oe)
-0.5
-1.0
(a)
M/MS
1.0
T=0.2TB
T=TB
T=2.07TB
0.5
0.0
-3000
-2000
-1000
0
1000
2000
3000
H(Oe)
-0.5
-1.0
(b)
Fig.3 Hysteresis loop (a) for isolated particles and (b) for magnetostatic interactions between
particles.
We considered that the interactions between particles are only dipolar
[4] and can be calculated using
ISING-TYPE SIMULATIONS ….


1
Hp = −
4π
125

pq
∑R
q
(4)
3
pq

M/MS
where R pq is the distance and pq the magnetic moment of the q particle.
We can see in Fig.3 b) the metastables states due to the formation of
clusters. The distance between particles was taken as d = 3r .
In the next figure we present some simulations for the first
magnetization curve after thermal demagnetization. The system was
thermally demagnetized starting from the superparamagnetic temperature
to room temperature.
1.0
0.8
0.6
isolated particles
interacting particles
0.4
0.2
0.0
0
500
1000
1500
2000
2500
H(Oe)
Fig. 4 First magnetization curve
The simulations were made for N=10000 particles, temperature
T=100 K and distance between particles d = 3r . We observe from Fig.4
that an isolated particle system reaches saturation at lower fields than in
interacting system.
4. Conclusions
We developed an Ising-Preisach model for systems with
magnetostatic interactions between particles. The model can be used to
study the magnetization curves and their temperature, field and time
dependence. The system relaxes from metastable states toward
thermodynamic equilibrium by activation over local energy barriers.
The hysteresis loops and first magnetization curves were studied
considering a nonlinear energy barrier based on the Stoner-Wohlfarth
model.
126
C. ROTARESCU ET AL
The effects of interactions on the magnetization processes in
nanostructured materials continue to pose a significant challenge due to
their technological use as magnetic memories.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
F. Preisach, Z. Phys. 94, 277 (1935)
N. Metropolis et al., J. Chem. Phys. 21, 6, (1953)
R. W. Chantrell et al., J. Appl. Phys. Vol.76, No.10, (1994)
R. Tanasa, C. Enachescu, A. Stancu, J. Linares, F. Varret, Physica B, Vol.343, (2004)
C. A. Viddal, R. M. Roshko, Physica B, (2007) doi: 10.1016/j. physb. 2007.08.038
I. Mayergoyz, Mathematical Models of Hysteresis (Springer-Verlag , New York, 1991).
E. Della Torre et al., IEEE Trans. Magn., Vol. 38, No. 5, pp.3409-3416 (2002).
R. M. Roshko and C. A. Viddal, Phys. Rev. B, Vol. 72, art. no. 184422, (2005).
E. Della Torre, L. H. Bennett, C. E. Korman, Physica B (2007), doi:
10.1016/j.physb.2007.08026
[10] L. Neel, J. Phys. Le Radium 11 (1950) 49
[11] J. Souletie, J. Phys. 44 (1983) 1095
[12] G. Bertotti, Hysteresis in Magnetism, Academic Press, New York, (1998)
[14]
T. Song, R. M. Roshko, E. D. Dahlberg, J. Phys.: Condens. Matter 13 (2001) 3443Wesley, 1990)
[8] A. Vilenkin, Phys. Rev. Lett. 53, 1016 (1984).
[9] J. Garcia-Bellido, Int. J. Mod. Phys. D2, 85 (1993) [hep-ph/9205216].
[10] G.W. Anderson and S.M. Carroll, ``Dark matter with time-dependent mass'', astroph/9711288.
[11] R. R. Caldwell, astro-ph/9908168
[12] P.H. Frampton, astro-ph/0209037 (2002)
[13] Y. Duan, Chinese Phys. Lett. 20, 12, 413-418, (2002)
[14] H. Wei, R. G. Cai and D. F. Zeng, Class. Quant. Grav. 22, 3189 (2005) [hep-th/0501160].
[15] J. D. Barrow, Class. Quant. Grav. 21, L79 (2004) [gr-qc/0403084];
J. D. Barrow, Class. Quant. Grav. 21, 5619 (2004) [gr-qc/0409062].
[16] I.L. Shapiro, Phys. Rep., 375, 113, (2001)
[17] B.B. Mandelbrot, Fractals, Forms, Chance and Dimensions, San Francisco (1989)
[18] F. R. Bouchet, High Resolution simulation of Cosmic String Evolution: Small ScaleStructure and Loops, in The Formation and Evolution of Cosmic Strings, Cambridge Univ. Press,
1989
[19] L. Guzzo, A. Iovino, G. Chincarini, R. Giovanelli, M. Haynes, Ap. J. 382 L5 (1991)
[20] J. P. Huchra, M. Geller, V. de Lapparent, Ap. J., S 72, 433-470 (1990)
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
DISPERSION AND ATTENUATION CHARACTERISTICS OF
SOME METAMATERIALS IN MICROWAVE RANGE
Iulia Andreea Mihai 1, Marian Gabriel Banciu 2, Teodor Petrescu 1
Telecommunications Department, Electronics, Telecommunications and
Information Technology Faculty, University Politehnica of Bucharest
1-3, Bd. Iuliu Maniu, 061071, Bucharest 6, Romania
[email protected]
2
National Institute of Materials Physics, Măgurele, Jud. Ilfov, Romania
1
Abstract
In this paper, some types of 1-D metastructures are investigated in microwaves. Even in a
restricted frequency range, a purely left-handed material does not exist and the composite right/left-handed
(CRLH) structure is a more appropriate model of practical implementations of left-handed materials. It is
shown that, above a certain cutoff frequency, the CRLH structures behave as a conventionally right-handed
material. In this paper, dispersion/attenuation diagrams are investigated for purely left-handed and CRLH
transmission lines. Moreover, the magnitude and phase of the S parameter are analyzed in the 1-10 GHz
range. The investigations were carried out in both balanced and unbalanced CRLH cases in order to determine
the most advantageous conditions for microwave applications.
1. Introduction
Metamaterials are artificial, effectively homogeneous electro-magnetic
structures with unusual properties not readily available in nature [1]. The
electrically small “artificial molecules” have been used either to contributes to
the electric flux density macroscopically by increasing the effective electric
permittivity as for artificial dielectrics used in some microwave antennas [2,3],
either to exhibit a generalized magnetic permeability function as for the
artificial magnetics [4].
Metamaterials are known for the anti-parallelism existing between the
phase and group velocities. The major feature, which distinguishes them from
other periodical structures, such as the backward wave-structures, is that the
average cell size is much smaller than the guided wavelength. While the
propagation along the backward wave-structures are dominated by
diffraction/scattering phenomena, the metamaterials can be characterized by
constitutive electric permittivity and magnetic permeability and could be
designed with simultaneously low loss and broad bandwidth due to their nonresonant nature.
The electromagnetic propagation in one direction through an effectively
homogeneous material can be essentially modeled by one-dimensional
128
I.MIHAI ET AL.
transmission-line, which can be approximated by a periodic ladder network as
shown in Fig. 1.
Fig. 1. General lossy transmission line consisting of periodic cells.
(Only the first and the last cells are presented.).
A small portion of transmission line of length zexhibits an impedance
Z ( f ) ⋅ ∆z and an admittance Y ( f ) ⋅ ∆z , where

1
Z ( f ) = R + jX ( f ) = RR + j  2πfLR −
2πfC L


 and


1
Y ( f ) = G + jB ( f ) = GL + j  2πfC R −
2πfLL


 .

(1)
All the circuit components shown in Fig. 1 correspond to the one cell length z.
Two particular situations can be obtained if the transmission line shown
in Fig. 1 is simplified. On one hand, when LL = CL = GL = 0, the propagation of
a signal along the transmission line corresponds to the wave propagation
through a conventional medium, associated with a right-handed triad between
the electric field E, magnetic field H and the wave vector k. This particular case
will be called purely right-handed (PRH) case. On the other hand, when only
the components LL , CL and GL are included, a new transmission line is obtained,
which is the dual of the PRH transmission line. The signal propagation along
the line is associated in this case with the wave propagation through an artificial
„left-handed” medium associated with a left-handed triad between the E, H and
k vectors. This case is referred to as the purely left-handed (PLH) case.
In the general case, the transmission line of Fig. 1 is called composite
right/left handed (CRLH) transmission line [5]. The complex propagation
constant of a CRLH line is given by
γ RL = α RL + jβ RL = ZY
(2)
DISPERSION AND ATTENUATION….
129
In this paper, the CRLH dispersion / attenuation diagrams were
computed by using the Agilent Advanced Design System (ADS) simulation
software [6].
2. Perfectly left-handed case
A conventional transmission line can describe the conventional righthanded materials. Such a ladder network exhibits a cut-off frequency
f cPR = 2 f R where f R =
1
, beyond which the structure attenuates
2π LR C R
even when RR=0. On the other hand, the ladder network associated with the lefthanded transmission line presents a cut-off frequency f cPL = f L / 2 ,
where f L =
1
. For frequencies f < fL the transmission line attenuates
2π LL C L
even in the loss-less case for GL = 0.
The magnitude of the S parameters for two different loss-less PLH
ladder networks is presented in Fig. 2. The first network consists of N = 10
cells, each cell having LL = 2.5nH, CL = 1pF, then, the cut-off frequency for this
network is fcPL=1.59 GHz. If the same piece if PLH transmission line is
analyzed when considering a n times shorter unit cell (of length z/ n), then the
new ladder network should have n N cells with components values nLL and nCL.
In this way, the total electrical length and the characteristic impedance
Z L = LL C L of the piece of the PLH transmission line will not change, but
the cut-off frequency for the ladder network will be fcPL/n, which is in
agreement with the results shown in Fig. 2. Moreover, the frequency responses
presented in Fig. 2, show that the ladder network is a good approximation of the
PLH transmission line only for frequencies higher than, and far from the cut-off
frequency fcPL.
In a similar way as above, when n times shorter unit cell is considered
for a loss-less PRH ladder network, the new PRH network with n N cells will
have LL/n and CL/n components values for the same characteristic impedance
and electrical length, but for a n times smaller cut-off frequency (fcPR/n). The
unwrapped S21 phase of the new network exhibits the same linear dependence
on the frequency as the phase of the initial PRH network only for small
frequencies smaller than the cut-off frequency fcPR/n, as can be shown in Fig. 3.
130
I.MIHAI ET AL.
Fig. 2. Magnitude of the S parameters versus frequency for a PHL transmission line. a) |S21| for
n = 1, b) |S11| for n = 1, c) |S21| for n = 2, c) |S11| for n = 2.
Fig. 3. Unwrapped phase of S21 versus frequency. a) initial PLH network with n = 1, b) PLH
network with n = 2 (values shifted up with 1801 deg in order to have the same phase reference at
high frequencies), c) initial PRH network with n = 1, d) PRH network with n = 2.
DISPERSION AND ATTENUATION….
131
Outside the band, the S21 phase dependence has not too much
significance [7]. Hence, when out-of band behavior of the phase is ignored, it
can be easily seen, that the phase of S21 tends to the infinity when the frequency
tends to zero (Fig. 3). Therefore, for the PLH transmission lines, the phase
reference should be established at very large (theoretically infinite) frequencies
rather than in 0. For this reason, one of the data trace in Fig. 3 was shifted up in
order to have the same reference. It can be seen that, the unwrapped phase is the
same for both PLH ladder network with n = 1 (fcPL=1.59 GHz) and with n = 2
(fcPL=796 MHz) for large frequencies, far from the cut-off frequency.
3. CRLH line - Unbalanced case
For a general CRLH transmission line, shunt and series resonance
frequencies fse and fsh are defined by
f se =
1
1
and f sh =
2π LR C L
2π LL C R
(3)
In the unbalanced case f se ≠ f sh and the frequency response presents a gap
between the shunt and series resonance. In our case fsh < fse (Fig. 4). However
they may change places without significant modification in the CRLH
properties. The maximum gap attenuation in the dependence of the |S21| versus
frequency corresponds to f 0 =
fR fL =
f se f sh . When CRLH ladder network
with identical cell, but different cell numbers N, are considered, both the cut-off
frequencies and the shunt and series resonance frequencies are better
emphasized for large N.
132
I.MIHAI ET AL.
Fig. 4. |S21| dependence versus frequency for three CRLH networks with identical cells.
a) N =10, b) N =20, c) N =80.
The real and imaginary part of the complex propagation constant  are
presented in Fig. 5 for a CRLH line with LR = 2 nH, CR = 1 pF, RR = 0, LL = 2.5
nH, CL = 0.75 pF, GL = 0. For comparison, the propagation constant of a PRH
line with LR = 2 nH, CR = 1 pF and of a PLH line with LL = 2.5 nH, CL = 0.75 pF
are presented. In this case, fse = 4.11 GHz, fsh = 3.18 GHz.
The dispersion/attenuation diagram for CRLH transmission line shows
a lower pass-band (f<fsh), a band-stop (fsh<f<fse) and an upper pass-band f>fse. In
the stop-band, the wave is attenuated despite the loss-less components of the
ladder network.
On one hand, at low frequencies, the propagation constant RL of the
CRLH transmission line approaches asymptotically by the propagation constant
L of the perfectly left-handed transmission line. On the other hand, at very
high frequencies, RL tends to R of the conventional right-handed transmission
line (Fig. 5).
DISPERSION AND ATTENUATION….
133
Fig. 5. Dispersion/attenuation diagrams a) |RL |, RL <0 for f< fsh and RL > 0 for f> fse
(fse = 4.11 GHz, fsh = 3.18 GHz)b) RL , c) propagation constant R of the PRH transmission
line,d) absolute value |L| propagation constant L <0, for the PLH transmission line.
Fig. 6. Real and imaginary part of the characteristic impedance of an unbalanced CRLH line.
a) Real(Zc), b) Imag(Zc) (fse = 4.11 GHz, fsh = 3.18 GHz).
134
I.MIHAI ET AL.
The gap between the left-handed and right-handed ranges of a CRLH
transmission line can be also observed in the dependence of the line
characteristic impedance
Zc = Z L






2
f 
 −1
f se 
2
f 
 −1
f sh 
(4)
versus frequency, as it can be noticed in Fig. 6. When the frequency approaches
to 0, the characteristic impedance of the CRLH line approaches to the ZL= 57.73
 of the left-handed line. In addition, the characteristic impedance has a pole at
fsh and a null at fse.
The dependence of the unwrapped phase of S21 the CRLH ladder
network versus frequency (Fig. 7) shows discontinuities in the first derivative at
both series and shunt resonance frequencies. Furthermore, neither the phase nor
the S11 are significantly influenced by moderate loses introduced in the CRLH
line (cell quality factor Q = 100 – 500). However, the magnitude of S21 is more
influenced by the losses especially for frequencies close to the cut-off
frequencies fcL and fcR (Fig. 8).
Fig. 7. Unwrapped phase of S21 versus frequency (fse = 4.11 GHz, fsh = 3.18 GHz).
DISPERSION AND ATTENUATION….
135
Fig. 8. Magnitude of the S parameters of a lossy un-balanced CRLH transmission line.
a) |S21| for Q =100, b) |S11| for Q =100, c) |S21| for Q =500, d) |S11| for Q =500.
4. CRLH line - balanced case
For the balanced case fsh=fse which corresponds to the closure of the gap
between the left-handed and right handed ranges. From the balanced case, result
immediately that the characteristic impedance Zc = ZL = ZR does not depend on
frequency. For the balanced case, the cut-off frequencies of the CRLH ladder
network are given by simpler expressions [7]
f cLbal = f R 1 − 1 +

fL
f 
and f cRbal = f R 1 + 1 + L  .

fR
f R 

(5)
The |S21| dependence on the frequency for the lossy (Q = 100 and Q =
500) lines are presented in Fig. 9. For the CRLH network, the parameters values
were LR=2.5 nH, CR=1 pF, LL=2 nH and CL=0.8 pF. The calculated cut-off
frequencies fcLbal=1.59 GHz, fcRbal=7.96 GHz are in agreement with the values
from the data presented in Fig. 9.
136
I.MIHAI ET AL.
Fig. 9. Magnitude of the S parameters of a lossy balanced CRLH transmission line.
a) |S21| for Q =100, b) |S11| for Q =100, c) |S21| for Q =500, d) |S11| for Q =500.
Fig. 10. Unwrapped phase of S21 versus frequency for the lossy balanced CRLH line a) Q =100,
b) Q =500
DISPERSION AND ATTENUATION….
137
For the balanced case, since fsh=fse , the dependence of the unwrapped
phase versus frequency does not show at these frequencies any discontinuities
in the first derivative as for the unbalanced case, as shown in Fig. 10.
5. Conclusions
At low frequencies, the CRLH transmission line becomes equivalent to
the perfect left-handed transmission line. On the other side, at high frequencies,
the equivalent parameters of the CRLH structure tend to the equivalent nondispersive parameters of conventional positive right-handed structure.
The balanced condition allows matching over an infinite bandwidth
because the characteristic impedance of the balanced transmission line is
frequency independent.
Due to their special dispersion properties, the CRLH structures are
attractive to applications in microwave range.
References
[1] N. Engheta, R.W. Ziolkowski, Metamaterials – Physics and Engineering
Explorations, Wiley-IEEE Press, New York, 2006.
[2] R.W. Ziolkowski, F. Auzanneau, “Passive artificial molecule realizations of
dielectric materials”, J. Appl. Phys, 82 (7), 1 Oct 1997, pp. 3195-3196.
[3] I. Awai “Artificial Dielectric Resonators for Miniaturized Filters”, IEEE
Microwave Magazine, 2008, pp. 55-64.
[4] P.M.T. Ikonen, S. Tretyakov, “Determination of Generalized Permeability
Function and Field Energy Density in Artificial Magnetics Using the
Equivalent-Circuit Method”, IEEE Trans. Microw. Theory Techn. Vol. 55, No.
1, 2007, pp. 92-96.
[5] C. Caloz, H. V. Nguyen, “Novel broadband conventional- and dualcomposite right/left-handed (C/D-CRLH) metamaterials”, Appl. Phys. A 87, pp.
309-316, 2007.
[6] ***, Advanced Design System 2008, Agilent Technologies, Santa Clara,
May 2008.
[7] G.V. Eleftheriades, K.G. Balmain, Negative Refraction Metamaterials:
Fundamental Principles and Application, Wiley-IEEE Press, 2005.
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
STRUCTURE AND MAGNETORESISTENCE OF
ELECTRODEPOSITED GRANULAR Co-Ni-N THIN
FILMS
Cristina SÎRBU, Lavinia VLAD, Sorin Iulian TĂNASE, Marius DOBROMIR
Violeta GEORGESCU
Abstract The granular Co-Ni-N films were electrodeposited onto an aluminium substrate, from a complex
solution based on CoSO4.7 H2O, NiSO4.7 H2O, and NiCl2.6H2O and some additional substances. The process
at the cathode interface includes the electrochemical reduction of NaNO3 to nitrogen (N2) by a mechanism of
cationic catalysis. The samples were investigated by X-ray Photoelectron Spectroscopy (XPS) for the
elemental composition, X-ray diffraction (XRD) for the crystalline structure and torsion magnetometer for the
magnetization behaviour. Magnetoresistance (MR) measurements at room temperature were carried out in dc
magnetic field applied perpendicular to the current, in plane of the film. Co-Ni-N granular films display
tunneling magnetoresistance effect (2-300%), which could be explained mainly by the elastic spin dependent
scattering of conduction electrons at the interface between magnetic granules (composed of Co-Ni solid
solution) and nonmagnetic regions (rich in N inter-granular frontiers and aluminium oxidized substrate).
1. INTRODUCTION
Spin-polarized transport in nano-structured systems has attracted much
attention due to its importance in both basic research and technology. In
heterogeneous granular films with ferromagnetic granules embedded in a
nonmagnetic metallic matrix, the resistance of films could exhibit large changes
upon applying a magnetic field (GMR effect) and has potential technical
applications in the micro-system technology [1-6]. Tunnelling
magnetoresistance (TMR) appears in the artificial layered or granular structures
composed of ferromagnetic metal-insulator-metal junctions, where electrons
tunnel through an insulating barrier. Our purpose was to obtain these effects in
electrodeposited Co-Ni with controlled impurities granular magnetic films. The
formation of granules in Co-Ni films was imposed by introducing nitrogen in
electro-deposit. There are publications dealing with the magnetoresistance of
electroplated Co-Ag granular films [1,3], but we have not found in the literature
any study referring to magnetic and transport properties in the case of
electrodeposited Co-Ni granular films with addition of nitrogen.
We used electrodeposition method for preparation of Co-Ni-N magnetic
thin films because it is a low-price procedure and the Co-Ni alloy deposits can
be easily prepared by electrodeposition related to other methods [7-9]. In the
same time, the Co-Ni alloy thin films could be used in the various magnetic
devices, due to their functional and technological properties.
STRUCTURE AND MAGNETORESISTANCE...
139
2. EXPERIMENTAL
The method for electrodeposition of Co-Ni thin films with additions of
Mg and N was described previously in [10-12]. The granular thin films were
electrodeposited on aluminium substrate, from a solution (labelled A)
containing: 0.10M CoSO4.7 H2O, 0.17M NiSO4.7 H2O, 0.04M NiCl2.6H2O,
0.15M Na2SO4.10H2O, 0.48M H3BO3, 0.68.10-2 C6H5Na3O7.2H2O, 0.17M
NaCl, 0.13.10-2M C12H25NaO4S in double distilled water. This solution is
considered as a base for introducing NaNO3 in different concentration (c), as a
source for nitrogen inclusion in the films. We aimed to study in this work the
influence of the c values and the pH values of the solution both on the deposit
morphology and, as a consequence, on the magnetic and transport properties of
the films.
The study of crystalline structure was performed by X-ray diffraction
(XRD) using Cu-Kα radiation (λ = 0.154 nm) in (θ - 2θ) standard geometry.
Elemental composition and nitrogen inclusion in the granular films were
identified by X-ray photoelectron spectroscopy (XPS) technique.
Magnetoresistance measurements were performed by using two-terminal
gold pressure contacts with a constant current of 7 µA applied in the plane of
the film (CIP configuration). The dc magnetic field (between −300 and +300
kA/m) was applied in the film plane, perpendicular to the current (transversal
CIP configuration). Measurements were carried out with an HM 8112-2
programmable multimeter at room temperature. Magnetoresistance (MR) was
defined as
R (H ) − R (H s )
(1)
× 100%
MR =
R (H s )
where R(H) denote the resistance of the sample measured in a field H and R(HS)
is the resistance measured in the maximum applied magnetic field.
Magnetic behaviour of the samples was investigated by torque
magnetometry, at room temperature.
3. RESULTS AND DISCUSSIONS
In a first stage of this work we electrodeposited a series of samples
from the base solution (A), by adding NaNO3 in different concentration (c) with
the aim to obtain granular films with different nitrogen inclusions. These
140
C.SIRBU et al.
samples were labelled S1, S2, S3, S4, S5, taking into account the values of c
(c=0, c=0.33, c=1, c=1.33 g/l, c=1.66 g/l, respectively). The electrodeposition of
films in this series was performed in the same conditions, with the following
plating parameters: T=65oC, pH=5, without stirring. The thickness of the films
was controlled by the electric charge passing through electrolyte during plating
process. The process at the cathode/electrolyte interface includes the
electrochemical reduction of NaNO3 to nitrogen (N2) by a mechanism of
cationic catalysis [12-15].
Nitrogen inclusion in the granular films was identified with the X-ray
photoelectron spectroscopy (XPS) technique. We investigated the samples in
this series, and we give as example in figure 1 the spectrum of the sample S1
deposited from the solution A without NaNO3 (the N1s line at a binding energy
of around 400 eV, characteristic for N, does not appear in this spectrum).
Co 2p3
Ni LMM
20000
0
1000
800
600
C 1s
Ni LMM
Ni LMM
40000
O 1s
60000
Co LMM
Ni LMM
80000
Ni 2p3
Co LMM
intensity (c/s)
100000
400
200
0
Binding Energy (eV)
Fig. 1 XPS spectrum of the sample S1, electrodeposited from the main solution (c=0)
From the XPS analyses of this series of samples we have found that the
nitrogen is included in the films in different percentage, as a function of the
NaNO3 concentration of the solution. We chose for exemplification of XPS
measurements the sample S5, electrodeposited from a solution containing
c=1.66 g/l. XPS spectrum of this sample is shown in figure 2. The sample S5
electrodeposited from the bath containing 1.66 g/l NaNO3 has 4.5at % nitrogen
on the frontier of the granules (fig. 2a), and 0.3 at % nitrogen when the spot is
focalized on the granules (fig. 1b). Taking into account that the information
depth of the XPS measuring technique is limited to 10 nm (or about 50 atomic
layers), it results that the nitrogen content in the granules frontier is larger than
in the granule itself.
The mechanism of the electrochemical reduction of nitrate is explained
by the theory of cationic catalysis [12-15]. The non–reacting cation of the
STRUCTURE AND MAGNETORESISTANCE...
141
30000
N 1s
5000
Co LMM, Ni LMM
20000
15000
10000
5000
0
b)
Co 2p3, Ni LMM
Co LMM
Ni LMM
10000
O 1s
C 1s
Co LMM, Ni LMM
15000
Ni LMM
intensity (c/s)
20000
intensity (c/s)
25000
C 1s
a)
O 1s
Co 2p3, Ni LMM
Co LMM
N 1s
25000
Ni 2p3
Ni 2p3
supporting electrolyte in the electrolyte-cathode interface, acts as an attracting
centre for the NO3 ions, by forming virtual ion pairs of the type Na….NO3,
which can not be repelled by the negatively charged electrode [13-15].
0
1000
800
600
400
200
1000
0
800
600
400
200
0
Binding Energy (eV)
Binding Energy (eV)
Fig. 2. XPS spectra taken on the frontier (a) and on the centre of granules (b) for the sample S5,
electrodeposited from the bath with c=1.66 g/l NaNO3
4000
2300
a)
b)
2200
3800
intensity (c/s)
intensity (c/s)
2100
3600
3400
3200
2000
1900
1800
1700
1600
3000
1500
2800
400
402
404
406
Binding Energy (eV)
408
410
1400
400
402
404
406
408
410
Binding Energy (eV)
Fig.3. N 1s XPS spectrum taken on the frontier between grains (a) and on the grain central part
(b) for the sample S5 electrodeposited from the bath containing c=1.66g/l NaNO3
We suppose that inclusion of N in Co-Ni films electrodeposited in our
complex baths is due to cationic catalysis processes mainly.
XRD data provided lattice parameters and confirmed the hcp nature of
the Co-Ni solid solution of granules in the films. As an example, the X-ray
diffraction pattern recorded on the sample S4 films is shown in fig. 4. From the
XRD patterns we calculated the following values of lattice parameters:
a=2.43÷2.51 Ǻ and c=4÷4.08 Ǻ for films in this series. The lattice parameters of
our films are very close to the parameters for the bulk samples of Co (a=2.5 Ǻ,
142
C.SIRBU et al.
c=4.08 Ǻ). The data are consistent with reach in Co composition of the films
(around 80-85 at% Co, 20- at % Ni, 0-4.5 at% N) determined by SEM
measurements.
Co (002)
100
80
intensity
Al(111)
Co (202)
60
Al (113)
Mg (202)
40
O2(115)
O2(334) Al (222)
20
0
20
40
60
80
100
120
2θ (degrees)
Fig.4 XRD pattern of the sample S4 electrodeposited from bath with c=1.33 g/l NaNO3
The addition of NaNO3 in the electrodeposition bath exhibits a strong
influence on the magnetoresistance of the thin granular films. All the samples
have significant values of magnetoresistance. The sample S3 exhibits the largest
MR effect in this series of samples (figure 5). Due to special morphology of the
film, composed of magnetic grains (of Co-Ni solid solution) and nonmagnetic
regions (with rich in N inter-granular frontiers and aluminium oxidized
substrate), this could be considered as tunneling magnetoresistance.
300
b)
a)
60
250
200
MR %
MR (%)
40
20
150
100
50
0
0
-50
-200
-100
0
H (kA/m)
100
200
-200
-100
0
100
200
H (KA / m)
Fig. 5 Magnetoresistance of the sample S3 (a) and P1 (b) as a function of applied magnetic field
To understand better the phenomena, the second stage of the work was to
study the influence of pH bath on the properties of the granular thin Co-Ni-N
STRUCTURE AND MAGNETORESISTANCE...
143
films. A new series of samples were electrodeposited in galvanostatic
conditions from the same solution as that used for electrodeposition of the
sample S3. The parameters were maintained the same for all samples in this
series (labeled P1, P2, P3, P4, for pH=2, pH=2.5, pH=2.8, pH=3.5 respectively).
The bath parameters had the values: T=30°C, t=9 min, c=1.33 g/l NaNO3
without stirring up the bath. The thickness of the samples was of about 200nm.
We present for exemplification in table 1 the MR values for these
samples:
Sample
pH
MR %
Table 1 MR values as function of the bath pH
S3
P1
P2
P3
5
3.5
2.8
2.5
60
300
110
75
P4
2
0.7
As we can see from the table 1, the magnetoresistance reaches a
maximum value for the samples deposited from solutions with pH around 3.5
(fig. 6). This could be explained by the occurrence of special catalysis effects
simultaneous with intensive H discharge in the interfacial region in this range of
pH. Therefore, the kinetic of the electrodeposition in interface is influenced by
pH and, consequently, the formation of grains and of grain frontiers (i. e. film
morphology) depends on these kinetic factors. The transport phenomena and
magnetoresistance of the samples is directly dependent on the sample
morphology. Although the MR values are larger for pH=3.5, the reproducibility
of the film properties is no so convenient in this region; therefore we
recommend for technological application at pH=5.0 electrodeposition of films.
The dependences of the magnetic properties of the films (i.e. coercive
field Hc fig. 7a, magnetic anisotropy constant Keff fig. 7b) were studied as a
function of the NaNO3 concentration (c) in the solution. The values of these
characteristics varied in the range Hc = (5 - 70) kA/m, Keff = -20*104 J/m3 to
+40*104 J/m3 (fig. 7).
300
250
MR %
200
150
100
50
0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
pH
Fig. 6. Dependence of the granular films magnetoresistance on the bath pH (for the second series
of samples deposited from the solution A, in galvanostatic conditions)
144
C.SIRBU et al.
50
80
(a)
(b)
Hc
70
40
60
30
20
Keff(a.u)
50
Hc(kA/m)
Keff
10
40
0
30
-10
20
-20
10
-0.5
0.0
0.5
1.0
1.5
c(g/l)
2.0
2.5
3.0
3.5
-30
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
c(g/l)
Fig.7. The dependence of (a) HC and (b) Keff of Co-Ni-N electrodeposited films on the NaNO3
concentration of the electrolytic solution
When the concentration of NaNO3 from the bath increases, coercive field and
the magnetic anisotropy constant increase too, as one can see from fig. 7, due to
modification of kinetic regime of electrodeposition.
3. CONCLUSIONS
We prepared by electrochemical method good quality granular Co-Ni-N
thin films on Al substrate. It was studied the influence of the deposition
parameters (in particular the additive concentration c, and pH of the bath) on the
transport properties and magnetic properties of the films. We have found that
the crystalline structure, transport and magnetic properties are strongly
influenced by the addition in the electrolytic bath of the NaNO3 sel, which
induces inclusion of N in electrodeposited film by cationic catalysis.
We obtained granular films with MR in the range 2 – 300%. TMR
effect could be explained mainly by the elastic spin dependent scattering of
conduction electrons at the interface between magnetic grains (composed of CoNi solid solution) and nonmagnetic regions (rich in N inter-granular frontiers
and aluminium oxidized substrate).
STRUCTURE AND MAGNETORESISTANCE...
145
The large TMR effect obtained for Co-Ni-N granular films make them
useful for technological applications.
5. REFERENCES
[1] T. Y. Chen, S. X. Huang, C. L. Chien and M. D. Stiles, Phys. Rev. Lett., 96 (20), 207203
(2006).
[2] A.E. Berkowitz, J.R. Mitekell, M.J. Corey, et al., Phys.Rev. Lett., 68, 3745 (1992).
[3] Ch. Wang, Zh. Guo, Y. Rong, T. Y. Hsu, J. Magn. Magn. Mater., 277, 273 (2004).
[4] T.-S. Chin, J. Magn. Magn. Mater., 209, 75 (2000).
[5] J.W. Judy, R.S. Muller, J. Microelectromechanical Sys., 6 (3), 249 (1997).
[6] T. Homma, Y. Sezai, T. Osaka, Y. Maeda, D.M. Donnet, J. Magn. Magn. Mater.,
[7] N.V. Myung, D. Y. Park, B. Y. Yoo, P. T. A. Sumodjo, J. Magn. Magn. Mater., 265, 189
(2003).
[8] S.N. Srimathi, S.M. Mayanna, B.S. Sheshadri, Surf. Technol. 16_1982.277–322.
[9] S.S. Abd El-Rehim, A.M. Abd El-Halim, M.M. Osman, J. Appl. Electrochem. 15_1985.107–
112.
[10] V. Georgescu, Mat. Sci. Eng., B27, 17 (1994).
[11] V. Georgescu, M. Georgescu, J. Magn. Magn. Mater., 242-245, 416 (2002).
[12] C. Sîrbu, V. Georgescu, J. Opt. Adv. Mater. 10 (9), 2396 (2008).
[13] Polatides, C., Kyriacou, G., J. Appl. Electrochem., 35, 421 (2005).
[14] I. Katsounaros, D. Ipsakis, C. Polatides, G. Kyriacou, Electrochimica Acta, 52, 1329 (2006).
[15] Kotov, V.Yu, Tsirlina, G.A., Russ. Chem. Bull., 52, 2393 (2003).
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
METAL NANOTUBES PREPARED BY SOL-GEL
METHOD
G. Călin, A. Coroabă, F. Brînză, N. Suliţanu
Faculty of Physics, Al. I. Cuza Univ., 11 Bv. Carol I, 700506 Iasi, Romania
[email protected]
Abstract
In this paper, we report an easy method for fabrication metal nanotubes within anodic aluminum
oxide (AAO) templates by a sol-gel method followed by a hydrogen reduction procedure.
Using sol-gel method procedure proposed in this paper for the preparation of metal nanotubes, tube
length, diameter, wall thickness and, most importantly, the chemical composition can be adjusted
conveniently. Metal nanotubes (Pb, Co, Ni and Mg) were prepared using anodic aluminum oxide (AAO).
The method for obtaining metallic nanotubes consists in electrochemical deposition of metals in
pores of a porous semiconductor template (matrix) under the action of voltage pulses. Application of the
voltage pulses in the electrochemical deposition allows us to obtain metallic nanotubes with wall thickness
electrical parameters dependent and by deposition time in a single electrochemical deposition process.
It is a great challenge to produce metal nanotubes with a designed length, wall thickness and
chemical components. This method can be used to fabricate a variety of metallic nanotubes, of different
chemical composition.
INTRODUCTION
The nanotubes represents a new generation of nanomaterials.
Martin and colleagues were manufactured for the first time Au
nanotubes by using Au electrodeposition in the pores of an anode of AAO
(anodized aluminum oxide) aluminum oxide.
Sol-gel method is simple and easy to use in preparation of onedimensional nanomaterials and has been used successfully in the manufacture
of nanotubes with oxides.
The method for manufacturing metallic nanotubes consists in
electrochemical deposition of metals inside the pores of a porous semiconductor
template (matrix, print) under the action of voltage pulses.
EXPERIMENTAL APPROACH
The AAO substrate preparation
An aluminum foil (Al) (99.99 % purity) was skimmed and then heattreated at the temperature of 500  C for about 4 hours. The foil obtained after
the treatment has been used as anode in an aqueous solution of oxalic acid
METAL NANOTUBES PREPARED BY SOL GEL METHOD
147
( H 2 C 2 O4 ) of 0,3M concentration, being subjected to a constant potential of
42V, at the temperature of 30  C for 9 hours.
The colloidal solution preparation
The colloidal solutions were obtained using ethylene glycol as solvent,
citric acid as corrosion agent and dissolved salts of the corresponding metals as
solutions. Metallic salts used to produce Fe, Pb, Ni, Co and Mg solutions are:
iron nitrate, lead acetate, nickel acetate, cobalt acetate and magnesium nitrate.
To obtain the colloidal solution, the solution was heated at 60  C for 12 hours
using a thermostat.
Nanotubes preparation:
Process 1
At the beginning the substrate was submerged in the colloidal solution
for 10minutes. With ultrasonic vibrations, the colloidal solution has penetrated
into the pores of AAO substrate.
Process 2
The AAO support are incorporated in the pores the colloidal solution,
has been removed from the colloidal solution and then the support was heated
from room temperature ( 28  C ) to 375  C with a warming rate of 1,5  C / min
on the air. After the second process was composed the nanotubes from the
metallic oxides into the pores of the AAO substrate.
Process 3
Finally, nanotubes have been obtained by additional burning at 600  C
in the hydrogen atmosphere for one hour, except lead nanotubes that were
burned at 350  C .
RESULTS AND DISCUSSIONS
Following the analysis of electronic microscope was observed a selforganized structure of the brick-wall type.
148
G. CALIN ET AL.
Figure 1. The self-organized structure of the aluminum oxide substrates.
Metallic nanotubes morphology was observed after the dissolution of
the substrate in the aqueous solution of NaOH and after washing several times
with distilled water.
Figure 2. SEM images for anodizing the aluminum in oxalic acid, for 10h at 42V:
A) with lead precipitation from the lead acetate precursor;
B) with nickel precipitation from the nickel acetate precursor;
C) with magnesium precipitation from magnesium nitrate precursor;
D) with cobalt precipitation from cobalt acetate precursor.
METAL NANOTUBES PREPARED BY SOL GEL METHOD
149
Figure 3. Cross-section SEM image of nickel nanotubes.
CONCLUSIONS
The aluminum anodizing method to obtain the matrix for nanotubes can
be realized in laboratory.
This method can be used to manufacture a variety of metallic nanotubes
with different chemical compositions.
The small amount of deposited material did not allow a structural analysis
of nanotubes.
REFERENCES
1. Zhenghe Hua, Shaoguang Yang, Hongbo Huang, Liya Lv, Mu Lu, Benxi
Gu and Yaouwei, Metal nanotubes prepared by a sol-gel method followed
by a hydrogen reduction procedure, Nanotechnology, vol. 17, no. 20, pp.
5106-5110, 2006.
2. Wang Q T, Wang G Z, Han X H,Wang X P and Hou J G, Controllable
Template Synthesis of Ni/Cu Nanocable and Ni Nanotube Arrays: A OneStep Coelectrodeposition and Electrochemical Etching Method, J. Phys.
Chem. B, vol. 109 (49), pp 23326–23329, 2005.
3. Mu C, Yu Y X, Wang R M, Wu K, Xu D S and Guo G L, Uniform Metal
Nanotube Arrays by Multistep Template Replication and Electrodeposition,
Adv. Mater. vol. 16, pp. 1550 – 1553, 2004.
4. Y. Fukunaka, M. Motoyama, Y. Konishi, and R. Ishii, Producing Shape–
Controlled Metal Nanowires and Nanotubes by an Electrochemical Method,
Electrochemical and Solid–State Letters, vol. 9, pp. C62-C64, 2006.
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII „AL.I.CUZA” DIN IAŞI
Tomul LIII-LIV, Fizica, 2007-2008
ON THE MECHANISM OF ELECTRICAL CONDUCTION IN
THIN-FILM METAL/ZnSe/METAL HETEROSTRUCTURES
Mihaela Diciu*, G.I. Rusu**,
*,,Traian’’ Technical College, 203 Traian Blvd., 800186, Galaţi,
Romania
** Faculty of Physics, ,,Al.I. Cuza’’ University, 11 Carol I Blvd.,
700506, Iaşi, Romania
ABSTRACT
Thin-film sandwich-type cells of metal/ZnSe/metal have been prepared by the thermal
evaporation under vacuum (quasi-closed volume technique).
The static current-voltage ( J − U ) characteristics for respective heterostructures have been
studied. It was established that for lower values of the applied electric field ( E < 10 4 V / cm ) the
characteristics are ohmic. Some non-ohmic effects have been observed for higher electric fields,
E > 10 5 V / cm .
It has been also shown that in function of the nature of the metal or the film thickness of
sandwich-type cells the barrier heights at metal-semiconductor interface are different. Different
Schottky-barrier heights have been found, ranging from 0.80eV to 0.84eV.
KEYWORDS: sandwich-type cells, Schottky-barrier, current-voltage characteristics.
1. INTRODUCTION
In the last years, theoretical and experimental investigations on the
electronic transport and optical properties of polycrystalline semiconducting
films have been much intensified [1-4]. This fact is due to important
applications of these films in various modern and future technologies of solidstate devices (sensors, detectors, solar cells, etc.). It is well known, that the
polycrystalline films are composed of a large number of crystallites (having
similar/different shape and size) joined together by crystallite (grain),
boundaries [1-8].
Several models are currently used for explaining the mechanism of
electrical conduction in polycrystalline semiconducting films. The basic models
were developed by Volger [5], Petritz [6], Seto [7, 8], etc.
Generally, in these models, the polycrystalline film is divided into similar
crystallites (grains), consisting of two distinct domains: the ,,bulk”
semiconductor at the centre of the crystallite (the kernel), and the boundary
(intercrystalline) domain which surrounds this centre. The crystallite boundaries
contain surface state (due to the impurities, defects, etc.) which can trap the free
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
151
carriers and consequently, a space charge region occurs. This process
determines the bending of energy bands near the crystallite boundary and the
potential barriers to the electronic transport results [9, 10].
In a series of previous papers [9, 11, 12], we showed that Seto’s model [7,
8] with several modifications proposed by Baccarani, Ricco and Spandini [13]
could explain the mechanism of electron transfer in thin films of binary
semiconducting compounds (ZnSe, CdS, Bi2O3, TiO2, ZnO, etc.) In all cases, the
measurements were performed using surface-type cells (planar configurations)
[14-16].
In this paper the mechanism of electrical conduction in ZnSe thin films is
studied on the thin-film sandwich structures of the metal/semiconductor/metal
type. In these systems, some nonohmic effects have been observed for high
electric fields [11, 12, 17, 18].
2. EXPERIMENTAL
The thin-film sandwich structures of the In/ZnSe/In or Au/ZnSe/Au type
were deposited onto glass substrate.
The quasi-closed volume technique was used [15]. Details on the
deposition equipment are reported in previous papers [9, 14].
The investigated samples were prepared using the following deposition
condition parameters: the source (evaporator) to substrate distance, D, was 8
cm; source temperature, Tev, was 900K-1300K, deposition rate, rd ranged
between 0.15 and 4.49nm/s; substrate temperature, Ts, was 300K.
Thicknesses of the films were measured interferometrically, using
multiple-beam Fizeau fringe method at reflection of a monochromatic light
(λ=551nm) [19] and for studied ZnSe films varied from 0.210µm to 1.050µm.
The films crystalline structure was investigated by X-ray diffraction
(XRD) technique (using a DRON-2 diffractometer and CoKα and CuKα
radiations λCoKα=1.7889Å, and λCuKα=1.5418Å).
Surface morphology was examined by means of atomic force microscopy
(AFM).
The temperature dependence of the electrical conductivity, σ, was studied
using surface type cells [9, 14]. In this case, the substrates were equipped with
two parallel thin-film indium or gold electrodes deposited, (by thermal vacuum
evaporation) before the deposition of ZnSe films. For studied samples, the thin
film electrodes were separated by a gap of 2-5mm and the width of each
electrode was 8-10 mm. The applied electrical fields have low intensities
152
M. DICIU et al.
( E < 1.73 × 10 3 V / cm ). Under these conditions non-ohmic effects in ZnSe
films were not observed [16, 22].
For study of the current- voltage ( J − U ) characteristics, the sandwichtype cells were prepared by deposition onto substrates of the metallic thin film
(bottom electrode), then of the investigated ZnSe film and of the second
metallic film (top electrodes).
Fig. 1 showed schematically sandwich arrangement of the samples for the
study of the current-voltage characteristics. The thickness of metallic electrodes
(for surface-type and sandwich-type cells) varied between 1.30µm-2.70µm. The
contacted area of films for sandwich arrangement was 3× 3 mm2.
The measurements were performed using a 6717 Keithley electrometer
and a 2010 Keithley multimeter. The experimental arrangements used for the
investigation of the temperature dependences of the electrical conductivity and
Seebeck coefficient (thermoelectric power) were similar to those presented in
[14-17].
2
Fig.1 Sandwich arrangement of the sample for
measuring the current-voltage characteristics (1substrate, 2-bottom electrode, 3-thin film, 4-top
electrode)
4
3
1
3. RESULTS AND DISCUSSION
For a large number of inorganic [9, 14, 16, 20] and organic [21]
semiconducting materials, we have experimentally established that thin film
samples with stable solid-state structure can be obtained if, after preparation, the
respective samples are subjected to a heat treatment. Generally, this consisted of
several heating/cooling cycles within a determined temperature range, ∆T . The
number of cycles, temperature rate, initial and final values of temperature range,
∆T , heating and cooling under vacuum or normal atmosphere, etc, strongly
depend on the semiconductor nature, deposition method and preparation
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
153
conditions, film thickness, structural characteristics of the sample, presence of
some exterior factors (electric fields, optical radiations, etc) [9, 14, 20, 22].
The study of the temperature dependence of the electrical conductivity
during this heat treatment may provide information on the processes taking
place in the respective films (removal of adsorbed and/or absorbed gases,
reduction in the concentration of the structural defects, modification of
crystallite size and shape in polycrystalline thin films, etc.) [9, 21, 22]. After
heat treatment, the temperature dependence of the electrical conductivity
becomes reversible. This fact indicated the stabilization of the film structure
within the respective temperature range. A detailed analysis of temperature
dependence of the electrical conductivity during the heat treatment is described
in a series of our previous papers [9, 14, 16, 22]. For ZnSe films, some obtained
results have been reported in [22]. All experimental date presented and
discussed in present paper have been obtained for heat- treated ZnSe films.
Typical shape of ln σ = f (10 3 / T ) curves during heat treatment is
illustrated in Figs.2 and 3 for two studied samples. The respective samples were
subjected to two successive heating/cooling cycles within temperature
ranges ∆T = 300 K − 600 K .
After heat treatment the temperature dependences of the electrical
conductivity becomes reversible.
-2
-3
ln[σ(Ω−1cm-1)]
-4
-5
-6
sample ZS.022
d=0.221µm,rd=0.15nm/s
first heating
first cooling
second heating
second cooling
-7
-8
-9
-10
-11
1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6
103/T[K-1]
Fig.2 The temperature dependence of the electrical conductivity during the
heat treatment for a ZnSe thin film (annealing temperature, Ta=600K)
154
M. DICIU et al.
-3
sample ZS.064
d=0.641µm, rd=0,71nm/s
first heating
first cooling
second heating
second cooling
-4
-5
ln[σ(Ω−1cm-1)]
-6
-7
-8
-9
-10
-11
1,8
2,0
2,2
2,4
2,6
2,8
3,0
3,2
3,4
3,6
103/T(K-1)
Fig.3 The temperature dependence of the electrical conductivity during the
heat treatment for a ZnSe thin film (annealing temperature, Ta=500K)
The XRD patterns indicate that ZnSe films are polycrystalline and have a
cubic (zinc blende) structure (Fig. 4 a, b). It was experimentally established that
as-deposited ZnSe thin films show a preferential orientation of crystallites with
the (111) planes parallel to the substrate surface. For heat-treated samples, the
preferential orientation of crystallites, with (111) planes, parallel to the substrate
is maintained (Fig.4b).
(111)
Intensity(arb.untis)
Intensity(arb.units)
(111)
(220)
20
30
40
50
60
20
30
40
50
60
Fig.4 Effect of the2θ(degrees)
heat treatment on XRD patterns for a ZnSe thin
film, (film thickness,
2θ(degrees)
d=0.820µm, deposition rate, rd=1.50nm/s, substrate temperature, Ts=300K), a. before
heat treatment, b. after heat treatment (annealing temperature, Ta=550K, annealing time,
ta=30min)
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
155
he average crystallite size of the films was calculated using DebyeScherrer formula [23, 24]
kλ
(1)
β cos θ B
where k denotes Scherer constant ( k = 0.90 [23, 24]), λ is X-ray wavelength,
D=
β
represents the full–width of half maximum of the respective diffraction peak
and θ B is Bragg diffraction angle.
The obtained values for crystallite size are indicated in Table 1.
Table 1 The values of the crystallite size for ZnSe thin films calculated from XRD
patternsa
2θ(degrees)
(hkl)
D(nm)
27.32
111
19.89
27.32
111
24.14
27.32
111
45.61
ZS.082B
0.820
1.50
300
53.60
220
28.70
ZS.022A
0.225
2.35
300
27.32
111
20.10
ZS.035A
0.350
1.71
300
27.32
111
24.92
ZS.082A
0.820
1.50
300
27.32
111
46.07
a
d, film thickness; rd, deposition rate; Ts, substrate temperature; θ, Bragg angle; (hkl),
Miller indices; D, crystallite size, B-before heat treatment, A-after heat treatment.
Sample
ZS.022B
ZS.035B
d(µm)
0.225
0.350
rd(nm/s)
2.35
1.71
Ts(K)
300
300
The AFM images (Fig. 5 a, b) confirm polycrystalline structure of the
films.
a
b
Fig.5 AFM images (scan size:2µm × 2µm) for ZnSe thin films with various thickness:
a. film thickness, d=0.268µm, deposition rate, rd=2,57nm/s, substrate temperature,
Ts=300K; b. film thickness, d=0.415µm, deposition rate, rd=2.14nm/s, substrate
temperature, Ts=300K.
156
M. DICIU et al.
The crystallite size increases with increasing the film thickness and
substrate temperature during film deposition [22]. Some correlations between
the structural characteristics of ZnSe films and their deposition conditions are
presented in [20, 22].
For a series of thin-film sandwich structure of the In/ZnSe/In or
Au/ZnSe/Au type the current-voltage characteristics (at room temperature) for
ZnSe films with different thickness have been investigated.
Generally, the shape of J − U characteristics depends on the metallic
electrode materials, thickness and structure of film, values of applied electric
fields, etc.
Some mechanisms of electrical conduction are possible to be active in
studied thin films [19, 25, 26].
The experiments showed that for lower values of applied electric field
( E < 10 4 V / cm ) the J − U characteristics (studied at room temperature) are
ohmic.
For three investigated samples these dependences are shown in Fig.6.
It can be observed that at greater values of E the current density
increased faster than linear.
In Fig. 7 for sandwich type structures with different metals (indium and
gold) are represented the current-voltage characteristics.
100
J(µA/cm2)
80
60
structures In/ZnSe/In
sample ZS.089,
d=0.890µm, rd=4.49nm/s
sample ZS.065
d=0.650µm, rd=2.80nm/s
sample ZS.032
d=0.320µm, rd=3.50nm/s
40
20
0
5
10
15
20
25
30
35
U(V)
Fig.6 Current –voltage characteristics of ZnSe thin-film sandwich
heterostructures
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
157
600
structure In/ZnSe/In
sample ZS.091
d=0.910µm, rd=3,20nmnm/s
structure Au/ZnSe/Au
sample ZS.105
d=1.050µm, rd=1,70nm/s
500
J(µA/cm2)
400
300
200
100
0
0
10
20
30
40
50
60
70
80
U(V)
Fig.7 Current –voltage characteristics of ZnSe thin-film
sandwich heterostructures for different metal (indium and gold)
Consequently, we try to explain the shape of the J − U characteristics in the
domain of higher applied fields.
By assuming that the electronic transport in ZnSe films is determined as
space-charge-limited the current density is given by the relationship (Mott and
Gurney square law [17, 18, 25])
J=
9 µε 0 ε r N c   Et  2
exp −
U
8 d 3 N t   kT 
(2)
where µ is carrier mobility, ε 0 is permittivity of the free space, ε r denotes the
relative permittivity of the semiconductor film, d is the film thickness, N c
represents the effective density of states of semiconductor, N t indicates the
concentration of traps of energy Et from the bottom of conduction band, k is
Bolzmann’s constant, T is absolute temperature and U is applied voltage
across the sample.
158
M. DICIU et al.
According to Eq.(2), the J = f (U 2 ) dependences must be linear,
supposing that the pre-exponential parameters do not depend on the intensity of
electric fields.
For three studied samples these characteristics are presented in Fig. 8.
30
25
J(µA/cm2)
20
15
structures In/ZnSe/In
sample ZS.082
d=0.457µm, rd=3.7nm/s
sample ZS.069
d=0.690µm, rd=3.1nm/s
sample ZS.082
d=0.821µm, rd=2.90nm/s
10
5
0
0
100
200
300
2
Fig.8 Dependences
400
500
600
700
2
U (V )
J = f (U 2 ) for intermediate values of applied electric field
It can be observed from Fig. 8 that these dependences are not linear.
Therefore, we may consider that the space-charge limited conduction does not
play an important role in studied films. If we admit for studied structures that
the potential barriers which are formed at metal-semiconductor (ZnSe) interface
can be described by Schottky potential function, the current intensity in studied
sandwich structures can be calculated using the well-known RichardsonSchottky formula [17, 18]
J RS
  Φ
= AT exp − 0
  kT
2
 1

 exp 

 kT

 e 3U 


 4πε 0 ε r d 
1/ 2



(3)
where Φ 0 is the Schottky barrier height of the metal-semiconductor interface
and A is effective Richardson-Dushman’s constant which can be calculated
from expression [17]
A=
4πme ek 2
h3
(4)
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
159
where me is the effective mass of charge carriers in thin semiconducting film,
e is electron charge and h is Planck’s constant.
In generally, the current-voltage characteristics could be representing in
Schottky coordinates, ln J versus U . For lower values of the intensities of
electric field, E < 10 4 V / cm , the characteristics are linear due to the
predominating of the thermionic emission and thus the experimental data are in
a good agreement with the Richardson-Schottky law.
In Fig. 9 for two studied samples are represented the dependences
ln J = f ( U ) in domain corresponding of the thermionic emission.
3,5
3,0
ln[J(µA/cm2)]
2,5
2,0
1,5
structure In/ZnSe/In
sample ZS.076,
d=0.760µm, rd=3.10nm/s
sample ZS.035,
d=0.350µm, rd=2.90nm/s
1,0
0,5
0
1
2
3
1/2
4
5
1/2
U (V )
Fig.9 The dependences ln J = f ( U ) in the voltage domain in which the
experimental data are in a good agreement with Richardson-Schottky law.
The Schottky barrier height ( Φ 0 ) can be determined by extrapolation of
experimental linear parts of the ln J = f ( U ) dependences for
U →0. The
Φ 0 values, for structures with various films thickness are indicated in Table 2.
Different Schottky-barrier heights have been found, ranging from 0.80eV to
0.84eV.
It can be observed from Table 2 that in function of the contact metalsemiconductor, nature of the metal and the film thickness of sandwich-type
cells, the barrier heights at metal-semiconductor interface are different. These
values are different due to the type of electron transition which depends on the
160
M. DICIU et al.
continuum of interface-induced gap states, metal work function, type of the
contact metal-semiconductor which is determined by diffusions of the metal
atoms in film, interactions occurring at the metal/semiconductor interface and
the native defects [27-30].
Christov and Vodenicharov [31-35] proposed a theory of the electron
emission of metals into insulators and semiconductors which takes into account
both the electron transitions through and over the potential barrier at the metalsemiconductor interface.
The type of emission (field, thermionic-field, thermionic emission) is
determined by the characteristic temperature, Tk which is defined as the
temperature at which the probabilities of electron transitions through and over
the potential barrier are equal.
The region of thermionic emission, from Schottky (rectangular) potential,
is determined by the characteristic temperature, Tk [17, 18, 31, 35]:
Tk =
h(eε 0 ε r )1 / 4
π kme
2
1/ 2
E 3/ 4
(5)
where me is the effective mass of charge carriers and E represents the
intensity of applied electric field, determined by the formula, E = U / d .
The condition Tk / T >2/3 defines the ,,extended region of thermionic
emission” where the current density has to be determined by the expression [17,
18]
(6)
J = QRS J RS
where Q RS represents a factor which indicates the probability for the transitions
upper part of the barrier achieved for higher electric fields, given by the formula
[17, 18, 34]
(π / 2)(Tk / T )
(7)
sin[(π / 2)(Tk / T )]
≈ 1 , therefore J ≈ J RS and thus the region of the ,,pure”
QRS =
At Tk / T <1/2, Q RS
thermionic emission is defined [17, 32-35].
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
161
Table 2 The values of Schottky barrier height for structures with various films
Structure
Sample
d(µm)
rd(nm/s)
Ts(K)
Φ0(eV)
In/ZnSe/In
ZS.021
0.210
2.70
300
0.82
In/ZnSe/In
ZS.076
0.760
2.90
300
0.83
In/ZnSe/In
ZS.089
0.890
4.49
300
0.84
Au/ZnSe/Au
ZS.098
0.980
1.50
300
0.80
Au/ZnSe/Au
ZS.105
1.050
1.70
300
0.81
thicknessa
a
d, film thickness; rd, deposition rate; Ts, substrate temperature; Φ 0 , Schottky barrier height.
The effective mass, me can be determined using the expression [35]
2
 h(eε 0 ε r )1 / 4 
3/ 2
me = 
 Ek
2
 1.76π kT 
(8)
where E k is a ,,characteristics field intensities” which corresponds to the
characteristic temperature, Tk for to the intersection point for thermionic
emission and thermionic-field emission parts of J − U characteristic (method
proposed by Vodenicharov [34, 35]).
In Fig.10 are indicated three dependences ln J = f ( U ) for a sample,
which represent: the theoretical dependences by supposing that in these
domains of electrical fields the Richardson-Schottky formula (Eq.3) is valid, the
experimental data, and the dependences after Christov-Vodenicharov theory.
This Figure indicates that to obtain a rather good agreement between theoretical
curves after the Christov-Vodenicharov theory and the experimental
dependences. Some deviations of the experimental points are observed for very
high values of the electrical field.
162
M. DICIU et al.
4,4
4,2
1
sample ZS.029
d=0.298µm, rd=3.10nm/s
4,0
ln[J(µA/cm2)]
3,8
2
3
3,6
3,4
3,2
3,0
2,8
2,6
2,4
1
2
3
4
5
6
U1/2(V1/2)
Fig.10 The dependences
ln J = f ( U ) for a ZnSe thin film: for dependences (1),
the values of J were calculated base don Christov- Vodenicharov theory; the
dependences (2) represent experimental data ; the dependences (3) are theoretical
dependences by supposing that in the respective domains of electric fields the
Richardson-Schottky equation is valid.
This fact shows that it can be possible both the electron transitions
through and over the potential barrier the metal-semiconductor interface.
4. CONCLUSIONS
The behaviors of the current-voltage characteristics could explain the
electron transfer mechanism through the studied thin-film sandwich structures.
The experimental results lead to the conclusions that the nature of the metal or
the film thickness of sandwich-type cells and the barrier heights at metalsemiconductor interface can influence the nature of the electrical conduction
mechanism.
For explanation of obtained experimental results, the electron transitions
through and over potential barrier at metal/semiconductor film interface are
taken into account.
ON THE MECHANISM OF ELECTRICAL CONDUCTION…
163
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