Download Quantum Transport Theory with Tight-Binding Hamiltonian Stefano Sanvito Department of Physics

Quantum Transport Theory with
Tight-Binding Hamiltonian
Stefano Sanvito
Department of Physics
Trinity College Dublin
[email protected]
September 25, 2002
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Overview
• Quick Review of Landauer-Büttiker formalism
• Scattering Theory I: simple 1D example
• Scattering Theory II: structure of the Green’s functions
• Scattering Theory III: General formalism
• Few Applications
• Beyond the Tight-Binding: LDA Hamiltonian
• Summary & Outlook
Most of the materials can be found in:
S. S., “Giant Magnetoresistance and Quantum Transport in Magnetic
Hybrid Nanostructures”, PhD Dissertation, University of Lancaster (UK)
(1999)
S. S., C.J. Lambert, J.H. Jefferson, and A.M. Bratkovsky, Phys. Rev. B
59, 11936 (1999)
1
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Length-Scale
System Length (L)
Fermi Wavelength (λ)
λF = 2π/kF =
q
2π/ns
(1)
Elastic Mean Free path (Lm)
Lm = vF · τm
(2)
Phase relaxation length (Lϕ)
Lϕ = vF · τϕ
if τϕ < τm
2
2
Lϕ = vFτmτϕ/2 if τϕ > τm
(3)
(4)
Spin-flip length (Ls)
Ls = vF · τs
Length
λF
Lm
Lϕ
Ls
Metals
a0
10-100 Å
10-100 Å
10-100 nm
Semicond. (2D)
35 nm
30 µm
Lm ≤ Lϕ Lm
up to 100 µm
(5)
Semicond. (3D)
35 nm
100 µm
Lm ≤ Lϕ Lm
up to 1 mm
Working Condition L ≥ Lϕ
2
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Landauer-Büttiker formalism
µ1
1
T
µ2
R
• The scattering leads
electrons
Basic Assumptions:
• (µ1 − µ2) → 0+
inject
uncorrelated
Current I emitted from the left lead:
∂n
I = ev
(µ1 − µ2)
∂E
(6)
but ∂n/∂E = ∂n/∂k · ∂k/∂E = 1/2π · 1/vh̄ and, ∆V = e∆µ
e2
I = ∆V
h
(7)
If an electron has probability T to be transmitted, the conductance Γ =
I/∆V becomes
Γ=
e2
hT
(T = 1 Sharvin Conductance)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Generalization to many scattering channels and many leads
µβ
µ
γ
µ
α
µδ
• All electrons injected from α-th lead have same µα
• N α scattering channels for α-th lead
• Tijαβ : probability to transmit i-th channel of α-th
lead into j -th channel of β -th lead
Then the fundamental current/voltage relation is
α
I =
nX
leads
Γ
αβ
β
µ
(8)
β
where
2
Γ
αβ
Nα Nβ
e X αβ
e2
αβ αβ†
=
Tij = Tr t t
h ij
h
(9)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Scattering Theory I: simple 1D example
0
−∞ ...... i0-3
0
γ1
i0 - 2
γ1
ε0
ε0
0
α
γ2
ε0
γ2
i0-1 ... i0+1 i0+2 i0+3
......
+∞
1) Green’s functions for infinite leads
eik(E)|j−l|
gjl (E) =
ih̄v(E)
the crucial point here is to calculate the inverse dispersion
E
−
0
−1
k = k(E)
(k = cos
)
2γ
(10)
(11)
2) Green’s functions for semi-infinite leads
Assume that the chain is terminated at i = i0 − 1 (no site for i ≥ i0).
Use boundary conditions: g new = g old + ψ with ψ a wave-function such
that ginew
= 0 for l < i0
0l
e−ik(j−2i0+l)
ψj (l, i0) = −
ih̄v
(12)
Note that the new Green’s functions has the same causality than the old.
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
3) Surface Green’s functions for decoupled chains
It is simply ginew
. Total GF’s for the two chains
0 −1,i0 −1

g=
eik1
γ1
0
eik2
γ2
0

(13)

4) Surface Green’s functions for coupled chains: Dyson’s equation
G=
1
γ1γ2e−i(k1+k2) − α2
e−ik2 γ2
α
α
e−ik1 γ1
(14)
5) Extract the t matrix
a) First note that the most general wave-function for electrons coming
from the left can be written as
ψl =









eik1 l
v1 1/2
+
r
e−ik1l
1/2
v1
l ≤ i0 − 1
(15)
t
1/2 e
v2
ik2 l
l ≥ i0 + 1
this introduces reflection r and transmission t coefficients.
b) Then construct the projector to map (14) onto (15). It is the same
for the infinite case:
eikl
glj P (j) = 1/2
v
for
l≥j
(16)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
c) Now, use the projector to extract r and t
lim
l→(i0 −1)
Gl,i0−1P (i0 − 1) =
1
v1
1/2
+
r
v1
1/2
(17)
gives
r = Gi0−1,i0−1P (i0 − 1)v1
lim
l→(i0 +1)
Gl,i0−1P (i0 − 1) =
1/2
t
−1
e
1/2
ik2 l
(18)
(19)
v2
gives
t = Gi0+1,i0−1P (i0 − 1)v2
1/2 −ik2
e
(20)
Since the incoming wave-functions have unit flux
2
2
|t| + |r| = 1
(21)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Scattering
wave−function
Stefano Sanvito
GF’s
infinite leads
Boundary Conditions
GF’s
semi−infinite leads
Interaction Matrix
V
Projector
Dyson’s Equation
Surface GF’s
full system
S MATRIX
Landauer−Buttiker
CONDUCTANCE
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Scattering Theory II: structure of the Green’s
functions
To understand the structure of GF’s for scattering problems,
consider the next simplest case: 2D simple cubic
z
The Green function for such a system can be shown to be
0
0
g(z, x; z , x ) =
M
X
Nn sin
n=1
ikzn(E)|z−z0|
nπ
nπ
e
0
x sin
x
M +1
M +1
ih̄vzn(E)
(22)
with kzn(E) and vzn(E) given by the dispersion
E = o + 2γ cos
nπ
M +1
n
+ cos kz
(23)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Schematically g(z, x; z 0, x0; E) is simply
M
X
0
n
eikz (E)|z−z | ∗ 0
0
0
βn(x )
g(z, x; z , x ; E) =
βn(x)
n
ih̄vz (E)
n=1
The
P
(24)
is over ALL POSSIBLE CHANNELS (open or closed)
rd
ld
lm
z|
rm
Four possible scattering channels
•
•
•
•
ld
lm
rd
rm
left-decaying channels
left-moving channels
right-decaying channels
right-moving channels
10
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Scattering Theory III: General formalism
GENERALIZATION
H1
H0
H1
H1
H0
H0
H1
z
The Hamiltonian for such a system is:






H =




...
...
...
...
...
...
...
...
H0
H−1
0
0
...
...
...
H1
H0
H−1
0
...
...
...
0
H1
H0
H−1
...
...
...
...
0
H1
H0
...
...
...
...
...
0
H1
...
...
...
...
...
...
0
...
...
...
...
...
...
...
...
...











(25)
with H0† = H0 and H−1 = H1† matrices M × M and M the number
of degrees of freedom in the “slice” described by H0
Note that this is identical to the linear chain case when we replace
0 −→ H0
γ −→ H1
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
GF’S FOR INFINITE LEADS
ψz is a column vector corresponding to the z -th slice. We introduce
the Bloch function (vk⊥ group velocity)
ψz = √
1
ik z
e ⊥ φk⊥
v k⊥
(26)
we obtain the “band structure” equation
ik⊥
−ik⊥
H0 + H1e
+ H−1e
− E φk⊥ = 0
(27)
First Problem: k = k(E)
The “band structure” equation can be solved by mapping the (27) onto
an equivalent eigenvalue problem
H=
−H1−1(H0
I
− E)
−H1−1H−1
0
(28)
where the eigenvalues are eik⊥z and the first half of the eigenvectors are
φk⊥ . Note that H1 CAN NOT be singular.
Now we calculate vk and sort out left- (φk̄ ) and right-moving (φk ) and
-decaying channels
12
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Green’s functions fundamental Ansatz
Similarly to the 2D simple cubic case we assume the following form
gzz0 =
 P
M
ikl (z−z 0 ) †

φ
e
wk

l=1 kl


PM

ik̄l (z−z 0 ) †

φ
e
wk̄
l=1 k̄l
l
z ≥ z0
(29)
l
z ≤ z0
wkl and wk̄l are computed by imposing
1. The Green’s equation [(E − H)g]zz0 = δzz0
2. Continuity for z = z 0
An expression for wkl and wk̄l which involves only φ, H1 and the “dual”
vectors φ̃ can be given (φ̃†k φkh = φ̃†k̄ φk̄h = δlh).
l
l
Then, by using convenient boundary condition, right (gR) and left (gL)
surface GF’s are calculated. The total surface GF for decoupled leads:
g(E) =
gL(E)
0
0
gR(E)
(30)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
EFFECTIVE V MATRIX
The idea is to reduces a complex V matrix to an effective coupling
matrix between two surfaces. We use the decimation technique. It
consists in redefining recursively the matrix V, by eliminating the degrees
of freedom which do not couple the external surfaces.
In V is an N × N matrix after the first iteration
(1)
Vij
= Vij +
Vi1V1j
E − V11
(31)
after l iterations
(l−1)
(l)
(l−1)
Vij = Vij
+
Vil
E−
(l−1)
Vlj
(32)
(l−1)
Vll
finally after N − 2M steps we end up with an M × M “effective
coupling matrix”
Veff (E) =
VL∗(E)
∗
VRL
(E)
∗
VLR
(E)
VR∗(E)
(33)
Note that Veff (E) is energy dependent. If the interaction is short range
the technique is very good for numerical optimization.
Now the total Green’s function is calculated via Dyson’s equation.
14
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
S MATRIX AND TRANSPORT COEFFICIENTS
It is a straightforward generalization of the case of the linear chain. The
Bloch state ψz is normalized to unit flux (from the expression of the
current)
1
ikz
ψz = √ e φk
vk
(34)
Note: if there is k-degeneracy the (34) does not diagonalize the current.
A rotation is needed.
It can be used to build up the general scattering wave-function
ψz =







ikl z
e√
vl φkl
+
P
rhl ik̄ z
√
e h φk̄h
h
v̄
z≤0
h
P
thl ik z
h φk
√
h vh e
h
(35)
z≥L
where:
• rhl = rk̄h,kl is the r coefficient for a rm electron with momentum
kl scattered in a lm electron with momentum kh
• thl = tkh,kl is the t coefficient for a rm electron with momentum
kl transmitted in a lm electron with momentum kh
Now, one defines the projector P as before, and the scattering
coefficients are computed.
15
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Beyond the Tight-Binding: LDA Hamiltonian
We use an LSDA-LCAO code to calculate the electronic structures of
heterojunctions. The final output is a tight-binding Hamiltonian,
then this scattering formalism can be applied.
Problems:
1. TB is non-orthogonal
2. H1 can be singular
3. Efficient treatment of large matrices is needed
Solutions:
1. Introduction of overlap matrix So in the form

S−1
 0
So = 
 ...
...
S0
S−1
0
...
S1
S0
S−1
0
0
S1
S0
S−1
...
0
S1
S0

...
... 

0 
S1
This does not add any particular complication.
For example the “band-structure” equation
becomes:
h
i
ik⊥
−ik⊥
(H0 − ES0) + (H1 − ES1)e
+ (H−1 − ES−1)e
φk⊥ = 0
(36)
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
THE H1 PROBLEM
“Lack of Bonding”
γ
ε0
H1 =
0
γ
0
0
“Excess of Bonding”
γ
ε0
H1 =
γ
γ
γ
γ
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Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Quick Fix
The dispersion relation k = k(E) can be calculated by solving the
generalized eigenvalue problem
αk
0
H−1
I
H0 − E
φk
χk
= βk
I
0
0
−H1
φk
χk
with
e
ik
= βk /αk
Complete Fix
Eliminate the singularities: DECIMATION
18
Quantum Transport Theory with Tight-Binding Hamiltonian
Stefano Sanvito
Digital Magnetic Heterostructures
In Plane transport
400
Tot
Ga s
Ga p
As s
As p
Mn s
Mn p
Mn dt
Mn de
2
σ(e /h)
300
200
100
0
-5
-4
-3
-2
-5
E (eV)
-4
-3
-2
E (V)
Perpendicular to Plane transport
200
Tot
Ga s
Ga p
As s
As p
Mn s
Mn p
Mn dt
Mn de
2
σ(e /h)
150
100
50
0
-5
-4
-3
E (eV)
-2
-5
-4
-3
-2
E (V)
19