Download Strong Temperature Dependence of the Quasi

Strong Temperature Dependence
of the Quasi-Particle Tunneling
between
Quantum Hall edges
Roberto D’Agosta
University of Missouri - Columbia
NEST-INFM
Max Planck Institute for Physics of Complex Systems
Cophen 04 - Dresden 25 June 2004
Outline
I. Edge physics in the IQHE
II. Edge physics in the FQHE: Chiral Luttinger
Liquid
III. Tunneling through a constriction
IV. Conclusions
The Quantum Hall Effect
Y
)
X
6
"
6(
For small magnetic field one
recovers the Classical Hall Effect
H.L. Stormer, Physica B, vol. 177, pag 401, 1992
For certain large values of the
magnetic field the Hall resistance
shows large plateaus at exact
quantized values...
...while the longitudinal resistance
drops dramatically to zero!
The Quantum Hall Effect
The motion of a particle in a
magnetic field is described by
the solutions of a Harmonic
Oscillator equation
In the plateau region:
!b
Landau
Levels
h̄
2
! =
eB
eB
!c =
m
1 h
RH =
,
R
=
0
2
!b e
is an integer or a fraction with an incredible precision!
If
!b
If
!b is a fraction we have the Fractional Quantum Hall Effect
is an integer we have the Integer Quantum Hall Effect
Edge states in IQHE
The physical edges of the Hall bar bend the Landau levels
Near the physical edges, 1D states are
generated (edge states)
The transport properties of the IQHE
are explained in terms of the edge
states
The edge states exist also in the FQHE but their origin is due
to the electron correlation!
Luttinger Liquid model
Two counter propagating
modes with
same velocity
and linear dispersion
g=
!
vF +V1 +V2
vF +V1 −V2
E
L
R
V2
V1
kF
!
"
z
!"(x), !#(x1) = −i g $",#%x&(x − x1)
Power law behavior of the density of states: the
exponent is determined by the interaction constant
k
Wen’s theory of the edge states
X.G. Wen PRB 41 12838 (1990), PRB 43 11025 (1991), PRB 44 5708 (1991)
FQHE
Gauge Invariance
Egde modes action
Boundary conditions
Edge modes
Density fluctuations in the boundary
Kac-Moody algebra
!
"
i%b z
!"#(x), !"$(x1) = −
'#,$(x!(x − x1)
2&
It will determine the
single particle properties
of the system!
The filling factor enters
the commutation
relations
Experiments on the Luttinger Liquid exponent
M. Grayson, D.C. Tsui, L.N. Pfeiffer, K.W. West, and A.M. Chang PRL 80 1062 (1998)
Tunneling from a 3D electron gas to an edge of the
FQHE
I !V
!
1
!=
"b
The experiments clearly
show that the Chiral
Luttinger tunneling
exponent varies
continuously with
the filling factor
Edge states as hydrodynamical modes
R. D’Agosta, R. Raimondi and G. Vignale PRB 90 (2003)
Hydrodynamic approximation
Lowest Landau Level
projection
q! ! 1
[!"(!r), !"(!r1)] Z= −i" #i j $i"0(!r)$ j !(!r −!r1)
2
HT =
i, j ∈ {x, y}
!"(r)V (r, r1)!"(r1) d!rd!r1
!t "#(!r,t) = " !i#0(!r)$i j
2
Z
V (r, r1)"#(!r1,t) d!r1
The density fluctuations are located in the region where the
equilibrium density has the large derivative
Edge states as hydrodynamical modes II
We integrate over
the shaded region to define
the edge excitation
!"#(x) =
Z
#
!, " ∈ {R, L}
!"(x, y) dy
y
!
"
%b z
!"#(x), !"$(x1) = −i '#,$(x!(x − x1)
2&
Z
1
H=
2
!"#(x)V#,$(x, x1)!"$(x1) dxdx1
Our edge excitations follow the same algebra of an interacting
chiral Luttinger Liquid but we have not assumed the presence
of a fully developed Quantum Hall phase
Solution of the interacting
Hamiltonian
!
$b
n
† n∗
!"#(x) =
bn'#(x) + bn'# (x)
Bosonization:
&
2% n>0
Eqn of motion:
"
!n (x)
form an orthonormal and complete
base for the Hilbert space
Single Particle operators:
•
Define
!"#(x) = $x%#(x,t)
•
Boson propagator D!,"(x, x1, #) = −2$i
•
Electron creation operator
Z
%
0
†
!"(x)
!
" i#t
d#" &!(x,t), &"(x1, 0) #e
e∗ 2#
−i e $ %"(x)
b
= U"e
Calculation of the conductance
#2
Definition
!Ii = " Gi j (V )!V j
j
#3
I
I
#1
Sum rules
#4
#6
#5
! Gi j = ! Gi j = 0
j
i
The conductances can be related to the boson propagator
2
ie
Gi j = −
h̄
lim D(xi, x j , %)
! $",i$#, j %→0
",#
V!,"(x, x1) = #(x − x1)V!,"
2
!be
Gi j =
"(xi − xi+1)
h
The Constriction
We model the constriction by assuming
VR,R(x, x1) = VL,L(x, x1) = V1
!
V2 |x| > a
VR,L(x, x1) =
1
V2 |x| < a
We solve the eqn of motion with
scattering boundary conditions and
calculate the transmission and
reflection coefficients
V2
a
L
R
1
V2
V2
The presence of the constriction does not modify the response
function at zero bias: we recover the exact quantization of the
conductance
Phenomenological theory of the tunneling
HT = !
!
†
"R(0)"L(0) +
∗
!
†
"L(0)"R(0)
- is assumed to be a phenomenological constant with no
dependence on temperature or energy scale
(we will reanalyze this assumption later!)
- is due to the non complete localization of the states in the
right or left edge
In linear response theory:
Rxx ! "#Im
G (t, x) =
−
Z
$
e ImG (t, 0)dt
i#t
−
0
†
†
"!L(t, x)!R(t, x)!R(0, 0)!L(0, 0)#
Results for the tunneling between two edges
If we remove the constriction we recover the Wen’s result
2"b−2
2"b−2
Rxx(T, 0) ! T
, Rxx(0,V ) ! V
X.G. Wen PRB 43 11025 (1991), C.L. Kane and M.P.A Fisher PRB 51 13449 (1995)
The presence of the interaction renormalizes the exponent
−2"
!b → !be
(tanh(2")
1
= V2 /V1)
The presence of the constriction introduces two different
behaviors (i.e. two characteristic exponents) in the time
domain. The separation is given by the time an edge wave
needs to travel trough the constriction.
R. D’Agosta et al. PRB 90 (2003)
Some experimental result
Recently the tunneling between two edges of the same
FQHE has been experimentally studied
S. Roddaro et al. PRL 90 046805 (2003)
dIT
G=
dVT
7
B=6 T
At high temperature the peak
seems agree with the theories
dIT/dVT [µS]
6
5
4
3
To address this
experimental result we
focus on the tunneling
amplitude!
T=30 mk
T=100 mK
T=200 mK
T=300 mK
T=400 mK
T=500 mK
T=700 mK
T=900 mK
!b = 1/3
2
-3
-2
-1
0
V [mV]
1
2
At low temperature the deep is
not expected! (Strong coupling)
3
Tunneling amplitude
The tunneling can be
due to the incomplete
localization of the wave
function in the edges
The tunneling can be mediated by
the presence of impurities that
break down the translational
invariance
! " !#R|#L"
Being proportional to the superposition of wave functions
belonging to different edges it is usually small and the
tunneling is usually treated as a perturbation
Tunneling amplitude
All the electrons occupy the Lowest Landau level
!"
Z
∗
#R(!x)#L(!x)
d2
− 4"2
d!x " e
G
A small variation of the edge distance
affects significantly
the tunneling amplitude!
ϕL
d
D
ϕR
G
Edge position: a simple model
"
F[n] = E[n] − T S[n]
}
d/2
In the incompressible region
!(x) ≡ !b
V(x)
"="b
x
}
!
In the compressible region
U
E[n] =
4!!2
Z
n(x) dx +
2
Z
kB
S=−
2!!2
V (x)n(x) dx
Z
!(x)
n(x) =
2"!2
"(x) ln "(x) + [1 − "(x)] ln[1 − "(x)] dx
!
"
!
"(x)
F[n] = U"(x) +V (x) + kBT ln
=µ
!n(x)
1 − "(x)
Edge position: a simple model II
The edge position
is determined by
d(T ) = d(0) + 2x0(T )
!(x0) = !b
Particle number conservation
In the simplest case
N=
Z
!
n(x) dx
x0
V (x) = eE x
kBT ln(1 − !b)
x0(T ) =
"
2
U
!b
kbT
µ(T ) = µ(0) +
[!b ln !b + (1 − !b) ln(1 − !b)]
!b
Edge position: a simple approach
At zero temperature the edge position is determined by the
minimization of the electrostatic energy
D.B. Chklovskii et al. PRB 46 4026 (1992), PRB 47 12605 (1993)
T != 0
minimize the free energy
1 2
2
F(d + !d) = F(d) + "d E(d)(!d) − T "d S(d)!d
2
!d " T
Temperature dependence of the tunneling amplitude
x0 (T )d(0)
− !2
|!(T )| = |!(0)| e
2
2
!
= |!(0)| e
2 T /T0
"
!
#b
#be2
kBT0 =
→ T0 # 600 mK
2" | ln(1 − #b)| $bd
G(0) = const × |!| T
2
T0 ! 140 mK
2"b−2
(!b = 1/3)
!2 !meV !"2
25
20
15
10
T0 ! 400 mK
5
0.2
0.4
0.6
0.8
T !K"
Recent experiments
S. Roddaro et al. cond/mat 043318 (2004)
!
"
2
!
#b
#be
kBT0 =
2" | ln(1 − #b)| $bd
25
T = 50 mK
! = 1/3
dV/dI(k#)
80
dV/dI(k#)
60
!=1
20
15
10
5
40
-80
!" = 1/5
-0.50 V
-0.40 V
0
-0.32 V
-0.26 V
-0.20 V
-0.18 V
80
V(µV)
Vg = -0.60 V
20
0
!" = 1/4
!" = 2/7
-600 -400 -200
0
200 400 600
VT(µV)
800
Extensions
•
A better treatment of the electrostatic energy or a better
approximation for the confining potential can improve
the estimate for the temperature scale but we expect that
it cannot modify the general scenario we have presented
here
•
The presence of different chemical potentials in the
edges implies a rigid shift of the edge positions thus
leaving their distance untouched
•
For very small temperature the prediction of the RG is
recovered because the tunneling amplitude goes to a
constant
Conclusions
•
The chiral luttinger liquid model can be used to study the
phenomenology of the edge states of the FQHE
•
We have extended the previous theory in order to
consider the possibility of continuous variation of the
filling factor
•
We have then considered the dependence of the
tunneling amplitude on geometrical factors and thus on
temperature. Our results are in qualitative agreement with
the available experimental data.
Acknowledgments
Roberto Raimondi (Universita’ di Roma Tre)
Giovanni Vignale (University of Missouri)
S. Roddaro
V. Pellegrini
F. Beltram
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