Quantum coherent transport in Meso- and Nanoscopic Systems

Philippe Jacquodpjacquod@physics.arizona.edu

U of Arizona

Conductance from transmissionOutline

The Landauer-Buttiker formula

Easy applications: conductance quantization

recovering Ohm’s law (incoherent)

resonant tunneling (coherent)

Anderson localization (coherent)

What is the conductance G of a perfect 1D conductor ?

Left e- reservoir Right e- reservoir

Current :

1D with p.b.c.

->Quantization

->Density of states

G=I/V

What is the conductance G of a perfect 1D conductor ?

e2/h : conductance quantumh/e2 = 25813 Ohms; “von Klitzing’s constant”

Quantized Hall Conductance

Answer:

What is the conductance of a perfect quasi-1D conductor ?

Left e- reservoir Right e- reservoir

D

Perfect, translationally invariant potential

-> Separable problem

Quantization in y -> channel index i=1,2,...N-> N determined by EF=EL+ET

EF

x

y

What is the conductance of a perfect quasi-1D conductor ?

Left e- reservoir Right e- reservoir

D

Answer:

What is the conductance of an imperfect quasi-1D conductor ?

Left e- reservoir Right e- reservoir

D

I.e. put scatterers with reflection,tunnel barriers etc.-> Landauer-Buttiker formula: transmission prob. 0<Tn<1

Tn: transmission probability in channel #n.

[001][100]

[010]two-dimensional electron gas(2DEG)

quantum point contact(QPC)

Example: Conductance Quantization (van Wees et al., ’88; Wharam et al., ’88)

Example: Conductance Quantization (Buttiker, prb ’90)

See also: Stone and Szafer, prl ’89

Saddle potential

Transmission probability

xy

T1 T2

Conductance as transmission : reproducing Ohm’s law

->1D ->2 tunnel barriers in series

Q.: What is the total transmissionA#1: Assume incoherent addition -> transmission probabilities T12=T1T2 + T1T2 R1R2 + T1T2 (R1 R2)2 +...

=T1T2/(1-R1R2) rewrite as (1-T12)/T12 = (1-T1)/T1 + (1-T2)/T2

for N identical barriers [1-T(N)]/T(N) = N (1-T)/T -> T(N)=T/(T+N(1-T)) N=n L with n=density of scatters; L0=T/(n (1-T))

T(L)=L0/(L+L0)~1/L for L>>L0

t1 t2D

Conductance as transmission : resonant tunneling

->1D ->2 tunnel barriers in series

Q.: What is the total transmissionA#2: Assume coherent addition -> transmission amplitudes

t12=t1 t2 + t1 t2 r1 r2 exp[2ikD] + t1 t2(r1 r2 exp[2ikD])2+...=t1 t2/(1-r1 r2 exp[2ikD])

T12=|t12|2=T1 T2/(1-2 (R1 R2)1/2 cos[φ]+R1 R2)

φ=2kD+phases from r1 and r2

t1 t2D

Conductance as transmission : resonant tunneling

->1D ->2 tunnel barriers in series

Q.: What is the total transmissionA#2: Assume coherent addition -> transmission amplitude

T12=|t12|2=T1 T2/(1-2 (R1 R2)1/2 cos[φ]+R1 R2)

φ=2kD+phases from r1 and r2

(Marcus lab.)

(Alhassid, RMP ‘01)

Example: Coulomb Blockade Resonance

Conductance as transmission : Anderson localization

T12=|t12|2=T1 T2/(1-2 (R1 R2)1/2 cos[φ]+R1 R2)

1D array with random D, i.e. random φ

1) average over φ

2) define , so that

3) small additional resistor in series with

->

T1 T2 T3D’D

Conductance as transmission : Anderson localization

T12=|t12|2=T1 T2/(1-2 (R1 R2)1/2 cos[φ]+R1 R2)

1D array with random D, i.e. random φ

-> or diff. eq.

T1 T2 T3D’D

Conductance as transmission : Anderson localization

T12=|t12|2=T1 T2/(1-2 (R1 R2)1/2 cos[φ]+R1 R2)

1D array with random D, i.e. random φ

Exponentially small transmission for L>>L0

This is due to disorder - in 1D it localizes all QM wavefunctions

T1 T2 T3D’D