Thermal Enhancement of Interference Effects in Quantum Point Contacts

Adel Abbout, Gabriel Lemarié and Jean-Louis PichardPhys. Rev. Lett. 106, 156810 (2011)

IRAMIS/SPEC CEA Saclay

Service de Physique de l’Etat Condensé, 91191 Gif Sur Yvette cedex, France

Electron Interferometer formed with a quantum point contact and another scatterer in a 2DEG

Interferences in one dimension 1d model with 2 scatterers

Scatterers with a weakly energy dependent transmission

L

Interferences with a resonance

L

2d model:Resonant Level Model for a quantum point

contact

From the RLM model towards realistic contacts

RLM model QPCs in a 2DEG

SGM imaging Conductance of the QPC as a function of the tip position

(Harvard, Stanford, Cambridge, Grenoble,…)Topinka et al., Physics Today (Dec. 2003)

)pwithout ti() tipwith( ggg

g falls off with distance r from the QPC, exhibiting fringes spaced by F/2

2DEG , QPC AFM cantilever

The charged tip creates a depletion region inside the 2deg which can be scanned around the nanostructure (qpc)

QPC Model used in the numerical studyLong and smooth adiabatic contact

Sharp opening of the conduction channels

y

x

xx

yy

xx

LL

LLnmmnU

LmLLnL

mn

100

]2231[10

),(

),(

41

322

),( mnU + TIP(Square Lattice at low filling, t=1, EF=0.1)

QPC biased at the beginning of the first plateau(Tip: V=1)

T=0 T = 0.01 EF

QPC biased at the beginning of the second plateau(Tip: V=-2)

T=0 T =0.035 EF

Resonant Level Model

2 semi-infinite square lattices with a tip (potential v) on the right side

coupled via a site of energy V0 and coupling terms -tc

Self-energies describing the coupling to leads expressed in terms of surface elements of the lead GFs

Method of the mirror images for the lead GFs. Dyson equation for the tip

• Transmission without tip ~ Lorentzian of width

• Transmission with tip(Generalized Fisher-Lee formula)

rlrlrl

lrlr

lr

iIRIIRRVE

IIET

,,,

220

04)(

0,0,11lim)exp(

12

)]2/2(exp[2/32

xGVVi

xO

xkxi

t

Rr

x

c

r

rrr

I4

FEE Narrow resonance:

Expansion of the transmission T(E) when is smallx

1

IRTTSsITTRTTTT

TISRSTsTIII

IRR

TTSTTT

..2..4

54

3

.......

1

2000

230

2023

02

0

20

2

2000

000

0

Out of resonance: T0 < 1, 1/x Linear terms

At resonance: T0=1; S0=0 1/x2 quadratic terms

(Shot noise)

T=0 : Conductance

• Out of resonance:

• At resonance:

00

0

2/30

0

1sin2

12cossin2

Ts

xO

kxkx

TET

Fringes spaced by (1/x decay)

2/52

2

0

1x

OkxT

T

2/F

Almost no fringes (1/x2 decay)

FETG

T > 0: Conductance at resonance

• 2 scales:

• Temperature induced fringes:

I

VL

TkVL

F

B

FT

4

Thermal length:

New scale:

2

0 22cos,

xkxkxk

LL

LxA

TgTg

FF

FT

Rescaled Amplitude

LLerfc

LL

LxA TT

8.,

1. Universal T-independent decay:

Lxexp2

2. Maximum for

TLx8

Bottom to top: increasing temperatureFL 2

Numerical simulations and analytical resultsIncreasing temperature (top to bottom)

20//2//4.0

10//20//40//2/

Fc

F

T

Vt

L

The thermal enhancement can only be seen around the resonance

RLM model QPC ?

• The expansion obtained in the RLM model can be extended to the QPC, if one takes the QPC staircase function instead of the RLM Lorentzian for T0(E).

• The width of the energy interval where S0=T0(1-T0) is not negligible for the QPC plays the role of the of the RLM model for the QPC.

Interference fringes obtained with a QPC and previous analytical results

assuming the QPC transmission function

Transmission ½ without tip, Red curve: analytical resultsBlack points: numerical simulations

Peak to peak amplitude

Similar scaling laws for the thermoelectric coefficients and the thermal conductance

Summary