Thermal Enhancement of Interference Effects in Quantum Point Contacts Adel Abbout, Gabriel Lemarié and Jean- Louis Pichard Phys. Rev. Lett. 106, 156810 (2011) IRAMIS/SPEC CEA Saclay Service de Physique de l’Etat Condensé, 91191 Gif Sur Yvette cedex, France
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Thermal Enhancement of Interference Effects in Quantum Point Contacts
Thermal Enhancement of Interference Effects in Quantum Point Contacts. Adel Abbout, Gabriel Lemarié and Jean-Louis Pichard Phys. Rev. Lett. 106, 156810 (2011). IRAMIS/SPEC CEA Saclay Service de Physique de l’Etat Condensé, 91191 Gif Sur Yvette cedex, France. - PowerPoint PPT Presentation
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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