Thermal Enhancement of Interference Effects in Quantum ：在量子干涉效应的热增强

r r r
r tc2
exp[i(2kx 2x
/
2)]
O
1 x3/2
exp(i)
limx
1V
V 1,0GrR
x,0
Expansion of the transmission T(E) when 1 is small
x
T T T0
S0 T01T0
(Shot noise)
R R
I
I I
I
T s.T0. S0.R S0.I T2 ...
T2
3T02 4
T02
T03 R2
5T02 4
T03 .I 2
s.
S0 T0 2T02 .R.I
Out of resonance: T0 < 1, At resonance: T0=1; S0=0
1/x Linear terms 1/x2 quadratic terms
T=0 : Conductance GTEF
Similar scaling laws for the thermoelectric coefficients and the thermal conductance
Summary
Topinka et al., Physics Today (Dec. 2019)
gg(with i) pg(withpo) ut ti
2DEG , QPC AFM cantilever
The charged tip creates a depletion region inside the 2deg which can be scanned around the nanostructure (qpc)
T > 0: Conductance at resonance
• 2 scales:
Thermal length:
New scale:
LT
V F k BT
L
V F
4I
• Temperature induced fringes:
g g 0T T A L x ,L L T c2 o 2 k kF F s x x k F x 2
• Transmission without tip
~ Lorentzian of width
4I
Narrow resonance:
E E F
T0(E)EV0Rr 4IR rIll2Ir Il2
l,r Rl,r iIl,r
• Transmission with tip (Generalized Fisher-Lee formula)
g falls off with distance r from the QPC, exhibiting fringes spaced by lF/2
QPC Model used in the numerical study Long and smooth adiabatic contact
Sharp opening of the conduction channels
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,…)
•
Out of resonance:
TE
T0
Βιβλιοθήκη Baidu
2
sin0
cos2k
k x
x
O
1
3/ 2
x
Fringes spaced by lF / 2
(1/x decay)
2
0
sin0 s 1T0
• At resonance:
Almost no fringes (1/x2 decay)
T
T0
2
k x2
Ox15/2
Thermal Enhancement of Interference Effects in Quantum Point Contacts
Adel Abbout, Gabriel Lemarié and Jean-Louis Pichard
Phys. Rev. Lett. 106, 156810 (2019)
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
Rescaled Amplitude ALx ,L LT .erfc8L LT
L 2lF
1. Universal T-independent decay:
2 exp
x L
2. Maximum for
x 8 LT
Bottom to top: increasing temperature
Numerical simulations and analytical results Increasing temperature (top to bottom)
assuming the QPC transmission function
Transmission ½ without tip,
Red curve: analytical results Black points: numerical simulations
Peak to peak amplitude
• 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
(Square Lattice at low filling, t=1, EF=0.1)
(n, m )
Lx n Lx Ly m Ly
U
(n, m )
m 10
2 [1
3
n Lx
2
2
2 Lx
3 1
]4
L x 100
Ly
U(n,m) + TIP
QPC biased at the beginning of the first plateau (Tip: V=1)
•
Interferences in one dimension
1d model with 2 scatterers
L
Scatterers with a weakly energy dependent transmission
Interferences with a resonance
L
2d model: Resonant Level Model for a quantum point
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
LT
lF /
2
//4
0/ /2 0/ /1
0
tc 0.4//V 2//lF 20
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).