We study three-terminal thermoelectric transport in a two-dimensional Quantum Point Contact (QPC) connected to left and right electronic reservoirs, as well as a third one represented by a scanning probe tip. The latter acts as a voltage probe exchanging heat with the system but no charges on average. The thermoelectric coefficients are calculated numerically within the Landauer–Büttiker formalism in the low-temperature and linear response regimes. We find tip-induced oscillations of the local and non-local thermopowers and study their dependence on the QPC opening. If the latter is tuned on a conductance plateau, the system behaves as a perfect thermoelectric diode: for some tip positions, the charge current through the QPC, driven by a local Seebeck effect, can flow in one direction only.

1.
F.
Menges
,
P.
Mensch
,
H.
Schmid
,
H.
Riel
,
A.
Stemmer
, and
B.
Gotsmann
, “
Temperature mapping of operating nanoscale devices by scanning probe thermometry
,”
Nat. Commun.
7
,
10874
(
2016
).
2.
D.
Halbertal
,
J.
Cuppens
,
M. B.
Shalom
,
L.
Embon
,
N.
Shadmi
,
Y.
Anahory
,
H.
Naren
,
J.
Sarkar
,
A.
Uri
,
Y.
Ronen
,
Y.
Myasoedov
,
L.
Levitov
,
E.
Joselevich
,
A. K.
Geim
, and
E.
Zeldov
, “
Nanoscale thermal imaging of dissipation in quantum systems
,”
Nature
539
,
407
(
2016
).
3.
A.
Harzheim
,
J.
Spiece
,
C.
Evangeli
,
E.
McCann
,
V.
Falko
,
Y.
Sheng
,
J. H.
Warner
,
G. A. D.
Briggs
,
J. A.
Mol
,
P.
Gehring
, and
O. V.
Kolosov
, “
Geometrically enhanced thermoelectric effects in graphene nanoconstrictions
,”
Nano Lett.
18
,
7719
(
2018
).
4.
N.
Gächter
,
F.
Könemann
,
M.
Sistani
,
M. G.
Bartmann
,
M.
Sousa
,
P.
Staudinger
,
A.
Lugstein
, and
B.
Gotsmann
, “
Spatially resolved thermoelectric effects in operando semiconductor–metal nanowire heterostructures
,”
Nanoscale
12
,
20590
(
2020
).
5.
J.
Park
,
G.
He
,
R. M.
Feenstra
, and
A.-P.
Li
, “
Atomic-scale mapping of thermoelectric power on graphene: Role of defects and boundaries
,”
Nano Lett.
13
,
3269
(
2013
).
6.
P.
Zolotavin
,
C. I.
Evans
, and
D.
Natelson
, “
Substantial local variation of the Seebeck coefficient in gold nanowires
,”
Nanoscale
9
,
9160
(
2017
).
7.
J.
Fast
,
E.
Barrigon
,
M.
Kumar
,
Y.
Chen
,
L.
Samuelson
,
M.
Borgström
,
A.
Gustafsson
,
S.
Limpert
,
A.
Burke
, and
H.
Linke
, “
Hot-carrier separation in heterostructure nanowires observed by electron-beam induced current
,”
Nanotechnology
31
,
394004
(
2020
).
8.
R.
Mitra
,
M. R.
Sahu
,
A.
Sood
,
T.
Taniguchi
,
K.
Watanabe
,
H.
Shtrikman
,
S.
Mukerjee
,
A. K.
Sood
, and
A.
Das
, “
Anomalous thermopower oscillations graphene-InAs nanowire vertical heterostructures
,” arXiv:2009.08882 (
2020
).
9.
G.
Benenti
,
G.
Casati
,
K.
Saito
, and
R. S.
Whitney
, “
Fundamental aspects of steady-state conversion of heat to work at the nanoscale
,”
Phys. Rep.
694
,
1
(
2017
).
10.
B.
Roche
,
P.
Roulleau
,
T.
Jullien
,
Y.
Jompol
,
I.
Farrer
,
D. A.
Ritchie
, and
D. C.
Glattli
, “
Harvesting dissipated energy with a mesoscopic ratchet
,”
Nat. Commun.
6
,
6738
(
2015
).
11.
H.
Thierschmann
,
R.
Sánchez
,
B.
Sothmann
,
F.
Arnold
,
C.
Heyn
,
W.
Hansen
,
H.
Buhmann
, and
L. W.
Molenkamp
, “
Three-terminal energy harvester with coupled quantum dots
,”
Nat. Nanotechnol.
10
,
854
(
2015
).
12.
F.
Hartmann
,
P.
Pfeffer
,
S.
Höfling
,
M.
Kamp
, and
L.
Worschech
, “
Voltage fluctuation to current converter with Coulomb-coupled quantum dots
,”
Phys. Rev. Lett.
114
,
146805
(
2015
).
13.
G.
Jaliel
,
R. K.
Puddy
,
R.
Sánchez
,
A. N.
Jordan
,
B.
Sothmann
,
I.
Farrer
,
J. P.
Griffiths
,
D. A.
Ritchie
, and
C. G.
Smith
, “
Experimental realization of a quantum dot energy harvester
,”
Phys. Rev. Lett.
123
,
117701
(
2019
).
14.
S.
Dorsch
,
A.
Svilans
,
M.
Josefsson
,
B.
Goldozian
,
M.
Kumar
,
C.
Thelander
,
A.
Wacker
, and
A.
Burke
, “
Heat driven transport in serial double quantum dot devices
,”
Nano Lett.
21
,
988
994
(
2021
).
15.
B.
Rutten
,
M.
Esposito
, and
B.
Cleuren
, “
Reaching optimal efficiencies using nanosized photoelectric devices
,”
Phys. Rev. B
80
,
235122
(
2009
).
16.
O.
Entin-Wohlman
,
Y.
Imry
, and
A.
Aharony
, “
Three-terminal thermoelectric transport through a molecular junction
,”
Phys. Rev. B
82
,
115314
(
2010
).
17.
T.
Ruokola
and
T.
Ojanen
, “
Theory of single-electron heat engines coupled to electromagnetic environments
,”
Phys. Rev. B
86
,
035454
(
2012
).
18.
B.
Sothmann
and
M.
Büttiker
, “
Magnon-driven quantum-dot heat engine
,”
Europhys. Lett.
99
,
27001
(
2012
).
19.
J.-H.
Jiang
,
O.
Entin-Wohlman
, and
Y.
Imry
, “
Thermoelectric three-terminal hopping transport through one-dimensional nanosystems
,”
Phys. Rev. B
85
,
075412
(
2012
).
20.
R.
Bosisio
,
G.
Fleury
,
J.-L.
Pichard
, and
C.
Gorini
, “
Nanowire-based thermoelectric ratchet in the hopping regime
,”
Phys. Rev. B
93
,
165404
(
2016
).
21.
R.
Sánchez
and
M.
Büttiker
, “
Optimal energy quanta to current conversion
,”
Phys. Rev. B
83
,
085428
(
2011
).
22.
A. N.
Jordan
,
B.
Sothmann
,
R.
Sánchez
, and
M.
Büttiker
, “
Powerful and efficient energy harvester with resonant-tunneling quantum dots
,”
Phys. Rev. B
87
,
075312
(
2013
).
23.
M.
Büttiker
, “
Role of quantum coherence in series resistors
,”
Phys. Rev. B
33
,
3020
(
1986
).
24.
Y.
Xing
,
Q.-F.
Sun
, and
J.
Wang
, “
Influence of dephasing on the quantum Hall effect and the spin Hall effect
,”
Phys. Rev. B
77
,
115346
(
2008
).
25.
P.
Roulleau
,
F.
Portier
,
P.
Roche
,
A.
Cavanna
,
G.
Faini
,
U.
Gennser
, and
D.
Mailly
, “
Tuning decoherence with a voltage probe
,”
Phys. Rev. Lett.
102
,
236802
(
2009
).
26.
M.
Kilgour
and
D.
Segal
, “
Inelastic effects in molecular transport junctions: The probe technique at high bias
,”
J. Chem. Phys.
144
,
124107
(
2016
).
27.
Q.
Ma
,
F. D.
Parmentier
,
P.
Roulleau
, and
G.
Fleury
, “
Graphene np junctions in the quantum Hall regime: Numerical study of incoherent scattering effects
,”
Phys. Rev. B
97
,
205445
(
2018
).
28.
D.
Sánchez
and
L.
Serra
, “
Thermoelectric transport of mesoscopic conductors coupled to voltage and thermal probes
,”
Phys. Rev. B
84
,
201307
(
2011
).
29.
F.
Mazza
,
R.
Bosisio
,
G.
Benenti
,
V.
Giovannetti
,
R.
Fazio
, and
F.
Taddei
, “
Thermoelectric efficiency of three-terminal quantum thermal machines
,”
New J. Phys.
16
,
085001
(
2014
).
30.
F.
Mazza
,
S.
Valentini
,
R.
Bosisio
,
G.
Benenti
,
V.
Giovannetti
,
R.
Fazio
, and
F.
Taddei
, “
Separation of heat and charge currents for boosted thermoelectric conversion
,”
Phys. Rev. B
91
,
245435
(
2015
).
31.
R.
Sánchez
,
B.
Sothmann
, and
A. N.
Jordan
, “
Effect of incoherent scattering on three-terminal quantum Hall thermoelectrics
,”
Physica E
75
,
86
(
2016
).
32.
P.
Streda
, “
Quantised thermopower of a channel in the ballistic regime
,”
J. Phys.: Condens. Matter
1
,
1025
1027
(
1989
).
33.
C. R.
Proetto
, “
Thermopower oscillations of a quantum-point contact
,”
Phys. Rev. B
44
,
9096
(
1991
).
34.
M. A.
Cipiloğlu
,
S.
Turgut
, and
M.
Tomak
, “
Nonlinear Seebeck and Peltier effects in quantum point contacts
,”
Phys. Status Solidi B
241
,
2575
(
2004
).
35.
A. M.
Lunde
and
K.
Flensberg
, “
On the Mott formula for the thermopower of non-interacting electrons in quantum point contacts
,”
J. Phys.: Condens. Matter
17
,
3879
(
2005
).
36.
A.
Abbout
, “
Thermoelectric transport in quantum point contacts and chaotic cavities: Thermal effects and fluctuations
,” Ph.D. thesis (
University Paris VI
,
2011
).
37.
R. S.
Whitney
, “
Nonlinear thermoelectricity in point contacts at pinch off: A catastrophe aids cooling
,”
Phys. Rev. B
88
,
064302
(
2013
).
38.
S.
Pilgram
,
D.
Sánchez
, and
R.
López
, “
Quantum point contacts as heat engines
,”
Physica E
74
,
447
(
2015
).
39.
S.
Kheradsoud
,
N.
Dashti
,
M.
Misiorny
,
P. P.
Potts
,
J.
Splettstoesser
, and
P.
Samuelsson
, “
Power, efficiency and fluctuations in a quantum point contact as steady-state thermoelectric heat engine
,”
Entropy
21
,
777
(
2019
).
40.
H.
van Houten
,
L. W.
Molenkamp
,
C. W. J.
Beenakker
, and
C. T.
Foxon
, “
Thermo-electric properties of quantum point contacts
,”
Semicond. Sci. Technol.
7
,
B215
(
1992
).
41.
L. W.
Molenkamp
,
T.
Gravier
,
H.
van Houten
,
O. J. A.
Buijk
,
M. A. A.
Mabesoone
, and
C. T.
Foxon
, “
Peltier coefficient and thermal conductance of a quantum point contact
,”
Phys. Rev. Lett.
68
,
3765
(
1992
).
42.
A. S.
Dzurak
,
C. G.
Smith
,
L.
Martin-Moreno
,
M.
Pepper
,
D. A.
Ritchie
,
G. A. C.
Jones
, and
D. G.
Hasko
, “
Thermopower of a one-dimensional ballistic constriction in the non-linear regime
,”
J. Phys.: Condens. Matter
5
,
8055
(
1993
).
43.
N. J.
Appleyard
,
J. T.
Nicholls
,
M. Y.
Simmons
,
W. R.
Tribe
, and
M.
Pepper
, “
Thermometer for the 2D electron gas using 1D thermopower
,”
Phys. Rev. Lett.
81
,
3491
(
1998
).
44.
B.
Brun
,
F.
Martins
,
S.
Faniel
,
A.
Cavanna
,
C.
Ulysse
,
A.
Ouerghi
,
U.
Gennser
,
D.
Mailly
,
P.
Simon
,
S.
Huant
,
M.
Sanquer
,
H.
Sellier
,
V.
Bayot
, and
B.
Hackens
, “
Thermoelectric scanning-gate interferometry on a quantum point contact
,”
Phys. Rev. Appl.
11
,
034069
(
2019
).
45.
C.
Yan
,
M.
Pepper
,
P.
See
,
I.
Farrer
,
D. A.
Ritchie
, and
J.
Griffiths
, “
Thermoelectric property of a one dimensional channel in the presence of a transverse magnetic field
,”
Appl. Phys. Lett.
115
,
202102
(
2019
).
46.
C. W.
Groth
,
M.
Wimmer
,
A. R.
Akhmerov
, and
X.
Waintal
, “
Kwant: A software package for quantum transport
,”
New J. Phys.
16
,
063065
(
2014
).
47.
This is valid when kBθ is much smaller than the typical energy scale ΔE associated to the oscillations of ταβ(E). While ΔE decreases with |xT|, the oscillations of ταβ(E) are expected to average out at large |xT|.
48.
M.
Büttiker
, “
Chemical potential oscillations near a barrier in the presence of transport
,”
Phys. Rev. B
40
,
3409
3412(R
(
1989
).
49.
Formally, |SLT| becomes even larger and larger when the QPC is gradually pinched off (τ00) upon decreasing μ, but this regime is out of reach experimentally.
50.
M. A.
Topinka
,
B. J.
LeRoy
,
S. E. J.
Shaw
,
E. J.
Heller
,
R. M.
Westervelt
,
K. D.
Maranowski
, and
A. C.
Gossard
, “
Imaging coherent electron flow from a quantum point contact
,”
Science
289
,
2323
(
2000
).
51.
C.
Gorini
,
R. A.
Jalabert
,
W.
Szewc
,
S.
Tomsovic
, and
D.
Weinmann
, “
Theory of scanning gate microscopy
,”
Phys. Rev. B
88
,
035406
(
2013
).
52.
The convergence is not obvious in Figs. 3(b) and 3(d) limited to |xT|<45 but we have checked it by considering larger values of xT (and of W to eliminate finite-size effects that arise away from the QPC).
53.
M. P.
Jura
,
M. A.
Topinka
,
M.
Grobis
,
L. N.
Pfeiffer
,
K. W.
West
, and
D.
Goldhaber-Gordon
, “
Electron interferometer formed with a scanning probe tip and quantum point contact
,”
Phys. Rev. B
80
,
041303
(
2009
).
54.
R. A.
Jalabert
,
W.
Szewc
,
S.
Tomsovic
, and
D.
Weinmann
, “
What is measured in the scanning gate microscopy of a quantum point contact?
,”
Phys. Rev. Lett.
105
,
166802
(
2010
).
55.
Within linear response, R can also be written as R=|SLT|/(|SLL|+|SLT+SLL|). With this formula, the discussion held after Eq. (13) can be rephrased as follows. On the QPC steps, |SLL||SLT| and R is tiny. On the QPC plateaus, there are some tip positions for which SLL(xT,yT)=0 and so R=1. In that case, SLL(xT,yT)=SLT(xT,yT) [compare Figs. 2(f)–3(f) and Figs. 2(h),,3(h)], hence R(xT,yT)=1 too and we recover the expected symmetry law R(xT,yT)=R(xT,yT).
56.
D. M.-T.
Kuo
and
Y.-C.
Chang
, “
Thermoelectric and thermal rectification properties of quantum dot junctions
,”
Phys. Rev. B
81
,
205321
(
2010
).
57.
J.
Matthews
,
D.
Sánchez
,
M.
Larsson
, and
H.
Linke
, “
Thermally driven ballistic rectifier
,”
Phys. Rev. B
85
,
205309
(
2012
).
58.
Z. H.
Zhang
,
Y. S.
Gui
,
L.
Fu
,
X. L.
Fan
,
J. W.
Cao
,
D. S.
Xue
,
P. P.
Freitas
,
D.
Houssameddine
,
S.
Hemour
,
K.
Wu
, and
C.-M.
Hu
, “
Seebeck rectification enabled by intrinsic thermoelectrical coupling in magnetic tunneling junctions
,”
Phys. Rev. Lett.
109
,
037206
(
2012
).
59.
G.
Rosselló
,
R.
López
, and
R.
Sánchez
, “
Dynamical Coulomb blockade of thermal transport
,”
Phys. Rev. B
95
,
235404
(
2017
).
60.
G. T.
Craven
,
D.
He
, and
A.
Nitzan
, “
Electron-transfer-induced thermal and thermoelectric rectification
,”
Phys. Rev. Lett.
121
,
247704
(
2018
).

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