Laurens W. Molenkamp Physikalisches Institut, EP3 Universitt - - PowerPoint PPT Presentation

Laurens W. Molenkamp

Physikalisches Institut, EP3 Universität Würzburg

Onsager Coefficients

• I

electric current density

• J

particle current density

• JQ

heat flux, heat current density

• µ

chemical potential

• T

temperature

• V

voltage, electrostatic potential difference

                                               T V e T L T L T L T L J J J e I

Q Q



2 22 21 2 12 11

2 11

e T L   e T e ST L       

2 12

 

2 2 22

S T T L    

21 12

L L 

From: R.D. Barnard Thermoelectricity in Metals and Alloys (1972)

Thermoelectric Properties

                           T e K M L G Q I / 

Onsager-relation: M = -LT “fluxes” “forces”

                              T I S R Q V 

                             

  

K GT S K T Q ST G M I Q

I T 2

1 

G L T e S

I

          

0

/ 

Diffusion Thermopower

Thermoelectric Properties

   

  

  

                        

2 2 2 2 2

) ( 2 ) ( 2 ) ( 2 T k E E E t E f dE e k h e T K T k E E E t E f dE e k h e L E t E f dE h e G

B F B B F B

Landauer-Büttiker-Formalism:

G L T e S

I

          

0

/ 

eT E S   

 

         E f T k E E

B F

• dd function in E

 L large for t(E) asymmetric around EF

Thermopower (S)

• Kelvin-Onsager relation (1931)
• Heike’s formula
• Mott relation

F

E B B

dE dG G T k q k S 3

2

  

eT E T G L S

I

     

0

 

i f B

g g k e e S ln ln 1 1       S

(spin) entropy contribution linear response

 

S    T Q

thermal energy to transfer one electron from a hot to a cold reservoir

Thermopower (S)

lim

  

   

I th T

T V S

Measuring Thermopower

?

reservoir 1 reservoir 2 µ,T

l l

µ ,T

r r

sample

cold hot

Thermopower Measurement

?

reservoir 1 reservoir 2 µ,T

l l

µ ,T

r r

sample

cold hot lim :

  

  

I th T

T V S lim :

  

  

I th T

T V S

Current Heating Technique

 



L e qpc dot th

T T S S V V V     

2 1

 



L e qpc dot th

T T S S V V V     

2 1

Current Heating Technique

• energy dissipation at the channel entrance
• nly hot electron gas within channel

(1 ps ≈ ee << eph ≈ 0.2 ns)

• energy relaxation in the reservoir
• diffusion thermopower

T = 10 mK, x = 500 nm  20 K/mm

• QD and QPC create thermovoltages

which can be measured as voltage difference between V1 and V2 V1-V2 = (SQD -SQPC) T = SQD T SQPC can be adjusted to zero

• ac-excitation and detection:

Pheat ~ [I sin(t)]2 ~ sin(2t) (z

 



L e qpc dot th

T T S S V V V     

2 1

 



L e qpc dot th

T T S S V V V     

2 1

First Experiments: Thermopower of a QPC

100 In semiconductors, at low T, ps. nearly thermalized hot electron distribution in the heating channel

e p

   

Step-by-step Barrier

Each channel in the point contact acts as a potential barrier, hence the thermopower shows a series of peaks

L.W. Molenkamp et al., Phys. Rev. Lett. 65, 1052 (1990).

Thermopower of a QPC

• H. van Houten et al., Semicond. Sci. Technol. 7, B215 (1992)

   

   

 

 

    

      

1 1 2

1 ln 1 ln 2 / exp 1 ln

n B B F B

n n

e e e k h e L T k E T k fdE

 

1 ; if 2 ln

2 1

     N E E N e k S

N F B

 

1 if 1

1

    N e k S

B



quantized thermopower

E1 E2

1 ; if    N E E S

N F

Reference QPC

• Voltage Probes have to be at same temperature and of the same material
• QPC can be used as a reference since TP of QPC is known (can be adjusted to zero)
• G of QPC is quantized – and therefore, so is S. This can be used as a method of

temperature calibration

V V VT,A VT,B VT,A‘ VT,B

L.W. Molenkamp et al., Phys. Rev. Lett. 65, 1052 (1990). L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992). A.A.M. Staring et al., Europhys. Lett. 22, 57 (1993).

• S. Möller et al., Phys. Rev. Lett. 81, 5197 (1998).

S.F. Godijn et al., Phys. Rev. Lett. 82, 2927 (1999).

• R. Scheibner et al., Phys. Rev. Lett. 95, 176602 (2005).
• R. Scheibner et al., Phys. Rev. B75, 041301(R) (2007).

Peltier Coefficient

L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992).

Theoretical estimate for Peltier coefficient is within factor of 2 from observed signal. Peltier heating/cooling linear in current, detect only 1f signal!

Thermal Conductance

L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992).

again within factor of 2 from the

• bserved signal.

Wiedemann-Franz yields thermal conductance quantum.

What about the Channel Resistance?

Signal amplitude fully in agreement with band model

0.4 V/K S  

Channel resistance reflects hydrodynamic Electron Flow

Signal amplitude fully in agreement with band model

0.4 V/K S  

Knudsen- and Poiseuille Flow-Regime

Molekularströmung Knudsen-Strömung Knudsen-Maximum Poiseuille-Strömung

[ ]

Teilchendichte n

li Vdrift

) , (

e ee

n T l n  ) , (

e ee

n T l n 

                                 1 2 ln ln 1

2 F B F F B F F ee

k q T k E E T k hv E l

Gurzhi, Shevchenko, JETP (1968) Guiliani and Quinn, PRB (1982)

20 40 60 80 100



e e

/ m 

1 2 3 4 5

T / K

3 x 1015 m-1 2.5 x 1015 m-1 2.0 x 1015 m-1

Electron-Electron Scattering Length in 2D

                                 1 2 ln ln 1

2 2 F D F B F F B F F ee

k q T k E E T k hv E l                                  1 2 ln ln 1

2 2 F D F B F F B F F ee

k q T k E E T k hv E l

Guiliani and Quinn, PRB (1982)

Hydrodynamic Electron Flow

120 µm 4 µm 4 µm 4 µm 20 µm

• 30
• 25
• 20
• 15
• 10
• 5

5 10 15 20 25 30 260 270 280 290 300 310

dV/dI ( ) I (µA)

60 µm

I (µA)

• 15
• 10
• 5

5 10 15

dV/dI 5  2

2 R W k e h W L R

F

    

eff F

L mv ne2  





W eff W dy F eff

y l mv ne L

2

) ( ~

Hydrodynamic Electron Flow

120 µm 4 µm 4 µm 4 µm 20 µm

• 30
• 25
• 20
• 15
• 10
• 5

5 10 15 20 25 30 260 270 280 290 300 310

dV/dI ( ) I (µA)

60 µm

2

2 R W k e h W L R

F

    

eff F

L mv ne2  





W eff W dy F eff

y l mv ne L

2

) ( ~

ee-Scattering in 2D

• 1. small angle scattering

1  

F B

E T k  

• 2. ee-scattering for

p p    

     

scattering angle

k=0 kF 1

2

kF

Energy and Momentum-Relaxation

Due to the different scattering processes there exist two different relaxation times for symmetric and asymmetric processes:

Gurzhi et al., Adv. Phys. 1987

Momentum-Relaxation:

F B

v T k q /  

4 2 

          T T kB

F ee a ee

  

4 2 

          T T kB

F ee a ee

   Energy-Relaxation:  

F B F F

T k k k k q 

       

  2 

  T

ee s ee

 

2 

  T

ee s ee

 

Additional hydrodynamic regime in 2D?

Three transport regimes:

• 1. Knudsen:
• 2. 1d-Diffusion:
• 3. Poiseuille:

a s

l l d  /

2 a s F B s

l l d T k l   / /

2



s a

l d l /

2



R.N. Gurzhi et al., Phys. Rev B 74, 3872 (1995)

Not so easy to observe in heating experiment….

Electron Beam in a 2DEG

Semiclassical collimation mechanisms

(Semi-classical) action is constant of the motion.

Electron beam Propagation in a 2DEG

An anisotropic electron momentum distribution remains present

• ver a long time, while the energy relaxation is fast.

e-beam injection

Electron Beam Propagation

kinetic equation: I: Integral of electron-electron scattering Integration over all p´p results in the scattering angle distribution function: g()

Considers all electron that scattered, but still preserve the momentum due to small angle scattering events.

Feedback

• H. Buhmann et al., Fiz.Nizk. Temp. 24, 978 (1998)

Energy Dependent Scattering

Model Calculation:

: angle dependent detection

Experiment

Sample Structure

Experimental Result

i. lee > L: Vd ~ Vi i. lee ≈ L: electron with  > 0 before reaching d ii. lee < L: increased ee-scattering causes heating of the 2DEG iii. increasing heating results in a thermovoltage

Thermoelectric Effect

Thermoelectric Effect

• H. Predel et al., Phys. Rev. B 62, 2057 (2000)

Angle resolved ee-scattering in a 2D system

Angle resolved ee-scattering in a 2D system

Angle resolved ee-scattering in a 2D system

i d L L 2 1 3 2 4 B x y

2

max

Measurement

Energy

F F i

i

+

Lock-In ref

i

Measurement

 

s

 Measurement ballistic part scattered electrons

From experiment at zero excess energy, corrected for lee Simply by subtracting the ballistic part

Result

4 2



s

15 10 5

B(mT) Experiment Theory

Opening Angle Distribution

1.0 0.8 0.6 0.4 0.2 0.0 6 5 4 3 2 1

2D 3D

Model allows for extraction of opening angle from experiment (Note: „3D“ Model does not include collimation effects)

• H. Predel, PhD Thesis 2001

Yanovski et al., Europhys. Lett., 56, 709 (2001)

Negative beam Signal – Vortices? Uuumh…

Thermopower of quantum dots

A E D C B F

2DEG Au-gates

QD

• hmic

contacts

electron heating channel

• GaAs/AlGaAs - 2DEG
• n = 2.3 1011 cm-2, µ = 106 cm2/Vs
• Ti/Au-surface electrodes
• (opt. and e-beam lithography)
• Au/AuGe - ohmic contacts

quantum dot

17 nm 38 nm 20 nm 0.4 GaAs 1.33 x 10

18

GaAs Al0.33Ga0.67As Al0.33Ga0.67As

surface valence band conduction band 2 DEG semiconducting GaAs substrate growth direction

cm Si

• 3

m

Quantum Dot (QD)

• Constant Interaction model:

– QD = small capacitor – energies depend linearly on Vgate – coefficients do not depend

• n N (number of electrons)
• Energy needed to add one electron:

– qm. Energy Eqm ~ 100 µeV – Coulomb Interaction EC = ½ e2/C ~ 2 meV – EC = Eqm + EC

• Parameters accessible in

conventional transport experiments

QD Vgate VSD CD Cgate CS

Transport Properties

linear transport

non-linear transport:

• capacitive coupling of leads and QD
• strong influence on hybridization of leads and QD

Thermopower of a QD

sequential tunneling

Vgate Vth e - like

N-1 N N+1

positive contribution to the thermovoltage zero thermovoltage

+

+

Thermopower of a QD

Vth e - like

N-1 N N+1

positive contribution to the thermovoltage zero thermovoltage

Vgate

negative contribution to the thermovoltage

sequential tunneling

h - like

+

Thermopower of a QD

Vth e - like

N-1 N N+1

positive contribution to the thermovoltage zero thermovoltage

Vgate

negative contribution to the thermovoltage zero thermovoltage

sequential tunneling

h - like

+

Thermopower of a QD

Vth

N-1 N N+1

Vgate

sequential tunneling

T E V

gap T 

h - like

Thermopower of a QD

sequential tunneling

Large, metallic-like QD N ~ 300 T ~ 230 mK EC ~ 0.3 meV E C / kBT ~ 15

A.A.M. Staring et al., Europhys. Lett. 22, 57 (1993).

Thermopower of a QD

sequential tunneling

small QD N ~ 15 T ~ 1.5 K EC ~ 2 meV E C / kBT ~ 15

Sample: Bo_I13C

• 1.0
• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 3.0
• 2.0
• 1.0

0.0 1.0 2.0 3.0 0.00 0.02 0.04

• VT (µV)

VE (V)

G (e

2/h)

S 

Thermopower of a QD

cotunneling contribution suppression of thermovoltage

Thermopower of a QD

cotunneling contribution

[M. Turek and K.A. Matveev, PRB, 65, 115332 (2001)]

• 1.0
• 0.8
• 0.6

8.0 6.0 4.0 2.0 0.0

• 2.0
• 4.0
• 6.0
• 8.0

VT* (µV)

VE (V)

N ~ 15 EC ~ 2 meV E C / kBT ~ 230

• R. Scheibner et al., PRB 75, 041301 (2007)

T ~ 100 mK T ~ 1.5 K

Thermopower of a QD

cotunneling contribution

• R. Scheibner et al., PRB 75, 041301 (2007)

no signatures of cotunneling processes in the CB regime

Chaotic Quantum Dot

ns = 3.4 x 1011 cm-2 Gqpc = 4 e2 / h  = 1 x 106 cm2 / (V sec) (Nqpc = 2 )

800 nm 700 nm

dot gate

Conductance Fluctuations

• 200
• 100

100 200

• 450
• 500
• 550
• 600
• 650
• 700
• 750

B / mT V

Gate / mV

2.0 1.9 1.8 1.8 1.7 1.6 1.5 1.5 1.4 1.3 1.2 1.2 1.1 1.0 0.92 0.84 0.77 0.69 0.61 0.54 0.46 0.38 0.30 0.23

e2/h

statistical ensemble Vgate = 10 mV  weak localization and short trajectories Vgate = - 600 mV

• 200
• 100

100 200 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

G / (e

2/h)

B / (mT)

T = 20 mK

Conductance Fluctuation Distribution

transmission probability distribution:  = 2: equal distribution

• nly for N1=N2 >> 1: gaussian distribution

here: N1 = 1, N2 = 2

1 , ) (

2 / 1 2 1

  

 

t t t p





 = 2 Problems:

• current transport in a QD is

accompanied by dephasing

• weak localisation and short

trajectories obscure effects of chaotic transport

Weak Localization

• 40
• 20

20 40 1.2 1.4 1.6 1.8 B / mT

G / (e / h)

2

averaged conductance

 

c

G G B G      

 

/ 1 ) (

1 2  



 

c

G G B G      

 

/ 1 ) (

1 2  



1/2

with and

ergodic c dwell

BA h e             Fit: G=2 = 1.3 e2/h G=1 = 0.3 e2/h c / A = 6 mT A = 0.3 m2 (Aeff = 1.15 A)

1 3 . 5 /

2

            

c ergodic dwell

e h   1 3 . 5 /

2

            

c ergodic dwell

e h  

i.e. the QD is chaotic!

Thermovoltage Fluctuations

Th

gate

statistical ensemble Vgate = 10 mV Vgate = - 550 mV T = 20 mK

B / mT V

Th / V

B / mT V

Th / V

Iheating = 40 nA T  235 mK

• 100
• 50

50 100 B / mT

• 450
• 500
• 550
• 600
• 650
• 700
• 750

V / mV

g a t e

  V = 0.46 V

t h

Thermopower Fluctuation Distribution

RMT: analytic form for N1 = N2 = 1 ) 1 ( ) (

2 1

G G c E G       

1 2 3 4 P(S)

• 60
• 40
• 20

0.0 20 40 60 S / ( V / K )



1 2 3 4 5 6 P(S)

• 60
• 40
• 20

0.0 20 40 60 S / ( V / K )



p(S) S / a.u.

 = 2  = 1

p(S) S / a.u.

 = 2  = 1

here: N1 = N2 = 2

• S. Godijn et al., PRL 82, 2927 (1999)

Residual Charging Energy

2DEG

dot

2DEG

characteristic time scale: Luttinger liquid theory:

h U h E

dwell erg

/ /

*

    

N

t U U ) 1 (

*

 

(Flensberg, 1993, 1994)

chaotic QD:

(Aleiner and Glazman, 1998)

 1 t

 

          E U U E t

2

ln 1

 

          E U U E t

2

ln 1

Thermopower in the Coulomb-Blockade Regime

limit) (classical /

2 C

e T k E

B

  

2

Theory

• 1
• 1

2

S / (kB/e)

0.0 0.5

• 0.5

1.0

G (arb.units)

171 173 175

EF / (e2/2C)

T=45mK

Experiment

• 938 -932 -926

V (mV)

gate

1.0 0.5 0.0

• 0.5

V ( V)

th 

0.2 0.1 0.0

• 0.1

G (e /h)

2

 

eT U E e C e N eT S

F ext

2 2 1

* 2 2 1

               

 

eT U E e C e N eT S

F ext

2 2 1

* 2 2 1

               

Beenakker et al., PRB 46, 9667 (92) Staring et al., Europhys. Lett. 22, 57 (93) Molenkamp et al., Semicond. Sci. Technol. 9, 903 (94)

Scaling Experiment

2 1

e L L

tunneling regime (G << 2e2/h) scanned G = 0.06...0.82 G0

Scaling Results

0.06 0.19 0.29 0.38 0.43 0.82

F 2

Theory kBT/U* = 0.22 0.25 0.30 0.33 0.37 0.45

Thermovoltage Thermopower Iheating = 40 nA Te = 255 mK, TL = 40 mK

Scaling

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

U* / U0

Luttinger liquid behaviour chaotic behaviour extrapolation

1  t

* exp

45 . ) 1 ( U t U  

* exp

45 . ) 1 ( U t U  

theory E = 23 eV U0 = 100 eV

2 *

49 . ln U E U U E U U          

2 *

49 . ln U E U U E U U          

• S. Möller et al., PRL 81, 5197 (1998)

Spin-Correlated QD

• existence of a magnetic moment
• n the QD can lift the CB
• transport mechanism: spin scattering
• hybridization of free electrons in the

leads with localized magnetic moment leads to resonance at the Fermi edge

Kondo Resonance

Spin-Correlated QD

0.4 0.6 0.8 1.0 1.2 1.4 1.6

• 1.0
• 0.5

0.0 0.5 1.0 1.5 2.0

• 10.0
• 5.0

0.0 5.0 10.0 15.0 20.0

• Vgate E / Volt

(dI/dV) / (e

2/h)

Conductance dI/dVBias Thermovoltage Vth

Vthermo / µV

strong coupling Kondo Regime weak coupling Cotunnel-Regime

0.8 0.9 1.0 1.1 1.2

• 0.4
• 0.2

0.0 0.2 0.4

• 4.0
• 2.0

0.0 2.0 4.0

(dI/dV) / (e

2/h)

Conductance dI/dVBias Thermovoltage Vth Mott - Thermopower (scaled to fit)

Vthermo / µV

Spin-Correlated QD

h - like e - like

Asymmetry between electron- and hole-like transport: Mixed-valence regime

Scheibner et al. PRL 95, 176602 (2005) ~ 1 eV

Spin-Entropy Transport

S = kB(ln gf - ln gi) = kB(ln 2 - ln 1) = kB ln 2 Entropy change S adding one electron to an empty site:

gs = 1 gs = 2 gs = 1

1. e-like transport from the hot to the cold reservoir S = kB(ln gf - ln gi) = kB(ln 2 - ln 1) = kB ln 2  SSE=-kB/e ln 2 2. h-like transport from the cold to the hot reservoir S = kB(ln gf - ln gi) = kB(ln 1 - ln 2) = -kB ln 2  SSE=-kB/e ln 2 3. e-like transport from the hot to the cold reservoir S = kB(ln gf - ln gi) = kB(ln 1 - ln 2) = -kB ln 2  SSE=kB/e ln 2 4. h-like transport from the cold to the hot reservoir S = kB(ln gf - ln gi) = kB(ln 2 - ln 1) = kB ln 2  SSE=kB/e ln 2

• R. Scheibner et al., PRL 95, 176602 (2005)
• R. Scheibner, PhD-Thesis, Würzburg 2007
• Vg
• 1. 2. 3. 4.

S0 SSE S

Spin entropy contributions

Thermal Rectifier

• R. Scheibner et al., NJP 10, 083016 (2008)

Thermal Rectifier

• R. Scheibner et al., NJP 10, 083016 (2008)

Thermal Rectifier

asymmetrically coupled states

Thermal Rectifier

• R. Scheibner et al., NJP 10, 083016 (2008)

 

) ( ) , ( ) , ( E t T E f T E f h dE J

R L tot

         

  

       (energy) (electron)  E e

   

     

 

     

       

dE df h e dE df Th e

E t dE E t E dE L L S ) ( ) (

2

11 12



     

T E E f E A E t

Z ,

2 / 2 / ) (

2 2 2

      

1E-3 0.01 0.1 1

• 2.20
• 2.15
• 2.10
• 2.05
• 1

1 2 3

N+2 N+1

GQD (e

2/h)

SQD (kB/e) VP (V)

250, 0, 400

Z

E eV    

Thermal Rectifier

thermal conductance, L22, for a temp. reversal

• R. Scheibner et al., NJP 10, 083016 (2008)
• 2.12
• 2.11
• 2.10
• 2.09
• 2.08
• 2.07

0.0 0.5 1.0 1.5

• 2.120
• 2.115
• 2.110
• 2.105
• 2.100
• 30
• 20
• 10

10 20 30

L22 (10

• 9e

2V 2/hK)

VP (V) VP (V)

L22 (10

• 12e

2V 2/hK)

0.0 0.2 0.4 0.6 0.8

• 250

250

• 2.2
• 2.1
• 2.0
• 2

2

GQD (e

2/h)

L12 (10

• 3 e

2kB/ h)

SQD (kB/e) VP (V)

thermal conductivity

 

11 21 12 22

/ L L L L   

1 2 2 1

/ 0.1

T T T T

         

• 2.12
• 2.11
• 2.10
• 2.09
• 2.08
• 2.07
• 10

10 20 30 40 50 60

• 2.125
• 2.120
• 2.115
• 2.110
• 2.105
• 2.100
• 2.095
• 4
• 3
• 2
• 1

1

 (fW/K)

VP (V) VP (V)  (fW/K)

Double Quantum Dot: towards multi-terminal thermoelectrics

17 nm 38 nm 20 nm 0.4 GaAs 1.33 x 10

18

GaAs Al0.33Ga0.67As Al0.33Ga0.67As

surface valence band conduction band 2 DEG semiconducting GaAs substrate growth direction

cm Si

• 3

m

• GaAs/AlGaAs - 2DEG
• n = 2.11011 cm-2, µ = 7 x 105 cm2/Vs
• Ti/Au-surface electrodes

(opt. and e-beam lithography)

• Au/AuGe - ohmic contacts

PC PG 2µm 300nm

QDG QDC

D S H D H S

Characterization

PG PC

All reservoirs at TC = 80 mK

• 450
• 425
• 400
• 375
• 350
• 325

0.3 0.6

VPC / mV VPG / mV

U: single QD charging energy

QDG QDC

ID G / e²/h

UQDC UQDG

Stability Diagram: Regions of high conductance delimit regions with fixed charge configuration

Characterization

PG PC

All reservoirs at TC = 80 mK

• 530
• 525
• 520
• 390
• 380
• 370

0.0 0.17 0.35

VPC / mV VPG / mV

G / e²/h

• 450
• 425
• 400
• 375
• 350
• 325

0.3 0.6

VPC / mV VPG / mV

ID

(N,M)

(N+1, M) (N+1, M+1) (N,M+1) G / e²/h

• ccupation numbers

QDC: M QDG: N

Characterization

PG PC

All reservoirs at TC = 80 mK

• 530
• 525
• 520
• 390
• 380
• 370

0.0 0.17 0.35

VPC / mV VPG / mV

G / e²/h

• 450
• 425
• 400
• 375
• 350
• 325

0.3 0.6

VPC / mV VPG / mV

ID

(0,0)

(1,0) (1, 1) (0,1) G / e²/h EC ~ 90 µeV EC

• ccupation numbers

QDC: M = 0 QDG: N = 0

µD

• 530
• 525
• 520
• 390
• 380
• 370

0.0 0.17 0.35

VPC / mV VPG / mV

G / e²/h

(0, 1) (1,1)

QDG QDC

S

D H

µH µS

Stability Vertex

• 530
• 525
• 520
• 390
• 380
• 370

0.0 0.17 0.35

VP2 / mV VP1 / mV

G / e²/h

• ccupation numbers

QDG: N QDC: M

µD

(1,1) (1, 1)

QDG QDC

S

D H

µH µS

Ec

Stability Vertex

Stability Vertex

• 530
• 525
• 520
• 390
• 380
• 370

0.0 0.17 0.35

VPC / mV VPG / mV

G / e²/h

µD

(1,1) (1, 0)

QDG QDC

S

D H

µH µS

Ec

(0,1) (1, 1)

Ec

Hunting an Energy Harvester

Allowing charge fluctuations on dot 1 enables (disables) charge transport through dot 2

What we really wanted is slightly different:

Energy harvesting

Original proposal:

• R. Sanchez, M.Büttiker, Phys. Rev. B 83, 085428 (2011)

and it actually occurs in between the triple points, where thermal gating is not dominant – but only for asymmetric barriers:

Energy harvesting