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 • J Q heat flux, heat current density • µ chemical potential • T temperature • V voltage, electrostatic potential difference  T  L 11 2   e L L    11 12             I J e V 2            T T e L L        12 21  L L  J     T J     21 22   Q Q    2  2  ST T T T     L 12 e e From: R.D. Barnard Thermoelectricity in Metals and Alloys (1972)       2 2 L T T S 22

Thermoelectric Properties        /  I G L e        “fluxes” “forces”              Q M K T Onsager-relation: M = -LT         V R S I                        Q T Diffusion Thermopower     / e L        Q M     S   ST      I G T G   0 T  0 I      2 Q S GT          1 K        T K  0 I

Thermoelectric Properties Landauer-Büttiker-Formalism:  2 2  e f    ( ) G dE t E  0 h E         E E f   2   F 2  odd function in E e k f E E       k T E ( ) B F L dE t E B  L large for t(E)  0 h e E k T asymmetric around E F B   2 2     2   2 K e k  f E E     ( ) B  F  dE t E    0   T h e E k T B     / E e L         S S    T G  0 eT I

Thermopower (S) • Kelvin-Onsager relation (1931)  E L      S G T eT thermal energy    0    I S Q T to transfer one electron from a hot to a cold reservoir • Heike’s formula   1 1       S ln ln S k g g B f i e e (spin) entropy contribution • Mott relation  2 k k T dG   B B S 3 q G dE E F linear response

Thermopower (S) Measuring Thermopower  V   lim th S  T   0  T 0 I reservoir 1 reservoir 2 cold ? hot µ,T µ ,T l l r r sample

Thermopower Measurement reservoir 2 reservoir 1 cold ? hot µ,T µ ,T l l r r sample V V     : : lim lim th th S S       0 0 T T T T   0 0 I I

Current Heating Technique                   V V V V V V S S S S T T T T 1 1 2 2 th th dot dot qpc qpc e e L L

Current Heating Technique                   V V V V V V S S S S T T T T 1 1 2 2 th th dot dot qpc qpc e e L L • energy dissipation at the channel entrance • QD and QPC create thermovoltages which can be measured as voltage difference between V 1 and V 2 • only hot electron gas within channel (1 ps ≈  ee <<  eph ≈ 0.2 ns) V 1 -V 2 = (S QD -S QPC )  T = S QD  T • energy relaxation in the reservoir S QPC can be adjusted to zero • diffusion thermopower • ac-excitation and detection: ~ [I sin(  t)] 2 P heat  T = 10 mK,  x = 500 nm  20 K/mm ~ sin(2  t) (  z 

First Experiments: Thermopower of a QPC    100 In semiconductors, at low T, ps. e p  nearly thermalized hot electron distribution in the heating channel

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 k        1 if 1 B S N 1 e E 2 E 1 H. van Houten et al., Semicond. Sci. Technol. 7 , B215 (1992)         ln 1 exp / fdE k T E k T B F B 0       2  2 e k   1         ln 1 ln 1 B L e e n n h e  1 n quantized thermopower ln 2 k        0 if ; 1 if ; 1 B S E E N S E E N  F N F N 1 e N 2

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 V T,A V T,B V T,B V T,A‘ 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 Theoretical estimate for Peltier coefficient is within factor of 2 from observed signal. Peltier heating/cooling linear in current, detect only 1f signal! L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992).

Thermal Conductance again within factor of 2 from the observed signal. Wiedemann-Franz yields thermal conductance quantum. L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992).

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 2         1 2 n  n  E k T E q ( ( , , ) )          ln ln 1 l l T T n n   F B F             ee ee e e l hv E k T k   ee F F B F Gurzhi, Shevchenko, JETP (1968) Guiliani and Quinn, PRB (1982) Knudsen-Maximum l i Knudsen-Strömung Molekularströmung Poiseuille-Strömung V drift [ ] Teilchendichte n

Electron-Electron Scattering Length in 2D 2 2                 2 2 1 1 2 2 D D E E k k T T E E q q                   ln ln ln ln 1 1     F F B B F F F F             100             l l hv hv E E   k k T T k k   ee ee F F F F B B F F 80 Guiliani and Quinn, PRB (1982) 3 x 10 15 m -1 60 m  / 2.5 x 10 15 m -1 e e  40 20 2.0 x 10 15 m -1 0 1 2 3 4 5 T / K

Hydrodynamic Electron Flow 310 120 µm 4 µm 4 µm dV/dI 4 µm 300 60 µm 20 µm  0 5 290 dV/dI (  ) 280 -15 -10 -5 0 5 10 15 I (µA)  L h 270    R R 0  2 2 W e k W F ne 2 260   L -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 eff mv F I (µA) 2 W ~ ne   ( ) dy L l y eff eff W mv 0 F

Hydrodynamic Electron Flow 310 120 µm 4 µm 4 µm 4 µm 300 60 µm 20 µm 290 dV/dI (  ) 280  L h 270    R R 0  2 2 W e k W F ne 2 260   L -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 eff mv F I (µA) 2 W ~ ne   ( ) dy L l y eff eff W mv 0 F

ee-Scattering in 2D     1. small angle scattering 2. ee-scattering for p p k=0 k F k F 2 1  k T scattering angle         1 B  E F

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: q k T v B F 2 2                     4 4 a a F F T T     ee ee ee ee     k B k B T T Energy-Relaxation:       k   q k T   B    k k F F F           2 2 s s T T ee ee ee ee

Additional hydrodynamic regime in 2D? R.N. Gurzhi et al., Phys. Rev B 74, 3872 (1995) Three transport regimes:  2 / 1. Knudsen: d l l s a    2 / / 2. 1d-Diffusion: l k T d l l s B F s a  2 / 3. Poiseuille: l d l a s Not so easy to observe in heating experiment….

Electron Beam in a 2DEG

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