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 V e T L T L T L T L J J J e I Q Q 1 1 11 11 e T L e T e ST L 1 S T T L 1 L 1 L From: R.D. Barnard Thermoelectricity in Metals and Alloys (197)
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 0 0 1 G L T e S I 0 / Diffusion Thermopower
Thermoelectric Properties 0 0 0 ) ( ) ( ) ( 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 odd function in E L large for t(e) asymmetric around E F
Thermopower (S) Kelvin-Onsager relation (1931) L S G T Heike s formula S 1 S e I 0 1 k e B ln E et g f ln g i Q TS (spin) entropy contribution thermal energy to transfer one electron from a hot to a cold reservoir Mott relation S 3 k q B kbt G dg de E F linear response
Thermopower (S) Measurement of the Thermopower S lim0 T V T th I 0 reservoir 1 reservoir cold? µ,t l l µ,t sample r r hot
Thermopower Measurement reservoir 1 reservoir cold? µ,t l l µ,t sample r r hot S : V th lim T T 0 I 0
Current Heating Technique V th V 1 V S dot S qpc T e T L
Current Heating Technique V th V 1 V S dot S qpc T e T L energy dissipation at the channel entrance only hot electron gas within channel (1 ps ee << eph 0. ns) energy relaxation in the reservoir diffusion thermopower T = 10 mk, x = 500 nm 0 K/mm QD and QPC create thermovoltages which can be measured as voltage difference between V 1 and V V 1 -V = (S QD -S QPC ) T = S QD T S QPC can be adjusted to zero ac-excitation and detection: P heat ~ [I sin(t)] ~ sin(t) (z
First Experiments: Thermopower of a QPC In semiconductors, at low T, 100 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, 105 (1990).
Thermopower of a QPC H. van Houten et al., Semicond. Sci. Technol. 7, B15 (199) 1 1 0 ln 1 ln 1 / exp ln 1 n B B F B n n e e e k h e L T k E T k fde 1 ; if ln 1 N E E N e k S N F B 1 if 1 1 N e k S B quantized thermopower E 1 E 1 ; if 0 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 one method of temperature calibration V V V T,A V T,B V T,A V T,B L.W. Molenkamp et al., Phys. Rev. Lett. 65, 105 (1990). L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (199). A.A.M. Staring et al., Europhys. Lett., 57 (1993). S. Möller et al., Phys. Rev. Lett. 81, 5197 (1998). S.F. Godijn et al., Phys. Rev. Lett. 8, 97 (1999). R. Scheibner et al., Phys. Rev. Lett. 95, 17660 (005). R. Scheibner et al., Phys. Rev. B75, 041301(R) (007).
Peltier Coefficient Theoretical estimate for Peltier coefficient is within factor of from observed signal. L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (199).
Thermal Conductance again within factor of from the observed signal. Wiedemann-Franz yields thermal conductance quantum. L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (199).
Next: quantum dots surface GaAs/AlGaAs - DEG n =.3 10 11 cm -, µ = 10 6 cm /Vs Ti/Au-surface electrodes (opt. and e-beam lithography) Au/AuGe - ohmic contacts growth direction 17 nm Al 0.33 Ga 0.67 As 38 nm 1.33 x 10 18-3 cm Si 0 nm 0.4 m GaAs Al 0.33 Ga 0.67 As GaAs DEG ohmic contacts QD semiconducting GaAs substrate B valence band conduction band C Au-gates electron heating channel DEG A F E D quantum dot
Quantum Dot (QD) Constant Interaction model: QD = small capacitor energies depend linearly on V gate coefficients do not depend on N (number of electrons) Energy needed to add one electron: qm. Energy E qm ~ 100 µev Coulomb Interaction E C = ½ e /C ~ mev E C = E qm + E C QD C S Parameters accessible in conventional transport experiments V SD C D C gate V gate
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 e - like sequential tunneling + positive contribution to the thermovoltage V th zero thermovoltage V gate N-1 N N+1
Thermopower of a QD e - like sequential tunneling + positive contribution to the thermovoltage V th zero thermovoltage h - like V gate N-1 N N+1 negative contribution to the thermovoltage
Thermopower of a QD e - like sequential tunneling + positive contribution to the thermovoltage V th zero thermovoltage h - like V gate zero thermovoltage N-1 N N+1 negative contribution to the thermovoltage
Thermopower of a QD sequential tunneling + V T E gap T V th h - like V gate N-1 N N+1
Thermopower of a QD sequential tunneling Large, metallic-like QD N ~ 300 T ~ 30 mk E C ~ 0.3 mev E C / k B T ~ 15 A.A.M. Staring et al., Europhys. Lett., 57 (1993).
Thermopower of a QD sequential tunneling S G (e /h) - V T (µv) 0.04 0.0 0.00 3.0.0 1.0 0.0-1.0 -.0-3.0-1.0-0.9-0.8-0.7-0.6-0.5 V E (V) small QD N ~ 15 T ~ 1.5 K E C ~ mev E C / k B T ~ 15 Sample: Bo_I13C
cotunneling contribution Thermopower of a QD suppression of thermovoltage
cotunneling contribution Thermopower of a QD [M. Turek and K.A. Matveev, PRB, 65, 11533 (001)] -8.0 V T * (µv) -6.0-4.0 -.0 0.0.0 T ~ 100 mk T ~ 1.5 K N ~ 15 E C ~ mev E C / k B T ~ 30 4.0 6.0 8.0-1.0-0.8-0.6 V E (V) R. Scheibner et al., PRB 75, 041301 (007)
cotunneling contribution Thermopower of a QD no signatures of cotunneling processes in the CB regime R. Scheibner et al., PRB 75, 041301 (007)
700 nm Chaotic Quantum Dot 800 nm dot gate n s = 3.4 x 10 11 cm - G qpc = 4 e / h = 1 x 10 6 cm / (V sec) (N qpc = )
V = 0.46 V th Thermovoltage Fluctuations T = 0 mk statistical ensemble V Th / V -750 V gate = 10 mv B / mt -700 V / mv gate -650-600 -550 Th -500 gate -450-100 -50 0 50 100 B/mT I heating = 40 na T 35 mk V gate = - 550 mv
Thermopower Fluctuation Distribution RMT: G E c( ) 1 G(1 G) = analytic form for N 1 = N = 1 p(s) = 1 here: N 1 = N = S / a.u. 4 3 6 5 4 P(S) 1 P(S) 3 1 0-60 -40-0 0.0 0 40 60 S / ( V / K ) 0-60 -40-0 0.0 0 40 60 S / ( V / K ) S. Godijn et al., PRL 8, 97 (1999)
Residual Charging Energy DEG dot characteristic time scale: Luttinger liquid theory: DEG erg dwell E / h U * / h * U 0(1 ) N U t (Flensberg, 1993, 1994) chaotic QD: (Aleiner and Glazman, 1998) E U 0 t 1 1 t ln U E 0
Scaling Experiment tunneling regime (G << e /h) L e L scanned 1 G = 0.06...0.8 G 0
Scaling Results Thermovoltage Thermopower 0.06 Theory k B T/U* = 0. 0.19 0.5 0.9 0.38 0.43 0.8 0.30 0.33 0.37 0.45 F I heating = 40 na T e = 55 mk, T L = 40 mk
Scaling 1.0 0.8 chaotic behaviour 0.6 extrapolation t 1 U* / U 0 0.4 0. Luttinger liquid behaviour 0 0 0. 0.4 0.6 0.8 1.0 theory E = 3 ev U 0 = 100 ev U * exp ( t 1) 0. 45 U 0 U * E U 0 U 0 ln 0.49 U 0 U 0 E S. Möller et al., PRL 81, 5197 (1998)
Spin-Correlated QD existence of a magnetic moment on the QD can lift the CB Kondo Resonance transport mechanism: spin scattering hybridization of free electrons in the leads with localized magnetic moment leads to resonance at the Fermi edge
Spin-Correlated QD.0 0.0 1.5 Conductance di/dv Bias Thermovoltage V th 15.0 1.0 10.0 (di/dv) / (e /h) 0.5 0.0 5.0 0.0 Vthermo / µv -0.5-5.0-1.0-10.0 0.4 0.6 0.8 1.0 1. 1.4 1.6 - V gate E / Volt strong coupling Kondo Regime weak coupling Cotunnel-Regime
Spin-Correlated QD e - like 0.4 4.0 0..0 (di/dv) / (e /h) 0.0-0. ~ 1 ev V thermo / µv 0.0 -.0 h - like Conductance di/dv Bias -0.4 Thermovoltage V th Mott - Thermopower (scaled to fit) -4.0 0.8 0.9 1.0 1.1 1. Asymmetry between electron- and hole-like transport: Mixed-valence regime Scheibner et al. PRL 95, 17660 (005)
Spin-Entropy Transport Entropy change S adding one electron to an empty site: S = k B (ln g f - ln g i ) = k B (ln -ln1) = k B ln -V g 1.. 3. 4. 1. e-like transport from the hot to the cold reservoir S = k B (ln g f - ln g i ) = k B (ln - ln 1) = k B ln S SE =-k B /e ln S 0. h-like transport from the cold to the hot reservoir S = k B (ln g f - ln g i ) = k B (ln 1 - ln ) = -k B ln S SE =-k B /e ln S SE S 3. e-like transport from the hot to the cold reservoir S = k B (ln g f - ln g i ) = k B (ln 1 - ln ) = -k B ln S SE =k B /e ln 4. h-like transport from the cold to the hot reservoir S = k B (ln g f - ln g i ) = k B (ln - ln 1) = k B ln S SE =k B /e ln g s = 1 g s = g s = 1 R. Scheibner et al., PRL 95, 17660 (005) R. Scheibner, PhD-Thesis, Würzburg 007
Spin entropy contributions
Thermal Rectifier R. Scheibner et al., NJP 10, 083016 (008)
Thermal Rectifier R. Scheibner et al., NJP 10, 083016 (008)
Thermal Rectifier asymmetrically coupled states
Thermal Rectifier G QD (e /h) 0.8 0.6 0.4 0. N+1 N+ J tot de h f ( E, T ) f ( E, T ) t( E) L R e (electron) E (energy) 0.0 S L L 1 11 e df Th de E t( E) de e df h de t( E) de S QD (k B /e) 0 - -4 / t( E) A f E EZ, T / E -.0 -.15 -.10 -.05 V P (V) R. Scheibner et al., NJP 10, 083016 (008)
Thermopower in (Ga,Mn)As First (Ga,Mn)As data by Shi group (Pu et al., Phys. Rev. Lett. 97, 036601 (006). - Signal too large compared to band model (claim Fermi level in impurity band) - Dependence on field direction mimics AMR, not band structure Data dominated by phonon drag?
Measure Diffusion Thermopower, only. Apply current heating technique to (Ga,Mn)As 0 nm (Ga,Mn)As (3% Mn) 60 nm n-gaas ( x 10 19 cm -3 ) ohmics 5nm Ti/ 30 nm Au No counter point contact: T el inferred from weak localization peak in channel
30nm Au 5nm Ti 0nm LT GaMnAs 1nm LT GaAs 60nm n+- GaAs 00nm HT GaAs GaAs
Large Device (no UTF) V T I I V T
Flashback: TAMR in (Ga,Mn)As C. Gould et al., Phys. Rev. Lett. 93, 11703 (004). Device Contact GaMnAs AlOx Au Au 3000 980 GaMnAs Device Contact Resistance () 960 940 90 900 880 860-60 -40-0 0 0 40 60 B (mt) A tunnel barrier between a non-magnetic metal (Au) and ferromagnetic (Ga,Mn)As can exhibit a huge magnetoresistance that can show the signature of a spin valve.
Spin-Valve like TMR in (Ga,Mn)As/AlOx/Non-magnet devices h c1 h c 0 deg. 180 deg. R () 3700 3600 3500 3400 3300 300 3100 3000 170 deg. 900-100 -80-60 -40-0 0 0 40 60 80 100 B(mT) R() 3700 3600 3500 3400 3300 300 3100 3000 350 deg. 900 100 80 60 40 0 0-0 -40-60 -80-100 B(mT) Dependence of the magnetoresistance effect on the in-plane field angle (angle with respect to [100]). Effect due to biaxial anisotropy and anisotropic d.o.s. Now back to thermopower experiment...
Tunnel Anisotropic Magneto Thermopower 90 T. Naydenova et al., Phys. Rev. Lett. 107, 19701 (011) 16 135 45 1 Thermo Voltage (µv) 48.8 48.6 48.4 48. 48.0 47.8 47.6 47.4 47. I=370pA (T=14K) h c1 0 60 h c Thermo Voltage (µv) 08 04 08 1 16 180 5 I=730pA (T=0K) Signal at saturation (300 mt), shows well-known (Ga,Mn)As symmetry... 70 0 315 Sweep Angle (Degrees) 0.6 47.0 0 0.5 46.8-00 -150-100 -50 0 50 100 150 00 B (mt) Temperature (K) 18 16 14 0.4 0.3 0. 0.1 I² (ma)² Signal with hysteresis, shows expected angle Thermo Voltage (µv) dependence...and is quadratic with current 1 0 50 100 150 00 0.0
hc1 and hc for various field directions 40 47.16 Magnetic field (mt) 0 0 0 49.6 48.8 48.0 47.04 46.9-0.03 0.00 0.03 40 47. -0.3-0. -0.1 0.0 0.1 0. 0.3 T. Naydenova et al., Phys. Rev. Lett. 107, 19701 (011) Model (originally for Fe/GaAs):
Thermopower Model and Simulation Energy (a.u.) DOS (a.u.) Energy (a.u.) J + E F J - E F DOS (a.u.) Signal amplitude fully in agreement with band model S 0.4 V/K (Ga,Mn)As 4. K GaAs:Si 4. K + T
Conclusions Current heating is a flexible technique for thermoelectric measurements on nanostructures. Avoids phonon drag, substrate effects. Many detailed investigations of quantum dot transport First observation of Kondo thermopower on a single impurity Discovered TAMT in a n-gaas/(ga,mn)as junction Collaborators: Stefan Möller, Sandra Godijn, Ralph Scheibner, Tsvetelina Naydenova, Holger Thierschmann. Hartmut Buhmann, Charles Gould Funding: DFG