Physics of Semiconductors

Transcription

1 Physics of Semiconductors 13 th Shingo Katsumoto Department of Physics and Institute for Solid State Physics University of Tokyo

2 Outline today Laughlin s justification Spintronics Two current model Spin injection Spin-orbit interaction Spin Hall effect Topological insulator

3 Review of IQHE Exact quantization with universal constants A sample with edges: Number of edge modes = filling factor n Hall conductance of a single pair edge mode: The edge current is scattering free for its strong chirality. Bulk-Edge correspondence A sample without edge: : TKNN formula Chern number is a topological invariant and an integer. ν c = 1 for single Landau subband?

4 Laughlin s discussion B R. Laughlin, Phys. Rev. B 23, 5632 (1981). Φ y Robert Laughlin x Landau gauge Magnetic flux Φ : X shift Chern number =1

5 Ch.7 Spintronics Two current model Spin injection

6 R x x ( h / e 2 ) Spin degree of freedom: A new paradigm Charge (kinetic) freedom Quantum confinement e+ e+ e+ e+ e+ e- e- e- e- e- e- J E J C V BC Semiclassical transport B V y n = 1 R H a l ( h / e 2 ) J x V x 3 n = 2 2 D E G 50 m K Quantum Hall and topology in solid state physics B ( T )

7 Spin degree of freedom: A new paradigm / e 2 ) a / e 2 ) R x Charge (kinetic) freedom Spin degree of freedom J E e+ e+ e+ e+ e+ e- e- e- e- e- e- J C Giant magnetoresistance spin valve V BC Spin injection Spin-manipulation of quantum information l ( h R H B J x V x 3 n = 2 n = 1 2 D E G 50 m K B ( T ) V y x ( h Topological insulators

8 The two current model Nevill Mott Divide a current to the one with spin and the one with spin. Drude: Condition: spin diffusion length λ s l mean free path (or other lengths) Spin polarized current: drift diffusion

9 Spin-dependent chemical potential Einstein relation for metals: ε s : local Fermi energy, δε s : Shift from thermal equilibrium Spin-dependent chemical potential

10 Spin current Spin current (simplest) definition: Angular momentum conservation: With spin relaxation: Charge conservation: Steady state: :spin diffusion equation :spin diffusion length

11 Spin injection j c μ FM μ μ NM FM μ μ 0 μ 0 μ NM E E E μ E ρ (E) ρ (E) ρ (E) ρ (E) ρ (E) ρ (E) ρ (E) ρ (E) M = F, N

12 Spin injection and detection j c FM1 μ μ μ 0F1 NM μ μ 0N FM2 μ 0F2 μ Jedema et al. Nature 410, 345 (2001).

13 Spin precession (review) Zeeman Hamiltonian From Heisenberg equation: z Larmor frequency x ω 0 y

14 DV (mv) Spin precession experiment j c V M g O N M F M F M N M S C G a t e H (Oe) H

15 Ch.7 Spintronics Spin-orbit interaction Spin Hall effect Topological insulator (quantum spin Hall effect)

16 Spin-orbit interaction (in electron motion) : Spin-orbit interaction BIA: Bulk inversion asymmetry SIA: Structure inversion asymmetry V III

17 SIA-SOI Rashba-type SOI Emmanuel Rashba (Actually through the valence band) E ± m α ħ 2 m α ħ 2 k

18 SOI and SdH oscillation 6 V g = V T = 0. 4 K r x x ( a r b. ) B ( T ) Nitta et al., Phys. Rev. Lett. 78, 1335 (1997).

19 Spin Hall effect k y Effective magnetic field k y spin k x k x effective field k

20 Spin Hall effect in an insulator Remember k p approximation Consider the case these are not zero. Then the discussion is in parallel with the TKNN formula.

21 Anomalous velocity and quantum spin Hall effect Wave packet: Bloch wave expansion TKNN Anomalous velocity Spin-subband Chern number Spin Chern number

22 Topological insulator: helical edge state E E F Charge conservation: y 0 Ordinary insulator Topological insulator k Helical edge mode: Extra spin flow at the edge Edge mode number = Chern number

23 Topologically insulating quantum well König et al., Science 318, 766 (2007). 7.3nm

24 Summary / e 2 ) a / e 2 ) R x Charge (kinetic) freedom Spin degree of freedom J E e+ e+ e+ e+ e+ e- e- e- e- e- e- J C Giant magnetoresistance spin valve Spin injection V BC Spin-manipulation of quantum information l ( h R H B J x V x 3 n = 2 n = 1 2 D E G 50 m K B ( T ) V y x ( h Topological insulators

25 Problem 1: Let us consider a pn-junction of Si at the temperature 300K. In the p-layer the acceptor (boron, B) concentration is m -3 and in the n-layer the donor (phosphorous, P) concentration is m -3. The doping profile is abrupt. (1) Obtain the built-in potential. (2) Calculate the depletion layer widths for p- and n-layers at reverse bias voltage -5V. (3) Calculate the differential capacitance at reverse bias voltage -5V for the area 1mm 1mm. Let put another p-layer and make a pnp transistor (gedankenexperiment). The hole diffusion length in the base is 10mm. (4) Calculate h FE for base widths 0.5mm and 0.1mm. (Ignore depletion layer widths, other non-ideal factors. Calculate under the simplest approximation.)

26 Problem 2: The left figure shows the Shubnikov-de Haas oscillation and the quantum Hall effect in two-dimensional electrons. (1) Calculate the electron concentration from the low (<0.5T) field data. (2) Something happened around 0.65T. What is it? Magnetic field (T)

27 Problem 3: Consider a double barrier resonant diode with GaAs as the well material and Al 0.4 Ga 0.6 As as the barrier material. Lets adopt E g =1.424 ev for GaAs and E g = x+0.265x 2 (ev) for Al x Ga 1-x As and DE c :DE v =6:4. The electron effective mass in GaAs is 0.067m 0 and ignore the change in Al x Ga 1-x As. Consider n-type electrodes (note that in the lecture we considered p-type). (1) Obtain the transfer matrix of 5nm thickness GaAs- Al 0.4 Ga 0.6 As. (2) Calculate the transmission probability of resonant diode with two 5nm barriers and a 5nm well region as a function of incident energy (from 0 to the top of the barrier with an appropriate interval) and plot in a figure.

28 Problem 4: Let us consider the rectangular potential illustrate in the left. (1) First consider the most coarse approximation. Choosing a kinetic energy E determines the effective potential with E/a. Now let us approximate the potential with a rectangular potential of width E/a, bottom V(0), infinite barrier height. Let m* be the effective mass and obtain the eigen energies from lower level with index n=1,2,.. (2) Compare the above result with more accurate one on Airy functions. (3) Also try comparison with Wenzel-Kramers-Brillouin (WKB) approximation for wavefunction penetration into the barrier.

29 Problem 5: In the left figure the green region indicates 2DEG, 1 to 6 are the electric contacts, the yellow regions are metallic gates. The structure has a quantum point contact in the middle. In the integer quantum Hall state with filling factor ν, the sample has ν edge modes. With applying gate voltage, we can tune the number of modes which transmit through the QPC, to χ. Other modes are completely reflected by the QPC. The current is through 1 and 4. (1) Obtain the longitudinal resistance R L, which is measured from the voltage between 2 and 3 V 23 or 6 and 5 V 65. (2) Obtain the Hall resistance R H, measured from V 26 or V 35.

30 Problem 6: Consider a 2DEG under IQHE with n =1. The edge modes can bring finite current without energy dissipation and the resistance is zero. The conductance of one-dimensional edge mode is then the inverse of the resistance and infinity. Let write the quantum resistance h/e 2 as R q. Two dimensional resistivity tensor: Then the two dimensional conductivity tensor defined by the inverse of resistivity tensor: That is, σ xx = 0! Does the calculation contain an error? If it does, what is the error? Or can you solve the puzzle?