Physics of Semiconductors

Shingo Katsumoto

Department of Physics

and

Institute for Solid State Physics

University of Tokyo 13th 2016.7.11

Outline today

Laughlin’s justification

Spintronics

Two current model

Spin injection

Spin-orbit interaction

Spin Hall effect

Topological insulator

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.

A sample without edge: : TKNN formula

Chern number is a topological invariant and an integer.

𝜈𝑐 = 1 for single Landau subband?

Bulk-Edge correspondence

Laughlin’s discussion

Robert Laughlin

B

Φ

x

y

R. Laughlin, Phys. Rev. B 23, 5632 (1981).

Landau gauge

Magnetic flux Φ : X shift

Chern number =1

Ch.7 Spintronics

Two current model

Spin injection

Spin degree of freedom: A new paradigm

Charge (kinetic) freedom

𝑉BC

e-

e- e-

e-

e-

e- e+

e+

e+

e+

e+

𝐽𝐸 𝐽𝐶

Semiclassical transport

Quantum confinement

Vx

B ( T )

R H a l l ( h

/ e 2 )

R x x

( h / e

2 )

E

5 m 2 D G 0 K

n = 1

n = 2

3

4 5

6

0 2 4 6 8 1 0

0 . 2

0 . 4

0 . 6

0 . 8

1

0

0 . 1

0 . 2

0 . 3

Vy

Jx

B

Quantum Hall and topology

in solid state physics

Spin degree of freedom: A new paradigm

Charge (kinetic) freedom

𝑉BC

e-

e- e-

e-

e-

e- e+

e+

e+

e+

e+

𝐽𝐸 𝐽𝐶

Vx

B ( T )

R H a l l ( h / e 2 )

R x x ( h

/ e 2 )

E

5 m 2 D G 0 K

n = 1

n = 2

3 4

5

6

0 2 4 6 8 1 0

0 . 2

0 . 4

0 . 6

0 . 8

1

0

0 . 1

0 . 2

0 . 3

Vy

Jx

B

Spin degree of freedom

Topological insulators

Spin-manipulation of

quantum information

Giant magnetoresistance

spin valve

Spin injection

The two current model

Divide a current to the one with ↑ spin and the one with ↓ spin.

Drude:

Condition: spin diffusion length 𝜆𝑠 ≫ 𝑙 mean free path (or other lengths)

Spin polarized current:

drift diffusion

Nevill Mott

1905-1996

Spin-dependent chemical potential

Einstein relation for metals:

Spin-dependent chemical potential

𝜖𝑠: local Fermi energy, 𝛿𝜖𝑠 : Shift from thermal equilibrium

Spin current Spin current (simplest)

definition:

Angular momentum conservation:

With spin relaxation:

Charge conservation:

Steady state:

:spin diffusion equation :spin diffusion length

Spin injection

FM NM

FM

NM

𝜇 𝜇

𝑗𝑐

M = F, N

𝜇↑

𝜇↑ 𝜇↓

𝜇↓

𝜇0

𝜇0

𝜌↑(𝐸) 𝜌↓(𝐸) 𝜌↑(𝐸) 𝜌↓(𝐸)

𝐸 𝐸 𝐸

𝜌↑(𝐸) 𝜌↓(𝐸) 𝜌↑(𝐸) 𝜌↓(𝐸)

𝐸

Spin injection and detection

FM1

NM 𝜇↑

𝜇↑

𝜇↓

𝜇↓

𝜇0F1

𝜇0N

𝑗𝑐

FM2

𝜇0F2

Jedema et al. Nature 410, 345 (2001).

Spin precession (review)

Zeeman Hamiltonian

From Heisenberg equation:

𝜔0

x

y

z

Larmor frequency

Spin precession experiment

H (Oe)

- - - - -

-

DV

(m

V)

F M F M N M N M

j c

V

S C

G a t e

M g O

H

Ch.7 Spintronics

Spin-orbit interaction

Spin Hall effect

Topological insulator (quantum spin Hall effect)

Spin-orbit interaction (in electron motion)

: Spin-orbit interaction

BIA: Bulk inversion asymmetry

SIA: Structure inversion asymmetry

V

III

SIA-SOI Rashba-type SOI Emmanuel Rashba

(Actually through the valence band)

𝐸±

𝑘

𝑚∗𝛼

ℏ2 −

𝑚∗𝛼

ℏ2

SOI and SdH oscillation

1 2 0

2

4

6

V g = 1 . 0 V -

- 0 . 7

- 0 . 3

0

0 . 3

0 . 5

1 . 5

B ( T )

r x

x ( a

r b . )

T = 0 . 4 K

Nitta et al., Phys. Rev. Lett. 78, 1335 (1997).

Spin Hall effect

Effective magnetic field

𝑘𝑥

𝑘𝑦

𝑘𝑥

𝑘𝑦

spin

effective field

k

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.

Anomalous velocity and quantum spin Hall effect

Wave packet: Bloch wave expansion

Anomalous velocity

TKNN

Spin-subband

Chern number Spin Chern number

Topological insulator: helical edge state

𝑘

𝐸

𝐸𝐹

Ordinary insulator

Topological insulator

0

y

Charge

conservation:

Extra spin flow at the edge

Helical edge mode:

Edge mode number = Chern number

Topologically insulating quantum well

7.3nm

König et al., Science 318, 766 (2007).

Summary

Charge (kinetic) freedom

𝑉BC

e-

e- e-

e-

e-

e- e+

e+

e+

e+

e+

𝐽𝐸 𝐽𝐶

Vx

B ( T )

R H a l l ( h / e 2 )

R x x ( h

/ e 2 )

E

5 m 2 D G 0 K

n = 1

n = 2

3 4

5

6

0 2 4 6 8 1 0

0 . 2

0 . 4

0 . 6

0 . 8

1

0

0 . 1

0 . 2

0 . 3

Vy

Jx

B

Spin degree of freedom

Topological insulators

Spin-manipulation of

quantum information

Giant magnetoresistance

spin valve

Spin injection

Problem 1:

Let us consider a pn-junction of Si at the temperature 300K. In the p-layer

the acceptor (boron, B) concentration is 1021 m-3 and in the n-layer the donor

(phosphorous, P) concentration is 1020 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 hFE for base widths 0.5mm and 0.1mm. (Ignore depletion

layer widths, other non-ideal factors. Calculate under the simplest

approximation.)

Problem 2:

Magnetic field (T)

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?

Problem 3:

Consider a double barrier resonant diode with GaAs as the well material

and Al0.4Ga0.6As as the barrier material. Lets adopt Eg=1.424 eV for GaAs

and Eg=1.424+1.265x+0.265x2 (eV) for AlxGa1-xAs and DEc:DEv =6:4. The

electron effective mass in GaAs is 0.067m0 and ignore the change in

AlxGa1-xAs. Consider n-type electrodes (note that in the lecture we

considered p-type).

(1) Obtain the transfer matrix of 5nm thickness GaAs- Al0.4Ga0.6As.

(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.

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.

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 RL, which is measured from the voltage between 2

and 3 V23 or 6 and 5 V65.

(2) Obtain the Hall resistance RH, measured from V26 or V35.

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/e2 as Rq.

Two dimensional resistivity tensor:

Then the two dimensional conductivity tensor defined by the inverse of

resistivity tensor:

That is, 𝜎𝑥𝑥 = 0! Does the calculation contain an error? If it does, what

is the error? Or can you solve the puzzle?