N.F. Mott

J. Langer V. Ambegaokar

B.I. Halperin

M. Pollak E. Abrahams B. Spivak

Lecture 1

Topics:

1. Impurity states in semiconductors

3. Impurity band structure

4. Electron transition between two localized states

6. Miller-Abrahams random resistor network

7. Percolation treatment

5. Mott’s law

8.

2. Basic experimental facts

Spin relaxation in the hopping regime

Impurity levels inside a forbidden gap

bottom of conduction band

top of the valence band

shallow donor

shallow acceptor

energy spectrum of conduction band

envelope of the donor wave function

binding energy

for anisotropic spectrum: at large distances

binding energy in the units of

D2 D3

asymptotic behavior of the envelope

∫ +

⋅

∝)(||

exp)(

pEE

rpi

pdrF

( )

++−∝

2/1222||2exp)( ymxmzmErF ttl

KT 10=101 =Tε

KT 2=

23 =Tε

501 =Tε 103 =

Tε

−

T1exp ε dominates

−

T3exp ε dominates

40≈

KT 1=

83/1

0 =

TT

Lightly doped

3 1DN a

Typically, an empty donor is the nearest neighbor of an acceptor

Some acceptors do not have a donor in the neighborhood- 0 complex

Some acceptors empty two donors

in their neighborhood- 2 complex

( )0 2 ( )N Nµ µ=Neutrality condition:

Positive energy shift by acceptor overweighs negative energy shift by second donor

increases decreases

For 0-comlex to exist there must be no donors in a sphere

with radius around a fixed acceptor

2-comlexes

( )0 2 ( )N Nµ µ= 0-comlexes constitute 1.3% of the acceptor concentration

Shift by acceptor Shift by donor

probability that two donors are the nearest neighbors of the acceptor

probability that at a donor is empty T µ

With increasing K

( )1/22 3~ D

e N rr

µ

( )1/233 ~ (1 )

DD

N rn N K

r= − for

r

r is determined by the condition:

excess concentration of charges

II. Long-range fluctuations:

I. Close pairs of donors:

RdRNNd D24π=

Reκ

ε2

=

εε

dNdg =)(

yields the same result

Fluctuation of number of donors in a cube with side

µ

Local density of states due to donor pairs

Long-range potential

activation energy for hopping

activation into conduction band

iE

jE

ijr

Phonon-assisted transition between two sites: phonon wave vector

sound velocity Golden rule

ji EE −

Matrix element

Planck distribution

overlap integral

deformation potential

density Fourier transform of the donor wave function

2V2V−

1E

2E12r

+−

−=

21

2 211212VEfVEfI γ

−−

+=

21

2 122121VEfVEfI γ

in equilibrium

−=

TEE 21

2112 expγγ

2112 II =

TV 21 ,

where

Resistance corresponding to a single hop

0=V

0for}max{ 212,1 >⋅= EEEEEa

0for|| 2121

Mott’s law

Transport resulting from energy levels within the strip

concentration of sites

constant density of states

within the strip

has a sharp minimum at

[ ]4/1

03/10 )(

== −

TTaTgr ε( ) 4/10

0

0 )(ln

=

TTT

ρερ

typical hopping distance increases with decreasing T

within the strip

variable-range hopping 30 )(1

agT

µ=

0|| εµε ≤−i

4/13

4/3

00 )()(

gaTT == εε

00 )(2)( εµε gN =

( )

+=

+=

TagTaN0

3/10

03/1

00

1exp)(

1exp εε

εερ

ρ

Numerical factor in log-resistance from percolation theory

Nearest-neighbor hopping:

TNe

TaNar D

D κε 3/12

3/1 ~1~ >

3/2322

)( aNa

eTa

eDκκ

>>

ar

R ijij2

ln =The sites with cij rr < are connected

1.07.23

4 3 ±== cDc rNBπ

Infinite cluster appears when

aNR

D3/1

73.1ln =average number of

inside a sphere with crr =

tunneling dominates over activation

neighboring donors

Numerical factor in log-resistance from percolation theory:

Bonding criterion

Variable-range hopping

dimensionless energy dimensionless distance ξ

ε

2/1

2

2

2

22

2

+

+=

bz

ayx ijijij

ijξ

( ) cijD BrdN =−∫ ξξθ

( ) cNNHjiijcVRH ndd

nij

=∆∆

∆−∫

Random resistor network Near the percolation threshold

ν

ξξξξξ

− c

ccaL ~)(

hopping length

critical exponent

9.034

32 ≈= νν

Resistance of the sample is dominated by infinite cluster with ( ) 1~cξξ −

“Unit cell” of the current-carrying network 1~ +νξcc aL

Hopping transport in amorphous film of a thickness d

rad c =>1~ +νξcc aL No thickness dependence if d exceeds

[ ] 20 adgTβ

=

( ) dL cd =ξ

+=

3/1

1ν

ξξdrDccd

3/14/10

)()(ln

ν

ρρ

=

∞ d

rTTDd

Dyakonov-Perel spin relaxation in diffusive regime

Initial spin orientation is forgotten after N steps: 1~)( 2δϕN

τvl =1

for free electrons in conduction band

στ

As

=1

ττ 2

1

sos Ω=relation can be cast in the form

mne τσ

2

=

Drude conductivity

2

2

nemA soΩ=

holds in the hopping regime

Relation

Electron rotates its spin only during time intervals when it tunnels between the sites

( )

Ω=

==

tun

hop

tunsotun

hoptunhops N τ

τττ

τδϕτ

ττ 221

tunneling time

waiting time 1−∝ hopσ

For localized electron the dominant mechanism of spin relaxation is interaction

N1ω N

2ω

with random hyperfine field created by nuclei inside the donor orbit

unlike SO-coupling, electron spin precesses

while electron waits for the hop

1212 ~ ξar

spin rotation is dominated by most resistive hops

( ) ( ) 1exp02/12 >>= c

Nc ξτωδϕIf

1~ +νξcc aLthe initial spin orientation is “forgotten” at distances less than

1== ξτωτωδϕ NhopN

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