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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