Download Lecture notes in Solid State 3 Eytan Grosfeld Introduction to Localization

Lecture notes in Solid State 3
Eytan Grosfeld
Physics Department, Ben-Gurion University of the Negev
Introduction to Localization
The Anderson model
We start with spatially separated atoms: electrons are localized near ions. Next
we add tunneling between atoms: in a clean crystal (perfectly ordered atoms),
Bloch theorem guarantees that our electronic states will be extended. The
question is what happens once we add disorder. We write the tight-binding in
the presence of on-site disorder
X
X
H=
i |iihi| + t
|iihj|
(6.1)
i
hiji
and
i ∈ [−W/2, W/2]
(6.2)
randomly and uniformly chosen, with W is the disorder strength,
i
=
i j
=
0
W2
δij
12
(6.3)
(6.4)
In the presence of magnetic field we add phases to the tunneling matrix elements
X
X
H=
i |iihi| + t
eiθij |iihj|
(6.5)
i
hiji
´j
e
where θij = − ~c
A · d`, known as the Peierls substitution, and guranteeing
i
that the electron picks up the correct phase over a closed loop, proportional to
the magnetic flux via the loop.
The Anderson model - implications for transport
Localized states decay away from their center, i.e. behave asymptotically like
ψ(r) = A(r)e−|r|/λ
(6.6)
where λ is the localization length, and λ → ∞ corresponds to an extended state.
The following results apply for the Anderson model: If the dimensionality
of the system d ≤ 2, then all eigenstates are localized (no matter how weak the
disorder is). For d = 3:
1
2
• Density of states forms tails of the band consisting of localized states;
• Interior of the band corresponds to extended states;
• Critical energies Ec , Ec0 separating localized from extended states are
called the mobility edges.
Implications for transport:
• Only extended states contribute to transport/conductivity significantly;
• Position of mobility edge depends on the magnitude of disorder;
• At critical magnitute of disorder: metal-insulator phase transition, i.e. all
states become localized (Anderson transition).
Current transport is by electrons with energy E ' EF . For EF < Ec (within
the hopping regime):
• Current transport by localized electrons, electrons tunnel/hop from one
localized state to the other;
• Tunneling probability between localized wavefunctions α → β: p(rαβ , Eβ , Eα ) ∝
e−αrαβ e−(Eβ −Eα )/kB T , leading to Mott’s law for conductivity: σ(T ) ∝
1/4
e−(T0 /T ) (but at T → 0 the system is insulating).
For EF > Ec : Fermi-energy within region of extended states: metallic regime
• Disorder leads to positive interference of certain terms in perturbation
expansion of conductivity: enhanced backscattering (weak localization)
leads to reduced conductivity;
• Externally applied magnetic field destroys positive interference and weak
localization (negative magnetoresistance, i.e. the decrease of resistance
with magnetic field).
Dirac materials: weak anti-localisation
• Backscattering is forbidden due to a Berry phase effect in k-space (leading
in the presence of magnetic field to positive magnetoresistance, i.e. the
increase of resistance with magnetic field).
Thouless time
As we know from the Einstein relation, diffusion comes hand in hand with
conductivity: in the absence of the first the second is certainly absent and
vice-versa. By now we have a qualitative understanding that a conductor is
associated with extended states, while an insulator is associated with localized
states. For extended states, the conducance can be written as
G = σLd−2
(6.7)
(while for localized states the conductance is of the form G ∝ exp(−L/ξ)). Using
the Drude result σ = ne2 τ /m, and the definition of the diffusion coefficient
D = vF2 τ /d, we can write
e2 ~D
n
d
2 nτ d−2
L
∼
L
(6.8)
G=e
m
h L2
mvF2
3
Fig. 6.1: Anderson metal-insulator transition: (a) Clean limit, all states in the
band are extended. (b) Disorder is turned on. Tails of localized states
appear, separated by the mobility edge from the region of extended
states. (c) Beyond a threshold, all states become localized.
4
Fig. 6.2: Combining two systems of size L.
where
ET =
is the Thouless time and
~
~D
= 2
τT
L
n
1
=
Ld
∆
mvF2
(6.9)
(6.10)
where ∆ is the mean level-spacing at the Fermi energy. We define the dimensionless conductance:
ET
G
g= 2 =
(6.11)
e /~
∆
The dimensionless conductance is thus the ratio between the Thouless energy and the mean level spacing. The thouless energy is a measure of sensitivity to boundary conditions: if we impose boundary conditions such that
ψ(L) = eiϕ ψ(0) with resulting spectrum En (ϕ), then the Thouless energy is the
geometric mean of the energy level shifts caused by a change of the boundary
conditions from periodic (ϕ = 0) to anti-periodic (ϕ = π) along the direction of
the current flow [Anderson and Lee, 1980]. To acquire further insight, we need
to consider what happens when we combine two subsystems A and B of size L.
When ET > ∆, states of the two subsystems combine to create complex states
that allow diffusion. When ET < ∆, states remain localized in either A or in
B. Therefore, when g & 1 we get metallic behavior, and when g . 1 we get an
insulator.
Scaling theory of localization
• Recommended reading: Abrahams, Licciardello, Ramakrishnan, Anderson
To proceed, we define the dimensionless conductance:
g=
G
e2 /~
(6.12)
5
where [G] = Ω−1 , and
G = σLd−2
(6.13)
where L is a typical length of a system in d dimensions.We consider the process
of enlarging the system through adding together bd cubes of size Ld to create a
cube of size (bL)d . Thouless hypothesized that the only parameter that decides
the properties of the system under such scaling is the conductance g:
g(bL) = f (b, g(L))
(6.14)
Taking the limit b → 1,
g((1 + ε)L) = f (1 + ε, g(L))
(6.15)
0
(6.16)
→ g(L) + εLg (L) = f (1, g(L)) + εf1 (1, g(L))
where f1 is the derivative of f with respect to its first argument, we get
L
d
g(L) = f1 (1, g(L))
dL
(6.17)
As usual, when written in the form
d log g
f1 (1, g(L))
=
= β(g(L))
d log L
g(L)
(6.18)
this defines the β function. when g is large, the system is metallic
g(L) = σ0 Ld−2 → β(g) = d − 2
(6.19)
When g is small, the system displays localization behavior in any dimension
g
g = g0 exp(−αL) → β(g) = log
(6.20)
g0
We assume that β(g) displays a monotonic behavior between these two limits.
Hence when β(g) > 0 the conductance increases with the size of the sample,
while when β(g) < 0 the conductance decreases with the size of the sample.
Glancing at Fig. 6.3, we arrive at the conclusion that all the states are
localized in 1D (which is also known from analytical calculations), and more
surprisingly, all the states in 2D are localized as well. In contrast, 3D is special:
necessarily there is some intermediate point for which β(g) = 0, defining gc .
This is an unstable fixed point between a conducting state and an insulating
state, known as the metal-insulator transition.
What happens then in 2D? One may wonder why all the states are localized, leading to the failure of the Boltzmann equation (no matter how weak the
disorder is). The only possible solution is that there exist divergent quantum
corrections to the Drude formula, that overwhelm the semi-classical physics.
This will be the subject of the next chapter, which deals with weak localization
corrections to the conductance, treated perturbatively in the disorder. Roughly
speaking, when these corrections are of the same size as the classical conductance, we get a transition to an insulator (strong localization).
6
Fig. 6.3: Conjectured behavior of β(g) for d = 1, 2, 3.
Fig. 6.4: Classical particle in a potential will always be reflected for E < Emax
and will always be transmitted for E > Emax . A quantum mechanical
particle may fully transmit even if E < Emax and may fully reflect
even if E > Emax due to interference effects.
7
Weak localization
For intuitive picture in 1D, consider the Hamiltonian
H=
p2
+ V (r)
2m
(6.21)
where V (r) is some potential (the disorder potential). At very low energies
all states are localized independent of dimension. The important question for
transport is what happens to states at EF , are they extended or localized?
In particular, suppose that the disorder potential maximum strength is Emax .
Classically, if a particle of energy E > Emax approaches the disordered region,
the particle will not be affected by the potential at all, while if E < Emax
it will necessarily reflect. When we go to quantum mechanics, deep wells will
clearly localize the electronic states. More surprisingly, it can be shown that all
the states in 1D get localized even for weak disorder. So even for E > Emax
repeated scatterings on the potential can lead to localization of the state due to
constructive interference effects. Recommended reading:
• Gorkov, Larkin, Khmelnitskii (1979)
• Anderson, Abrahams, Ramakrishnan (1979)
In the limit of weak disorder, we may consider electron paths from point r1 to a
point r2 . Quantum theory requires summing the amplitudes over the different
paths connecting the two points. These paths are not simple paths because they
involve multiple scatterings on the disorder. If we denote the amplitudes by Ai ,
the probability for going from a point r1 to a point r2 is
X 2
X
X
(6.22)
+
Ai A∗j
Ai =
|Ai |2
P = i
i
i6=j
| {z }
| {z }
classical probability interference effects
X
X
=
|Ai |2 + 2
|Ai ||Aj | cos(ϕi − ϕj )
(6.23)
i
i<j
Usually the contribution of the interference term is not important because when
we take many different paths of different
´ lengths, each
´ pair of paths may have
a phase difference ∆ϕij = ϕi − ϕj = ~1 Ci p · dr − ~1 Cj p · dr that will generically be finite, and when we average over many pairs, we get that the second
term averages to zero. However, there are special paths: those that are selfintersecting. For these paths, there are two ways to traverse the loop (clockwise
or anti-clockwise) and since we need to take dr → −dr and dp → −dp to get
from clockwise to counter-clockwise, we get that∆ϕ = 0, hence
P = |A1 |2 + |A2 |2 + 2|A1 ||A2 | = 4|A1 |2
(6.24)
is twice larger than the classical probability. In paricular, for a path that starts
at the origin, there is an increased probability that it will return to the origin,
leading to larger resistivity. We are therefore led to the conclusion that quantum mechanically the probability of return is increased compared to a classical
diffusive process.
8
So why 1D and 2D are special? This is related to the probability for a path
to self-intersect in lower dimensions. Quantum mechanically, the path of the
electron has width λ ∼ p~F in all transverse directions to its motion. If the
electron moves diffusively, then at time t > τ it can reach any point in a typical
distance r ∼ (DF t)1/2 from the origin where DF = vF2 τ /d. Therefore, the total
area occupied by the diffusion path is (DF t)d/2 . The probability that during
an additional time dt it will intersect with the initial point is therefore the ratio
between
dV = λd−1
(6.25)
F vF dt
to the total volume
(DF t)d/2
(6.26)
We are interested in the total probability, therefore we need to integrate over
time. This gives an estimate of the correction to the Drude conductivity
ˆ τϕ
∆σ
∼−
vF λd−1 (DF t)−d/2 dt
(6.27)
σD
τ
where the sign is chosen by realizing that the interference leads to an increased
probability for scattering and therefore reduces the conductivity. The two limits
of integrations are taken at the time scales where the picture described above
breaks. The lower limit is chosen as for t < τ we cannot discuss diffusive
processes anymore. The upper limit is the dephasing time or the inelastic mean
free path. It is decided by non-elastic processes that destroy phase coherence
and lead to suppression of quantum effects at larger times; It increases when
the temperature is decreased and becomes infinite for T = 0.
Performing the integrations one arrives at the following results

1/2 
1
τ

d=3

 (kF `)2 1 − τϕ
δσ
τ
ϕ
1
(6.28)
∼−
d=2

σD
i
hkF ` log τ


 τφ 1/2 − 1
d=1
τ
2
τ
For d = 2, ∆σ ∼ − eh log τϕ . Since τϕ → ∞ when T → 0 quantum corrections
diverge.
At finite magnetic field
Φ
= Aei2π Φ0
A1
A2
= Ae
so that
2
|A1 + A2 | = 2A
2
(6.29)
−i2π ΦΦ
(6.30)
0
4πΦ
1 + cos
Φ0
(6.31)
hence, compared to the usual weak-localization correction we get 4A2 −2A2 1 + cos 4πΦ
=
Φ0
2A2 1 − cos 4πΦ
leading to
Φ0
∆σ(B)
σD
ˆ
=
vF λd−1
F
τ
τϕ
dt
(DF t)d/2
4πBDF t
1 − cos
Φ0
(6.32)
9
which in 2D leads to
∆σ(B)
1
'
σD
kF `
ˆ
0
x
dy
(1 − cos y)
y
(6.33)
where x = 4πBDF τϕ /Φ0 . For weak B this leads to ∆σ(B) ∝ B 2 . Magnetic
field breaks time reversal symmetry and therefore increases the conductivity by
decreasing weak-localization effects.