March 12, 2007

Figure 1: Representations of the electron wavefunction in the Anderson model for a) extended and b) weakly localized cases. [3] |

It is well known in theory that bands of electrons of
certain energies propagate perfectly through perfect crystals; however,
in reality a perfect crystal is a myth. All crystals contain disorder,
in the form of impurities, lattice defects, or fluctuations of atomic
position due to finite temperature. This report will briefly and
qualitatively introduce the scaling theory of localization, which has
been very successful at explaining basic phenomena of conductance and
magnetoresistance in disordered conductors. It will also review the
first important experimental evidence gathered in support of this
theory, namely the observation at low temperatures of a
metal-to-insulator transition to very low conductivities in a
three-dimensional sample of silicon doped with
phosphorus.^{1}

Prior to 1958 the traditional view had been that
disorder-induced scattering in a metal causes an electron to lose phase
coherence after propagating a distance on the order of its mean free
path *l*, but nevertheless the fundamentally extended state of the
electron is preserved. Anderson in 1958 argued that this is not the
case; rather, for some critical disorder the electrons become localized
in the metal, after which conduction, if it occurs, does so by hopping
from localized state to localized state. Crossing this critical
disorder threshold marks a fundamental transition from metal to
insulator.^{2} A graphical illustration of this transition is
shown in Figure 1. Since then a number of theories have attempted to
further quantify this metal-insulator transition. One particularly
important one by Mott^{3} predicted a discontinuous
metal-insulator transition and a finite minimum conductivity at zero
temperature proportional to the interelectronic spacing in a sample.
This theory was based in part on the criterion that an electron's mean
free path must exceed the distance between atoms.

In contrast, Abrahams et. al. introduced the scaling
theory of localization in 1979.^{4 } The theory basically models
a disordered system as a d-dimensional hypercube, made by putting small
cubes of side length L>>*l* together into a sample of size
(2L)^{d}. The eigenstates of the final sample are linear
combinations of the eigenstates of each L^{d} block. If the
eigenstates are mainly localized, they will be insensitive to the
boundary conditions used for putting the smaller blocks together; if
extended, the opposite. Sensitivity to changing the boundary conditions
from periodic to antiperiodic, therefore, is correlated with the nature
of the eigenstates as the system gets larger. The crucial hypothesis is
that the (dimensionless) conductance g of the system at scale L is
directly proportional to and completely characterizes the average of the
energy-level differences obtained by modifying the boundary conditions,
small for localized systems and larger for extended
systems.^{4,5,6}

Figure 2: Schematic of experiment described in [1]. |

Abrahams et. al. further argued that since the change in disorder when the system becomes a little bigger is determined by its value at the previous length scale, with the only reasonable measure of this disorder being the conductance, the logarithmic derivative

called the scaling parameter, is a monotonic function
of g alone. β(g)=0 corresponds to the transition between extended
and localized states. In contrast with the previous theory of Mott, for
a given sample g is expected to evolve smoothly - with no discontinuity
- from a microscopic conductance g_{0} to its asymptotic form
for large L. This asymptotic form will in general depend on L, the
dimensionality of the system, and on the amount of system disorder. In
the low scattering limit where Ohm’s law for a d-dimensional hypercube,
g(L)=σL^{d-2}, is valid with no further correction,
clearly β(g)=(d-2), implying d=2 is a critical dimension, and that
in 2D or below, only localized behavior is possible at large enough
length scales.^{4}

For large g, it is possible to calculate corrections
to the classical β(g)=(d-2) in g^{-1} using diagrammatic
perturbation theory.^{5,7} The first order correction,
proportional to g^{-1}, is found to be negative, i.e., –a/g.
Its origin is singular backscattering, which is interference between
electron waves and their scattered analogues traveling across
time-reversed paths. The Green's function describing propagation of an
electron can be constructed as a path integral over all paths connecting
the start and endpoints. In the presence of impurities (but no magnetic
field) most electrons will arrive at the endpoint with random phase,
with the exception of self-intersecting paths. The closed loop in such
paths may be traversed in either direction, and so in the presence of
time-reversal symmetry the paths will interfere. This interference
further localizes the states leading to a first order correction that
decreases the value of β(g).

In three dimensions, β(g)=(1-a/g) so the
conductance of any disordered system is always less than ohmic. The
behavior of a given system will be metallic if its microscopic
conductance (its conductance on the length scale *l*) g_{0}
> g_{c} , where g_{c} is the microscopic conductance for
which β(g)=0. Conversely the system will be insulating if
g_{0} < g_{c}. Since current in a sample is transported
by electrons at the Fermi energy, g_{0} is the conductance at
this energy. If disorder is kept fixed and instead the Fermi energy is
varied, g_{0} will change smoothly and will become g_{c}
when the Fermi energy reaches a value defined as the mobility edge. For
values near the mobility edge, β(g) may be approximated as linear,
or β(g) ≈ δ(g)/v. Then, integrating, g(L)=σL,
where σ=g_{c}/R, is the macroscopic conductivity at the
phase transition. R, with units of length, is in this notation

with A a constant of order unity. Note R goes to zero
at the phase transition. Extrapolated from an analysis in 2+ε
dimensions with ε small, v in 3 dimensions is roughly unity, and
it should be the same on either side of the critical point.^{5}
In sum, at the mobility edge and at temperatures approaching 0K, the
scaling theory of localization predicts a continuous transition to zero
of the macroscopic conductivity from some finite value. This is
inconsistent with Mott's theory both in the continuity of the
transition and in the existence of a finite minimum conductivity.

Using modern semiconductor process technology, it is
straightforward to modify the Fermi level of a highly crystalline
semiconductor with great accuracy, by doping. Doping in this context is
the process of replacing the primary atom at a few lattice sites in the
crystal with an atom containing typically one more or one fewer valence
electrons. The excess electrons or holes introduced into the crystal
via this process modify the density of states in the crystal and shift
the Fermi level up or down respectively.^{8} Measurement of the
conductivity of semiconductors doped to span the metal-to-insulator
transition, at a temperature low enough that the effects of thermal
disorder are negligible by comparison, should therefore provide a
rigorous test of the scaling theory of localization and the behavior it
predicts for the macroscopic conductivity in this regime.

In 1980, Rosenbaum et al. performed just such an
experiment with an array of samples of silicon doped with phosphorus (an
electron donor).^{1} The silicon crystals themselves were grown
via a Czochralski process, in which a seed silicon crystal is dipped
into molten, ultrahigh purity silicon with the relevant concentration of
dopant added, and drawn out under very controlled conditions to produce
a crystalline ingot.^{9} The donor density n in the samples was
varied from about 3.76x10^{18} to about 7x10^{18} atoms
per cubic centimeter. Resistances were measured using four probes, to
minimize the effect of contact resistance; a lock-in amplifier; and a
frequency of 10 Hz, to enable phase coherence of voltage and current.
Moveable probes were used to ensure homogeneity in n over the relevant
length scale. Measurements were obtained at temperatures down to 1 mK
using a dilution refrigerator, with the samples attached to a copper
cold finger, and the 0K behavior was extrapolated from this data.
Further experimental details are shown in the schematic in Figure 2, and
a semilog plot of the resulting conductivity obtained as a function of n
is shown in Figure 3.^{1}

The results in Figure 3 show a change of
10^{3} in the zero-T conductivity for a 1% change in doping
density. The transition is sufficiently sharp that its continuity
cannot be determined within experimental error. However, the minimum
conductivity anticipated by Mott et. al., for the system in question is
approximately 20 (ohm cm)^{-1}, and three metallic samples in
the experiment have conductivities significantly lower than this. While
density inhomogeneities could, according to the Mott theory, account for
the depressed minimum conductivity, it is unlikely they do in this case
based on experimental estimations of macroscopic inhomogeneity of
~0.04%.

Further, for n farther away from the phase
transition, the data fits reasonably well with the result from 3D
scaling localization theory. The conductivity in the form
σ=g_{c}/R as obtained from the theory above, fits with

where n_{c}=3.74x10^{18}
cm^{-3} and g_{c}/R_{0}=260 (ohm
cm)^{-1}. The value obtained for v, 0.55, is inconsistent with
theory. However it is found to be the same on either side of the
transition, using dielectric susceptibility data from insulating samples
from a previous experiment.^{10} Thus the work in [1] serves as
convincing evidence in favor of localization theory in general, and as
compelling evidence in favor of a scaling theory of localization in
particular.

Neither the Mott theory nor the scaling theory of
localization fully explain the dynamics of metal-insulator transitions
in imperfect crystals. A more detailed review of subsequent experiments
investigating metal-insulator transitions in phosphorus-doped silicon
specifically, including the effects of fine-tuning n_{c} by
applying mechanical stress, may be found in [11]. Indeed a good deal of
controversy has developed and much has been written on the subject since
these initial theories were proposed and experiments
performed.^{12-14} In particular, the scaling theory discussed
in this work has made a number of assumptions, for example that T=0, and
that the electrons in the system are noninteracting and in a static
random lattice. Some of the most important corrections to the theory
presented here are those that introduce effects of temperature and
especially Couloumb interactions between electrons.^{13} Further
investigations have made predictions for the effect of electric and
magnetic fields on disordered conductors. The scaling theory of
localization and its derivatives remain highly influential in modern
solid state physics and electronics to this day.

© 2007 S.L. Hellstrom. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] T.F. Rosenbaum, K. Andres, G.A. Thomas, R.N. Bhatt,
"Sharp Metal-Insulator Transition in a Random Solid," Phys. Rev. Lett.
**45**, 1723 (1980).

[2] P.W. Anderson, "Absence of Diffusion in Certain
Random Lattices," Phys. Rev. **109**, 1492 (1958).

[3] N.F. Mott, E.A. Davis, *Electronic Processes in
Non-Crystalline Materials*, (Oxford, 1971).

[4] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V.
Ramakrishnan, "Scaling Theory of Localization: Absence of Quantum
Diffusion in Two Dimensions," Phys. Rev. Lett. **42**, 673 (1979).

[5] P.A. Lee, T.V. Ramakrishnan, "Disordered Electronic
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[6] D.J. Thouless, "Electrons in Disordered Systems and
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[7] J.S. Langer, T. Neal, "Breakdown of the
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[8] R.S. Muller, T.I. Kamins, M. Chan, *Device
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2003).

[9] J.H. Wang, "Quality Evaluation of
Resistivity-Controlled Silicon Crystals," J. Cryst. Growth **287**, 258
(2006).

[10] M. Capizzi, G.A. Thomas, F. DeRosa, R.N. Bhatt,
T.M. Rice, "Observation of the Approach to a Polarization Catastrophe,"
Phys. Rev. Lett. **44**, 1019 (1980).

[11] T.F. Rosenbaum, R.F. Milligan, M.A. Paalanen, G.A.
Thomas, R.N. Bhatt, "Metal-insulator Transition in a Doped Semiconductor,"
Phys. Rev. B. **27**, 7509 (1983).

[12] N.F. Mott, "The Minimum Metallic Conductivity in
Three Dimensions," Phil. Mag. B. **44**, 265 (1981).

[13] B.L. Altshuler, A.G. Aranov, P.A. Lee,
"Interaction Effects in Disordered Fermi Systems in Two Dimensions," Phys.
Rev. Lett. **44**, 1288 (1980.

[14] B.L. Altshuler, D. Khmel'nitzkii, A.I. Larkin,
P.A. Lee, "Magnetoresistance and Hall Effect in a Disordered Two-Dimension
Electron Gas," Phys. Rev. B. **22**, 5142 (1980).