The mean free path of electrons in metals*
E. H. Sondheimer{
Royal Society Mond Laboratory, Cambridge
Contents page
1. Introduction 499
2. Conduction in thin ®lms and wires 505
3. Magnetic e ects in thin conductors 515
4. The anomalous skin e ect 524
Acknowledgement 536
References 536
1. Introductio n
1.1. The foundations of the modern electron theory of metals were laid at the
beginning of the present century, when the existence of a gas of free electrons was
postulated by Drude in order to explain the conducting properties of metals; the
behaviour of the electrons was subsequently analysed by Lorentz by means of the
statistical methods of the dynamical theory of gases. The chief success of the Drude±
Lorentz theory was the prediction of the Wiedemann±Frenz law connecting the
electrical and thermal conductivities, but later developments revealed an increasing
number of serious di culties, outstanding among them the inability of the theory to
explain why the conduction electrons do not contribute appreciably to the speci®c
heat of a metal. This paradox was not resolved until the advent of quantum
mechanics, when Pauli and Sommerfeld applied the Fermi±Dirac statistics to the
free electrons in a metal, showing that in this way most of the contradictions could
be reconciled.
The Drude±Lorentz±Sommerfeld theories are essentially formal in character.
They involve as arbitrary parameters the number n of free electrons per unit volume,
which is assumed to be of the same order as the number of atoms per unit volume,
and the mean free path l of the electrons which is to be determined from a
comparison of theory and experiment. So far as the electrical conductivity
¼
0
is
concerned, the results of the Sommerfeld theory are summarized by the formulae
n ˆ
8p
3
m·vv
h
3
;
…1a†
¼
0
ˆ
n
°
2
l
m·vv
;
…1b†
Advances in Physics, 2001, Vol. 50, No. 6, 499±537
Advances in Physics ISSN 0001 ±8732 print/ISSN 1460±6976 online # 2001 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080 /0001873011010218 7
* This article was the ®rst to appear in Advances in Physics, originally published in volume 1,
1952. It had attracted 1030 citations by October 2001, and is ranked 5 in the index of articles attracting
more than 100 citations.
{ Fellow of Trinity College, Cambridge.
in which h is Planck’s constant, ¡
°
is the charge and m is the mass of an electron, and
·vv is the velocity of an electron at the surface of the Fermi distribution.
¼
0
therefore
depends on n, l and fundamental constants only. In order to obtain the correct order
of magnitude for the conductivity and the correct temperature variation, it is
necessary to assume that l is of the order of several hundred interatomic distances
at ordinary temperatures, and increases rapidly in a pure metal towards very low
temperatures.
Such long free paths whic h vary with temperature are very di cult to explain on
classical theory, bu t they can be understood on the basis of the recent developments
in the theory of metals. These were initiated by Bloch and elaborated by a great
many authors; they have been concerned with a detailed quantum-mechanica l
analysis of the motion of electrons in a crystal latice , and they have made it possible
to give precise meanings to the two fundamental concepts of the `number of free
electrons’ and the ` mean free path’ and to obtain numerical estimates of these
quantities in certain cases. The electrons in a metal are regarded as distributed over a
number of energy bands, ®lling most of them completely. All the electrons are free to
move through the lattice, but only those which are contained in incompletely ®lled
energy bands can contribute to the resultant current and are to be regarded as free
electrons for the purposes of conduction theory. The number of free electrons in a
metal is of the same order as the number of atoms: the precise number, however,
depends on the detailed con®guration of the energy bands and need not be a simple
multiple or submultiple of the number of atoms. The variation of n w ith temperature
is negligible, since at ordinary temperatures the fre e electrons form a highly
degenerate Fermi±Dirac gas.
An electron can move freely through a perfect and rigid crystal lattice and ther e is
no resistance. In a pure metal a ®nite free path is caused by the thermal vibrations of
the lattice and is of the same order as the average wavelength of the sound waves in
the metal, which is large compared with the interatomic distance and is increased by
lowering the temperature. The free path does not increase inde®nitely, however, as
the temperature is lowered, and at very low temperatres it tends to a constant
`residual’ value l
r
which is determined by static lattice imperfections such as the
presence of impurity atoms, and which is of the order of the distance between the
impurities.
The following theoretical formula may be deduced for the electrical conductivity
of a metal, subject to many simplifying assumptions concerning the interaction
between the electrons and the lattice vibrations (see, for example, Wilson 1936,
Chapter VI):
1
¼
0
ˆ
m·vv
n
°
2
l
r
‡
m
2
± ²
1=2
9ph
2
C
2
8n
D°
2
Mk
Y±
3=2
T
Y
5
…
³=T
0
z
5
dz
…e
z
¡ 1†…1 ¡ e
¡z
†
:
…2†
Here k is Boltzmann’s constant,
±
is the Fermi energy level (
±
ˆ
1
2
m·vv
2
†,
Y
is the
Debye temperature, M is the mass of an atom,
D
is the volume of the unit cell, and C
is a constant which determines the interaction between th e electrons and the lattice.
According to this formula the `ideal’ and `residual’ resistances are additive, and the
ideal resistance is proportional to T at high and to T
5
at very low temperatures. (For
commercially pure metal s the residual resistance can be neglected at ordinary
temperatures.) The temperature variation of the resistance predicted by (2) is,
generally speaking, in good agreement with observation, particularly for the
E. H. Sondheimer500
monovalent metals. The absolute value of the free path at any temperature can be
estimated by combining (1) and (2) and substituting reasonable value s for the
parameters. The chief uncertainty concerns the magnitude of the interaction
constant C, estimates of which can be obtained by numerical integration if the wave
functions of the conduction electrons are known; it is of the same order as the Fermi
energy
±
. Th e estimates of the free path are of the right order of magnitude, but
precise numerical values cannot be obtained in this way.
1.2. An important conclusion to be drawn from the detailed quantum-mechan-
ical theory is that the simple Sommerfeld treatment of the conduction phenomena
remains co rrect within certain limits. In the Sommerfeld theory the electrons are
regarded as perfectly free, their energy being proportional to the square of the
velocity; this remains approximately true for electrons moving in a lattice, but th e
mass m which appears in equations (1) and (2) must be regarded as an e ective mass
which is of the same order of magnitude as, but not necessarily equal to, the mass of
a free electron. This model of `quasi-free’ electrons, which is implicit in the derivation
of (2), applies most closely to the monovalent metals in which the conduction
electrons are all contained in a single energy band; for these metals, moreover, the
number of conduction electrons should be precisely one per atom. In multivalent
metals, in which the electrons occupy more than one band, the model may still be
used to give a semi-quantitative description of the simpler conduction phenomena,
but quantities such as n and m must then be regarded as representing certain
averages of the numbers of electrons and the e ective masse s of the electrons in the
various bands, and the precise numerical values have no immediate physical
signi®cance. It would, of course, be possible to consider more complicated models,
for example one in which the conduction electrons are contained in two overlapping
energy bands. In the present state of the theory of metals, however, it is impossible to
work out a theory which fully takes into account the electronic structure peculiar to
any particular metal, and instead of introducing a large number of parameters of
doubtful physical signi®cance it is best to work with the simplest model which gives
reasonable results. For more complicated conduction phenomena, however, par-
ticularly those which are associated with the presence of a magnetic ®eld or with
anisotropy e ects, the free-electron model is entirely inadequate (it leads, for
example, to a zero magneto-resistanc e e ect), and eve n a qualitative theory can
only be obtained by using a model in which the energy surfaces do not form a singly-
connected set of spheres.
In the Sommerfeld theory the free path is most conveniently introduced through
the time of relaxation
½
, which is de®ned as follows. Let
v
ˆ …v
x
;
v
y
;
v
z
† be the
velocity of an electron, and let 2…m
=
h†
3
f …
v; r
†d
r
d
v
be the number of electrons in
the volume element d
r
ˆ dx dy dz which have their velocities in the range
d
v
ˆ dv
x
dv
y
dv
z
; f is the distribution function as usually de®ned in Fermi±Dirac
statistics. Suppose that some non-equilibrium distribution function is set up by a
system of external forces which are suddenly removed; the rate of approach
to equilibrium under the in¯uence of collisions alone is then supposed to be
given by
@
f
@
t
µ ¶
coll
ˆ ¡
f ¡ f
0
½
;
…3†
Mean free path of electrons in metals 501
where f
0
is the equilibrium distribution function. The time of relaxation need not be a
constant and may depend, for example, on the velocity; if v is the mea n velocity of
those electrons to which
½
refers, the corresponding free path l is de®ned by l ˆ v
½
.
In general ‰
@
f
=@
tŠ
coll
takes the form of an integral operator which does not reduce to
the simple form (3); it is then impossible to de®ne a free path in any natural way. The
detailed theory of the conduction mechanism in metals shows, however, that a free
path does exis t in the above sense under certain conditions for quasi-free conduction
electrons; thus it can always be de®ned for scattering by randomly distributed
impurity atoms, and it also exists for scattering by lattice vibrations if the
temperature is above the Debye temperature. So far as the electrical conductivity
is concerned, it is a reasonble approximation to assume that a free path can be
de®ned for all temperatures, so that equation (1) holds and the temperature variation
of the free path is the same as that of the electrical conductivity. An approximate
theoretical expression for l in terms of the atomic constants of the metal can be
obtained by combining equations (1) and (2). It must be borne in mind, however,
that the free path associated in this way with the electrical conductivity is not
necessarily the same as the free path associated, for example, with the thermal
conductivity.
1.3. With all the provisos mentioned, then, the simple Sommerfeld picture of the
free electrons in a metal and their mean free path retains its validit y and may be used
to discuss the conduction phenomena. It is therefore an important problem to
determine the various parameters of the theory by as many independent methods as
possible; such estimates are of interest in themselves in providing information about
the electronic structure of metals, and they serve as a valuable check on th e
consistency of the free-electron theory. Furthermore, since the calculation of the
free path from fundamental principles is highly complicated and involves many
drastic approximations , it is desirable to have methods by which l may be estimated
directly from observational data.
The simplest procedure (Mott and Jones 1936) is to compare equation (1) with
the observed electrical conductivity, but this gives only n
2=3
l and l cannot be
obtained unless n can be estimated independently.{ It is, however, possible to obtain
l directly by measuring the conductivity under conditions where the fre e path may be
compared with some other characteristic length in the metal. E ects of this type have
attracted much attention in the last few years, and it is with them that we shall
henceforth be concerned.
The most obvious method is to us e a thin ®lm or wire and to arrange that the free
path is comparable in magnitude with the thickness or diameter of the specimen; the
arti®cial limitation of the free path by the boundaries of the specimen causes an
increase in the resistivity above its value in the bulk metal, and this may be used to
deduce the ratio of free path to thickness or diameter. This topic i s reviewed in
section 2.
In section 3 we consider the more complicated e ects which occur when a thin
specimen is placed in a magnetic ®eld. The ordinary bulk magneto-resistance e ect
depends in a complicated way on the binding of the electrons in the lattic e and is zero
E. H. Sondheimer502
{ It should be remarked that the value of l de®n ed by Mott and Jones and tabulated on p. 268 of
their book is twice the free path de®ned here; the latter is the physically relevant quantity.
for quasi-free electrons; in thin specimens, however, where boundary scattering of
electrons is important, the alteration of the electron trajectories in a magnetic ®eld in
general leads to a non-zero change of resistance even if the electrons are regarded as
free. These curious `geometrical’ e ects, being classical in nature, are entirely
di erent from the bulk e ect and are essentially simpler to understand. The details
of the phenomena vary with the shape of the specimen and with the relative
con®gurations of specimen, curren t and magnetic ®eld; their analysis in all cases
involves a new quantity with the dimensions of length, namely the radius
r
0
ˆ m·vvc
=°
H of a free-electron orbit in a magnetic ®eld H. The experiments therefore
give, in addition to the free path, a direct estimate of the momentum m
·
vv of the
electrons at the surface of the Fermi distribution, and hence of the number of free
electrons according to equation (1a). The method is, however, severely restricted in
practice by the disturbing e ect of the bulk magneto-resistance phenomenon.
In section 4, ®nally, we consider the so-called anomalous skin e ect in metals.
This is a more sophisticated side e ect in which the free path is compared, not with
the physical dimensions of the specimen, but w ith the distance to which a high-
frequency electric ®eld penetrates into the metal. Experiments on the high-frequency
skin resistance of metals allow values of l
=¯
to be deduced, where
¯
is the classical
skin penetration depth, an d sinc e
¯
depends only on the frequency and the d.c.
electrical conductivity, the free path can again be obtained directly from experi-
mental magnitudes.
1.4. These phenomena present interesting problems from both experimental and
theoretical points of view. Let us consider the orders of magnitude of the various
characteristic lengths. The free path in metals at room temperatures is of the order of
10
¡5
cm or less, but in a pure metal at liquid-helium temperatures the high values of
the conductivity indicate that it may be as large as 10
¡2
cm. It is clear, therefore, that
extremely thin ®lms would be required in experiments on the size e ect at normal
temperatures; such ®lms are di cult to prepare and usually show subsidiary resistive
e ects, often tim e dependent, which tend to obscure the pure geometrical limitation
of the free path with which we are alone concerned. In order to obtain results which
are free from ambiguity, it is therefore essential to carry ou t the experiments at very
low temperatures where relatively large specimens (of thickness 10
¡3
cm) may be
used. The necessity for using specimens of this order of thickness becomes even
greater when we consider the e ects which take place in a magnetic ®eld. These
e ects show up when the orbit radiu s r
0
is comparable with the thickness of the
specimen. Since ·vv is of the order of 10
8
cm/sec, a magnetic ®eld of reasonable
magnitude (several kilogauss) corresponds to r
0
10
¡3
cm; and since r
0
is inversely
proportional to H, impracticably large magnetic ®elds would be require d in the case
of ®lms much thinner than this. Finally, in the anomalous skin e ect l must be large
compared with
¯
; the ratio l
=¯
is proportional to l
3=2
and to the squar e root of the
frequency, so that both very low temperatures and high frequencies are required. For
a pure metal at liquid-helium temperatures and for microwave frequenci es, l
=¯
is of
the order of 100.
One may say, therefore, that the size e ects are essentially low-temperature
phenomena. Since the free path at low temperatures varies from specimen to
specimen and is not a characteristic property of the metal, it is usual to measure
(or estim ate in some way) the value of the bulk conductivity
¼
0
which corresponds to
the free path l, and to express the results of the experiments in terms of the ratio
¼
0
=
l.
Mean free path of electrons in metals 503
