1

The Drude Model of Metals

This material outlines the model of metallic conductivity developed by P. Drude ca.

1900. The model produced a number of successes in describing the properties of metals,

and even today is useful for simple descriptions of metallic conductivity, as well as for

roughly estimating properties. The Drude model, however, failed to describe a number of

experimental observations, and also presented a number of conceptual problems that

defined the directions for the theory of metallic conductivity for the next two decades. The

resolution of these issues required the development of the quantum theory of solids.

Basic Assumptions of the Model

Drude based his model of metallic conductivity on applying the kinetic theory of

gases to a metal which he regarded as a "gas" of electrons. (Note: The electron was

discovered by J.J. Thompson in 1897.) The simplest form of the kinetic theory of gases

treats the molecules of a gas as identical solid spheres that move in straight lines until they

collide with each other (or with the walls of the containment vessel). The collision time is

assumed negligible, and other than the force that acts during each collision, no other forces

are assumed between the particles.

For simple gases, there is only one kind of particle present, but as we have

discussed a metal must possess at least two. The electron was known to be negatively

charged, but the metal was neutral. Thus a compensating particle with positive charge was

needed to maintain charge neutrality. Drude assumed that the positive charge was attached

to very heavy particles which were immobile. At the time, there was no precise idea about

the origins of either the light, mobile electrons or the heavy immobile positively charged

particles. This was solved by the quantum theory of solids. In this treatment of the Drude

model, however, it is simply assumed that when atoms of a metallic element are brought

together, the valence electrons become delocalized and free to move through the metal,

while the positively charged ion-core remains intact and fills the role of the heavy

immobile particle. This is shown qualitatively below. An isolated atom of the metal has a

Valence Electrons

Core Electrons Nucleus

Conduction Electrons

Core Electrons Nucleus Ion{

Figure 1 Schematic of the isolated atom (left) and metal (right).

2

3 23(electrons/cm ) 6.02 10 ( / ).mn Z A

nucleus of charge eZa, where Za is the atomic number and e is the magnitude of the electron

charge (taken to be a positive number). The nucleus is, of course, surrounded by Za

electrons with charge -eZa. Some of these electrons, Z, are relatively weakly bound and are

known as valence electrons. The remaining Za-Z electrons are more tightly bound and

therefore play a much less important role in binding. These are known as core electrons

and they will be regarded as fixed to the nucleus in the metal forming a metal ion. The

valence electrons, on the other hand, are free to move far from their parent atoms. In a

metal, these electrons are often referred to as conduction electrons. It is these electrons

which Drude applied the kinetic theory of gases to with the modification to the theory of

the presence of a background of immobile positive metal ions.

The density of the electron gas is determined as follows. An element contains

6.021023 atoms per mole (Avogadro's number), and m/A moles per cm3, where m is the

mass density (g/cm3) and A is the atomic mass. Since each atom contributes Z conduction

electrons, the conduction electron density is

Typical conduction electron densities are between 1022-1023 electrons/cm3. Another useful

measure of electron density is given by the radius of a sphere with volume equal to the

volume per conduction electron

Typical values of rs lie between 1-2 Å.

These densities are much greater (about a thousand times greater) than molecular

gas densities at normal temperatures and pressures. Nonetheless, and in spite of the

presence of strong electromagnetic interactions for such model systems, Drude boldly

applied the kinetic theory of a dilute neutral gas to this dense conduction electron "gas"

using the following assumptions:

1) Between collisions, the conduction

electrons do not interact either with

each other or with the metal ions. In the

absence of an externally applied field,

the electrons move in straight lines with

constant velocity. In the presence of an

applied field the conduction electrons

move according to Newton's laws of

motion in the presence of the externally

applied field. Neglect of the

electron-electron interactions is known

as the independent electron approximation, and neglect of the electron-ion interactions is

known as the free electron approximation.

2) Collisions are instantaneous and abruptly alter the electron velocity. Drude attributed

these collisions to conduction electron scattering from the metal ion cores. (The true

13

31 4 3; .

3 4s sr r

n n

Figure 2 Conduction electron trajectory

according to Drude.

3

collision analogue to kinetic gas theory would be electron-electron collisions.) We will

learn later that electron-electron scattering is indeed usually an unimportant scattering

mechanism in a metal, however, the picture of a conduction electron scattering off the ion

cores (shown above) is incorrect. However, a qualitative (and sometimes quantitative)

understanding of metallic conduction is possible by simply assuming some scattering

mechanism without being precise about the exact nature of this mechanism.

3) An electron undergoes a collision with probability per unit time 1/. This means that the

probability of an electron experiencing a collision in any (infinitesimal) time interval dt is

dt/. This time τ is known as the relaxation or collision or mean free time. This collision

time is taken to be independent of the electrons position or velocity.

4) The conduction electrons achieve thermal equilibrium with their surroundings only

through collisions (given the approximations in 1 no other mechanism remains). These

collisions preserve local thermodynamic equilibrium as follows: when the electrons

emerge from a collision, they do so with a speed that is entirely determined by the local

temperature and in a random direction. This velocity is assumed to be entirely independent

of the electron velocity prior to collision.

Applications The remainder of this discussion will concentrate on several important applications

of the Drude model, and its successes and failures in each case.

DC Electrical Conductivity Any model of DC electrical conductivity must reproduce Ohm's Law which says

that the current, I, flowing in a wire is proportional to the potential drop, V, along the wire:

V = IR, where R is the resistance of the wire. The geometrical dependence of the resistance

is usually eliminated by the introduction of a quantity characteristic of the wire material

only. This quantity is the resistivity, ρ, defined as the constant of proportionality between

the electric field E and the current

density j induced by the field

The current density is parallel to the flow of charge with a magnitude given by the total

charge per unit time crossing a unit area perpendicular to the charge flow. For example, if

a uniform current I flows in a wire of length L and cross-sectional area A, then the current

density is j=I/A. The potential drop along the wire is given by V = EL, and we therefore

have V = ρjL= ρIL/A. Thus, in this case R = ρL/A.

Now, if we have n electrons per unit volume all with velocity v , then the current

density is parallel to v , and in time dt the electrons move a distance vdt along this direction.

Therefore, n(vdt)A electrons cross area A perpendicular to the flow, and since each carries

charge -e the charge crossing A in time dt is -nevAdt. The current density is then given by

Since at any point in the metal electrons move with a variety of velocities, the net current

density is given by the above equation provided v is the average electronic velocity. In the

absence of an electric field, the electrons are just as likely to be moving in one direction as

any other and v averages to zero (there is no net current density). However, in the presence

.E j

.j nev

4

of a field E , there will be a mean electronic velocity, known as the drift velocity, directed

opposite the field (the electron charge is negative) which we can compute.

Consider an electron at time t that underwent its last collision at t = 0. Its velocity at

time t is then the velocity immediately after the collision 0v plus an additional term -e E t/m

that it acquired by moving in the field for time t. The direction of the electron motion after

a collision is assumed to be random so there is no net contribution to the current density

from 0v , and the current density must be determined by the average of -e E t/m which is

simply -e E/m. Therefore

We introduce the conductivity = 1/ρ and rewrite this as

2

; .ne

j Em

This verifies that we have a linear relationship between E and j (as required by Ohm's

Law) and allows us to estimate the conductivity in terms of known quantities (with the

significant exception of ). Alternatively, we could use measured resistivities to estimate

the relaxation time. If we express the resistivity in units of microhm-cm we can write the

relaxation time as

3

14

0

0.2210 sec.sr

a

The quantity ao is the Bohr radius and is approximately 0.5 Å. At room temperature, this

gives relaxation times of 10-14 to 10-15 seconds. To better understand what this means,

consider a quantity known as the mean free path defined as the average distance traveled

between collisions. This is given by l = v0, where v0 is the average electronic speed. At

the time, this speed was estimated using the results from the kinetic theory of gases (or

from classical energy equipartition) mv02 = 3kBT, where kB is Boltzmann's constant. This

gives a speed of about 107 cm/sec at room temperature, and therefore a mean free path of

1-10 Å. This is therefore consistent with Drude's supposition that the collisions were with

the metallic ions since this is roughly equal to the ionic spacing typical of metals.

Unfortunately for Drude this is completely wrong. The estimate of the average electronic

speed is an order of magnitude too low (as we will see later), and, contrary to these findings,

the relaxation time in real metals is temperature dependent and at low temperatures can be

over an order of magnitude greater than our estimate. These facts can produce mean free

paths of 1000 Å or more. This is certainly difficult to reconcile with the picture of the

conduction electrons bouncing off the ion cores.

Most of our problems stem from our ignorance of the scattering mechanism and

hence not being able to know . In the absence of any theory regarding the origin of the

2

/ ; .avg

nev eE m j E

m

5

collisions, we need to find quantities that are independent of to test the predictions of our

description of metallic conductivity. We shall next examine two cases which meet this

requirement, namely the calculation of electrical conductivity in the presence of a spatially

uniform magnetic field, and when the electric field is spatially uniform but time dependent.

We begin with the following observation.

The average electronic velocity v at any given time is ( ) /p t m , and the current

density is therefore

( ).

nep tj

m

We would like to know the momentum at a time (infinitesimal) dt after time t, or ( )p t dt .

The probability of an electron undergoing a collision in that time dt is simply dt/, so the

probability of the electron not suffering a collision in the same time is 1-dt/. Those

electrons that do not suffer a collision do evolve under the effects of the force ( )f t of the

electric and/or magnetic fields. These electrons then acquire an additional momentum (to

first order in dt) of simply ( )f t dt . Thus the contribution of all non-colliding electrons to

the momentum is the fraction that do not collide (1-dt/) times their average momentum

( ) ( )p t f t dt or

2( ) 1 ( ) ( ) ( ) ( ) ( ) ( )dt dtp t dt p t f t dt p t p t f t dt O dt

.

Here O(dt)2 refers to terms of order (dt)2. The correction to the above due to electrons that

do undergo a collision is of O(dt)2 because the fraction dt/ collide and their velocities are

randomly directed so that their contribution to the average net velocity is only that acquired

due to the applied force ( )f t . These electrons therefore contribute ( )dt f t dt

to

( )p t dt which is of higher order in dt than the linear terms we are retaining. We can then

write

2( ) ( ) ( ) ( ) ( ) ,dtp t dt p t p t f t dt O dt

where the contribution from all electrons is now included. If we now divide by dt and take

the limit as dt →0, we get

Thus, we see that the effect of the electron collisions is simply to introduce a frictional

damping term into the equation of motion. Let's apply this result to some interesting cases.

The Hall Effect (see also Kittel Ch. 6 section titled Motion in Magnetic Fields)

Consider the situation shown below in Fig. 3. An electric field is applied to a wire

( ) ( )( ) .

dp t p tf t

dt

6

along the x-direction inducing a current density along the same direction in the wire. In

addition, a magnetic field H lies in the positive z-direction. The magnetic force

deflects electrons in the negative y-direction (remember, the drift velocity of an electron is

opposite the current flow). Eventually the electrons encounter the side of the wire where

they accumulate. This produces an electric field in the y-direction that opposes the electron

motion and (further) accumulation at the wires side. Once a steady state is reached, the

force from the transverse (or Hall) electric field will exactly balance the magnetic force

resulting in current flow being limited to the x-direction only.

We are interested in two quantities. The first is the ratio of the field along the wire Ex to the

current density jx,

This quantity is known as the (transverse) magnetoresistance which Hall found to be

independent of H. The second quantity of interest is the size of the transverse field Ey. We

expect that this field will depend on both H and jx since it balances the magnetic force. We

therefore define a quantity called the Hall coefficient as

ev H

c

( ) .x

x

EH

j

.y

H

x

ER

j H

+ + + + + + + + + + + +

+ - - - - - - - - - - - - - - - -

Ey

H

Ex jx

-evH

vx x

y

z

Figure 3 Schematic of the Hall effect experiment.

7

Since the Hall field is along the negative y-direction, the Hall coefficient should be

negative (provided of course that the charge carriers are negative as we have assumed). If,

however the sign of the Hall coefficient is positive, we must conclude that the charge

carriers are positively charged. Thus measurement of this coefficient is extremely

important as it identifies the sign of the charge carriers in a material. It is an experimental

fact that some metals (Be, Mg, In, Al) have positive Hall coefficients. This puzzle can only

be solved by a quantum theory of solids. Despite this problem, the Drude analysis gives

fairly good agreement with a number of experiments on other metals.

The force acting on each electron is Hf e E vc

, and the momentum per

electron is given by

In the steady state, the current is time independent and the x- and y-components of

momentum satisfy the following

where c is called the cyclotron frequency and is given by

Multiplying by -ne/m and using the expression we derived earlier for the components of

the current density we find

where o is the DC conductivity in the absence of a magnetic field that we calculated in the

previous section. The Hall field Ey can now be found by setting jy to zero in the second

equation (we require there to be no transverse current). This gives

The Hall coefficient is therefore

.dp p p

e E Hdt mc

0 ,

0 ,

xx c y

y

y c x

peE p

peE p

.c

eH

mc

0

0

,

,

x c y x

y c x y

E j j

E j j

0

.cy x x

HE j j

nec

1.HR

nec

8

This says that the only parameter with metal dependence in the Hall coefficient is the

electron density. This then represents a means of testing the assertion that the electron

density is determined by assuming that the weakly bound atomic valence electrons become

the conduction electrons in a metal. This task is complicated by the general finding that,

contrary to our prediction above, the measured Hall coefficients are dependent on the

magnetic field and the temperature. However, at very low temperatures for carefully

prepared samples in high magnetic fields, the measured Hall coefficients approach the

limiting value given above.

Finally, we can find the magnetoresistance by setting jy to zero in the other equation

above to find jx = 0Ex confirming Hall's observation that the resistance is independent of

the magnetic field. Note that this is simply the expected result for the conductivity in zero

magnetic field. More recent experiments demonstrate, however, that there can be a

magnetic field dependence (in some cases this dependence can be quite dramatic) to the

resistance. Again, we will require a quantum mechanical treatment to explain this

behavior.

AC Electrical Conductivity (Supplements Kittel Ch. 10 section titled Dielectric Function

of the Electron Gas)

We want to calculate the current induced by a time-dependent electric field, so we

write the field in the form

( ) .i tE t Ee

The equation of motion from our discussion above is given by

We seek solutions of the form

( ) .i tp t pe

Substituting into the equation of motion gives

Recall that /j nep m , and realize that j will have the same time dependence as p , and

we see

Recalling our earlier definition of the conductivity

.dp p

eEdt

.p

i p eE

2

.1

nenep mj Em i

9

,j E

and comparing, we see that the AC conductivity is frequency dependent (as should be

expected) and is given by

Let us discuss some complications that we have heretofore ignored. First, the electric field

of an electromagnetic wave is accompanied by a transverse magnetic field of the same

magnitude which we have not included in our calculation. We can get away with this

because, as we have seen, the magnetic field gives an additional term in the equation of

motion ( / ) .ep mc H This is smaller than the electric field term by the factor v/c. Even

for large current densities, v is no greater than about 0.1 cm/sec and the magnetic field term

can obviously be ignored. Secondly, electromagnetic fields vary in space as well as in time,

but we have proceeded assuming a spatially uniform field. This is a valid assumption

provided that the field does not vary appreciably over a length comparable to the electron

mean free path. This is because the current density at any point is determined entirely by

the interaction with the electric field since the electrons last collision. The distance

traveled since the last collision is almost always a few mean free paths at most. Our result

is therefore valid when the wavelength λ of the field is large compared to the electron mean

free path l. The wavelength of visible light is 1000 to 10,000 Å and this assumption is

satisfied. Clearly when this is not the case we must resort to other means to determine the

conductivity.

Now, proceeding in the long wavelength limit, in the presence of current density

j we can write Maxwell's equations (assuming the induced charge density ρ vanishes) as

Again, we require a solution with time dependence e-it. We can substitute for j in terms

of E using our familiar conductivity equation .j E Using some (what should be)

familiar vector identities and doing some algebraic manipulation we find

or

This is a wave equation with form

2

00( ) , .

1

ne

i m

1 4 10 ; 0 ; ; .

H EE H E H j

c t c c t

2 4,

i i iE E H E E

c c c c

22

2

41 .

iE E

c

22

2( ) .E E

c

10

with a dielectric constant (complex) given by

Substituting our earlier result for () and assuming » 1, we find that

where p2 = (4ne2/m), and p is known as the plasma frequency. For < p, is real and

negative and the solutions to the wave equation decay in real space (that is no radiation can

propagate in the metal), but when > p, is positive and the solutions to the wave

equation are oscillatory (radiation can propagate and the metal becomes transparent). This

is, of course, only true if near p our high-frequency condition, »1, is satisfied. This

turns out to be true for metals. In fact, the alkali metals are observed to become transparent

in the ultraviolet (λ ~ 1000 Å), and the observed wavelengths below which alkali metals

become transparent agree quite closely with those predicted by the Drude model. For other

metals, however, the actual dielectric constant is far more complicated than we have

calculated and can differ considerably from the Drude result.

Another consequence of the Drude dielectric function is that a gas of electrons can

sustain charge density oscillations (i.e. has an oscillatory time dependence e-it). The

equation of continuity (charge conservation) gives

That, together with Gauss's Law,

gives

This has a solution provided

which is the condition we found for the onset of electromagnetic wave propagation in the

metal. It is also the condition required for the propagation of a charge density wave. This

wave is known as a plasma oscillation or plasmon and can be understood in terms of a

simple model. Suppose we displace the entire electron gas a distance d with respect to the

4( ) 1 .

i

2

2( ) 1 ,

p

, ( ) .j j it

4 ( ) ,E

( ) 4 ( ) ( ) .i

4 ( )1 0 ,

i

11

ionic background as shown in the figure below. This results in a surface charge which

gives rise to an electric field of magnitude 4nde. As a consequence, the electron gas

collectively obeys the equation of motion:

which gives the oscillatory behavior of the plasma.

Thermal Conductivity The Wiedemann-Franz law states that the ratio κ/σ of the thermal and electrical

conductivities of a (majority of) metal(s) is directly proportional to the temperature with a

proportionality constant that is roughly the same for all metals. The ability of the Drude

model to explain this empirical law was its greatest achievement.

We begin by assuming that the thermal current in a metal is carried entirely by the

conduction electrons. (Metals, after all, conduct heat much better than insulators.)

Thermal conduction by the ions (due to vibrations about their equilibrium positions) which

would be present in both metals and insulators is neglected with respect to thermal

conduction by the conduction electrons (only present in metals).

Consider a metal bar along which the temperature varies slowly. In the absence of

heat sources or sinks, thermal energy will flow opposite the temperature gradient (hot areas

cool and cool areas warm up). If we now supply heat to the hot end as fast as it flows away,

we can produce a steady state with both a temperature gradient and a uniform flow of

thermal energy. We define the thermal current density qj as a vector parallel to the heat

flow with a magnitude given by the thermal energy per unit time crossing a unit area

perpendicular to the heat flow (in analogy with the electrical current density). In analogy

with Ohm's Law we can write the thermal current to be proportional to the temperature

(Fourier's Law):

The thermal conductivity is represented by κ. It is positive since the thermal current flows

opposite the temperature gradient. Consider a one-dimensional example in which the

2

2

2

( )4 4 4 ,

d dNm Ne Ne nde ne Nd

dt

.qj T

+

+ +

+ +

+

- - - - - - - -

σ = +nde

N/Z ions σ = - nde

E = 2πσ + 2πσ = 4πnde

N electrons

Figure 4 Model of a plasma oscillation.

12

temperature drop is uniform in the x-direction. In the steady state, the thermal current also

flows in the x-direction. Note that after each collision an electron emerges with a speed

determined by the local temperature, with higher local temperatures at the collision site

producing more energetic electrons. Thus, even though the mean electronic velocity can

vanish at a given point, electrons arriving at a point from the high temperature side will

have higher energies than those arriving at the same point from the low temperature side.

This leads to a net flow of thermal energy toward the low temperature side.

Suppose that at a point x half the electrons arrive from the high temperature side

and half arrive from the low temperature side. Let (T) be the thermal energy per electron

in a metal at equilibrium temperature T, then an electron that underwent its last collision at

x’ will, on average, have a thermal energy (T[x’]). Electrons arriving from the high

temperature side on average have had their last collision at x-v, and therefore have thermal

energy per electron given by (T[x – v]). The contribution of all such electrons to the

thermal current density is the number of such electrons, n/2, multiplied by their velocity

multiplied by this energy, or (n/2)v(T[x – v]). Those electrons arriving from the low

temperature side with contribute (n/2)(-v)(T[x + v]) since their velocities are in the

opposite direction. We can add these to get the total thermal current density

.2

q nj v T x v T x v

If the temperature variation is small on the scale of an electron mean free path we can

expand this about x to get:

Generalizing to the three dimensional case requires only replacing v with the x-component

of the electronic velocity v and averaging over all directions. Note that <vx2> = <vy

2> =

<vz2> = (1/3)v2, and that ( / ) ( / )( / ) ( / ) / vn d dT N V d dT dE dT V c , the electronic

specific heat. Now we can write

and therefore

21 1.

3 3v vv c lvc

Before continuing, we need to discuss some issues that this derivation has dodged thus far.

For example, if the thermal energy of an electron depends on the direction it comes from

why then doesn't the electron speed as well? This would lead to a non-vanishing electronic

velocity and thus an electrical current. This in fact occurs immediately after the

temperature gradient is applied, but in analogy to the Hall field, a steady state is reached by

accumulation of charge at the surface of the metal that produces an electric field that

2 .q d dTj nv

dT dx

21( )

3

q

vj v c T

13

opposes further charge accumulation and precisely cancels the effect of the temperature

gradient on the electronic mean velocity. This field is called the thermoelectric field and

the effect goes under the name of the Seebeck effect. We have also been sloppy in

replacing certain quantities with their thermal averages. Nonetheless, this calculation

produces some remarkable results.

Using the above equation and dividing by the DC electrical conductivity to

eliminate τ we get that

Again, Drude applied the classical ideal gas law to estimate the electronic specific heat and

mean square velocity

to get the result

This agrees with the Wiedemann-Franz law with a proportionality constant (known as the

Lorenz number and equal to κ/σT) that depends only on the universal constants kB and e.

Using the above we calculate a Lorenz number of 1.1110-8 watt-ohm/K2. This is roughly

half the observed value. Originally, Drude made a factor of two error in the calculation of

DC electrical conductivity which serendipitously led to the "correct" value for the Lorenz

number. Nevertheless, this success led to continued investigations using the Drude model.

There are, however, a number of puzzles that cannot be resolved in this model. For

example, no electronic contribution to the specific heat at all close to (3/2)nkB was ever

observed. In fact, at room temperature there appeared to be no electronic contribution to

the specific heat at all! Drude's ability to correctly predict the Wiedemann-Franz law

(aside from the factor of two error) was due to two errors of factors of 100 that canceled.

More specifically, at room temperature the electronic contribution to the specific heat is

approximately 100 times less than the classical prediction, and the mean square electronic

velocity is about 100 times larger than predicted. These results can only be obtained with a

quantum theory of the electron gas. This is the topic of Chapter 6 in Kittel.

2

2

1

3 .vc mv

ne

23 1 3, and

2 2 2v B Bc nk mv k T

23

.2

BkT

e