1 The electron as a particle

1.1 Introduction
A solid consists of atoms, and an atom consists of electrons and a nuclei. In discussing electrical properties, we can treat a solid as many positive ions, which are xed, immersed in an electron gas. The electrons follow the same statistical distribution as gas molecules, i.e., Maxwell-Boltzmann distribution. The average kinetic energy 1 of each degree of freedom is kT . 2 So we may say that the mean thermal velocity of electrons is given by

1 2 3 mvth = kT 2 2
because particles moving in 3D have 3 freedoms.

(1)

The scenario, xed positive ions immersed in electron gas, is a very good approximation of electronic structure of solids. The model is depicted in Fig. 1. In this model, positive ions (atoms deprived of electrons) are piled in a 3D periodical arrangement. Free electrons are moving in the interatomic space. They can be accelerated or decelerated by external electric eld, and can be scattered by the ions. The electrons are far away from each other, so the interaction between them can be ignored (characteristic of gas).

Figure 1: Ions immersed in electron gas

1.2 Eect of an electric eld - conductivity and Ohm's law

is applied between the two ends of a solid whose

Then an electric eld E = U/L exists at every point in the solid, and e a = m E . Thus the electrons will acquire a velocity in the direction of the eld. Note that meanwhile the electrons also have random thermal

L.

causes an acceleration

velocity. Then we assume the directed velocity is lost after every collision, because electron is much lighter than ion. Suppose

is the average time between two successive collisions, the nal velocity

of an electron will be

and the average velocity is

1 vaverage = a 2
The correct formula is

(2)

In fact, this is statistically wrong, because we can not simply use  average time .

vaverage = a
where

(3) The

is the

collision time , vaverage is drift velocity , also denoted by vD .

relation between

vD =

e E m

(4)

where the proportional constant is called the  mobility , indicating the speed of the device. Assume all electrons drift with their drift velocity density of electrons,

vD ,

the total number of elec-

trons crossing a plane of unit area per second is given by multiplying

vD

by the

Ne .

Multiplying further by the charge on an electron we obtain

the electric current density:

J = Ne evD
Note: the random velocities (diusion velocity) do not contribute to they average out to zero. They give rise to or device size is small,

(5)

noise

electrical noise . When current is small

because

becomes very interesting. The useful working device

(size, frequency, etc.) is limited by noise. Furthermore, the current density and the external electric eld is related by

J=

N e e2 E m

(6)

This is the microscopic picture corresponding to Ohm's law:

J = E
where

(7)

is the electrical conductivity, given by

e Ne e = e Ne e m

(8)

Thus the conductivity is determined by two factors: charge density

Ne e and mobility
must be

e .
Ohm's law implies that

E . In this model, it is reasonable to assume that l, the distance between collisions (usually called the mean free path ), rather than , is independent of external eld. Here l must be related to by
independent of the external eld

is constant independent of

E.

That is,

l = (vth + vD ) l is a constant, vD vth >> vD .

(9) must vary with

E,

so

= l/(vth + vD )

unless

The thermal velocity at RT is

vth =
In a typical metal,

3kT /m 105 m/s

(10)

e = 5 103 m2 /(V.s). For an electric eld of 1V/m, the 3 drift velocity is vD = 5 10 m/s. Thus the above inequality vth vD holds, i.e., the conductivity and the mobility e are independent of external eld. In semiconductor, e is much higher than in metal. At very high eld (e.g., in 7 today's transistor, 3 V across a 200 nm channel, the eld is 1.5 10 V/m), the
mobility will be saturated, the so-called

channel eect s.

high eld eect .

It is one of the

short

1.3 The Hall eect

= az v . Now apply a v B is given by

magnetic eld

in the

+y

direction. The force on (11)

e( B) v
Note

is negative. In this case, the force is

ax evB .

So electrons will be deected

upwards, in the

+x direction.

They will accumulate at the top end of the slab. Then

the bottom will be positive due to loss of electrons. An electric eld is thus built between the top (-) and the bottom (+). This internal eld, denoted by

EH , prevents
(12)

the electron from moving upwards. The electrons will stop moving upwards when

EH = vB
Since

is proportional to current density

J,

the above condition is written as

EH = RH JB
where

, RH =

1 Ne e

(13)

RH

is the so-called

Hall coecient .

Figure 2: Hall eect By measuring

EH , J

and

B , RH

as well as

Ne

can be determined.

If the carriers are positively charged, i.e.,

is positive, then it will still move

upwards and build an electric eld in the x direction. This is opposite to what the above case. Experiment show some conductors (e.g., zinc) and semiconductors have this behavior. So in those materials, the current is conducted by positively charged carriers, called  holes . For a typical metal,

vD = 5 103

m/s,

B=1

T, the transverse electric eld is

EH = vB = 5 103 V/m