CHARGE-CARRIER DRIFT IN SEMICONDUCTORS

CHARGE-CARRIER TRANSPORT IN SEMICONDUCTORS

* In chapter 4 we studied semiconductors in equilibrium and determined the free

electron and free hole concentrations.

* The knowledge of free carriers densities is very essential toward the

understanding of the electrical and optical properties of semiconductor materials and devices.

* This is because current, which are the flow of free electrons and holes, is strongly

dependent on these free carriers densities. The process, by which electrons and holes flow, is called transport.

* In this chapter, we will study the two basic transport mechanisms in

semiconductor materials : drift which is the movement of charge due to an applied electric field, and diffusion which is the flow of charge due to density gradients.

* We will also develop the first electronic device which is a simple resistor and

explain the theory behind its operation.

CURRENT DENSITY : GENERAL DEFINITION

* Consider the volume element of a unit cross-sectional area shown taken in an n-type semiconductor of electron concentration n.* Under the application of an electric field E the acquired electron velocity is vn as shown.* The electric current density developed, Jn, is defined as the current per unit area and given by

 

Jn = -envn* Similarly for a p-type semiconductor the hole current density Jp for a hole density p and an average hole velocity vp is

 

Jp = epvp

DRIFT IN ELECTRIC FIELD* When electrons and holes move in a force field they are said to drift in the field. In general, drift is a type of transport available to electrons and holes in a semiconductor.* When drift is the cause of the average velocities given to electrons and holes, these velocities are termed drift velocities in this specific case.The question that now arises is what is the relationship between drift velocity and electric field E.* Before we tackle this question, let us consider the situation in the semiconductor when it is in thermodynamic equilibrium, which means when there is no voltage V applied across the semiconductor.* In thermal equilibrium and when no voltage is applied and on the average, half the electrons in the conduction band must be moving in one direction and half in the opposite direction in thermodynamic equilibrium.* The same must be true for the holes in this situation. That is, vn must be zero because there is no current density, Jn, for conduction band electrons in thermodynamic equilibrium and vp must be zero because there is no current density, Jp, for valence band holes in thermodynamic equilibrium.* Now what happens when the electric field E is turned on ?

* To determine these values of and in the presence of the electric field, we consider electrons first and note that the effective mass approximation, allows us to write

 

mn* dv

dt= -eE

* Since the velocity of an electron, written here as without the subscript n to distinguish it from the average electron velocity, is made-up of , its field-caused component, and , its random component, this expression tells us how the additional velocity component given to an electron's random velocity changes with time in the presence of our external electric field E. Upon integration

 

v =-eEt

mn*

+ constant

* The constant in this equation is a particular electron's random velocity at t=0.The electron concentration does not increase indefinitely but is limited by collisions with other electrons, phonons, impurities and crystal defects

* The average velocity, vn, one particular electron has at any particular time is hence governed by an average time between collisions, tn, and may be written as

 

vn =-eEt nmn

*

* This may be written as

 

vn = -mnE

and

mn =et nmn

*

where mn is called the electron mobility.

* Similar algebra for holes shows that

 

v p =eEt p

mp

*

which may be re-written as

 

v p = mpE

and

mp =et p

mp

*

where mp is called the hole mobility.

Experimentally measured mobilities at 300 K for some single crystalline pure semiconductors.

* The drift velocity of electrons in an electric field is proportional to the electric field. It is in the direction opposite to E.* The drift velocity of holes in an electric field is also proportional to the electric field. It is in the direction of E.If these expressions are substituted into the basic definitions of current density, the result is a model

 

Jn = enmnE =snE

where sn = enmn is conductivity component contributed by electrons

for electron drift current density and a model

 

Jp = enmpE =spE

where sp = enmp is conductivity component contributed by holes

for hole drift current density.

Conductivity and Resistivity

s = enmn + epmp

and

r =1

s= enmn + epm p( )

-1

Copper

Si (undoped)

Silicon dioxide

6x105

5x10-6

10-14 to -16

Material s in W-1cm-1

At room-temperature

Problem (1)Electrons are injected into one end of 1.5 mm long semiconductor and are

found to require a transit time of 10-2 seconds to reach the opposite end at room temperature, when the sample is subjected to a voltage of 0.1 mV. What is the electron drift velocity and the electron mobility for this semiconductor ?Solution

The average drift velocity of the electrons vn is given by

 

vn =L

t(1)

where L is the length of the semiconductor and t is the transit time. Hence

 

vn =1.5X10-4( )

10-2( )=1.5X10-2 cm /s (2)

The drift velocity and electron mobility are related as given by

 

vn = mnE

Where E is the electric field. Therefore

 

mn =vn

E

However the electric field and the potential difference, V, across the semiconductor are related by

 

E =V

L

(4)

(5)

Therefore from Eqs. (3) and (4)

 

mn =vnL

V

or

mn =1.5X10-2( ) 1.5X10-4( )

0.1X10-6( )= 22.5 cm2V -1s-1 (6)

DRIFT IN ELECTRIC AND MAGNETIC FIELDS : THE HALL EFFECT

N-TYPE SEMICONDUCTOR

* We will explore the motion of electrons and holes in a semiconductor that is subjected to an applied electric field E and the semiconductor is placed in a magnetic field B. We will first study the case of electrons in an n-type semiconductor.* Let us consider an n-type semiconductor biased as shown the figure below. Before the magnetic field is turned on the electrons drift with the average drift velocity

 

vn = -mnE

* A charge moving in a magnetic field experiences a magnetic force Fm. This magnetic force is referred to as Lorentz force.* The magnitude of this force is proportional to the magnitudes of the charge, velocity, and magnetic field.* The magnetic force is in a direction perpendicular to the plane containing the magnetic field and velocity vectors.

* When the magnetic field is turned on, an electron drifting in our n-type semiconductor would be acted upon by a Lorentz force Fm given by

 

Fm = -evn ´B

in which vn, B, and Fm constitute a right-handed system for a moving positive charge.* Using the right-hand rule to figure out how Fm is directed, we see that it is a force pushing the electrons toward the surface we have labeled "Top" in the figure.* As the electrons are drifting due to the net interaction of the drift force and the collision force, they are also subjected to a Lorentz force orthogonal to their drift direction.* This will cause the top surface to become negatively charged. Since the semiconductor is space-charge neutral and will try to stay that way, a corresponding positive charge (holes and positive ion cores) will develop on the "Bottom" surface in the figure.* This accumulation of charges in both surfaces will continue until an electrostatic field is developed between these two charges such that the force on charge carriers caused by this field is equal to the Lorentz force.

* This electrostatic field is called the Hall field, EH, and it will be oriented from bottom to top as seen in the figure above. The magnitude of this Hall field is

 

evnB = eEH

* We can see from the figure that

 

EH =VH

Hwhere VH is the resulting Hall voltage and H is the sample's thickness.

* Using the equation above and the relation

and from the figure, and for a uniform electric field applied to the semiconductor the electric field, E = V/L ; L is the length of the sample) we get

 

vn = mnV

L

 

mn =L

H

æ

è ç

ö

ø ÷ VH

V

æ

è ç

ö

ø ÷

1

B

which provides us with a way of measuring the mobility of the electrons in the material.

* If we use the relation

(only the magnitude of Jn) for an electron drift current we get

 

n =BJnH

eVH

which provides us with a way of finding the density of electrons in the material.

* The above equation is often written as

 

VH =1

ne

æ

è ç

ö

ø ÷ BHJn

or

VH = RHBHJn

where

 

RH =1

ne

æ

è ç

ö

ø ÷

and RH is called the Hall coefficient.

 

vn = mnV

L

P-TYPE SEMICONDUCTOR

We can similarly derive relations similar to those for n-type for a p-type semiconductor as follows :

 

mp =L

H

æ

è ç

ö

ø ÷ VH

V

æ

è ç

ö

ø ÷

1

B

and

 

VH =1

pe

æ

è ç

ö

ø ÷ BHJp

or

VH = RHBHJp

where

 

RH =1

pe

æ

è ç

ö

ø ÷

is the Hall coefficient.

Problem (2)A Hall sample is biased as shown below. The magnetic field Bx = 1 weber/m2 is

applied along the positive x-axis.(i) If the charge carriers in the sample are electrons, what will be the direction of the Hall field EH ?(ii) The magnitude of the Hall voltage Vz is measured to be 10 mV. Evaluate the free electron concentration n in the material.(iii) For the small Hall voltage Vz in (ii) above, find the electron mobility, mn, in the material.

Solution(i) Using the right-hand rule for vn, B, and the force Fm we see, from the figure, that the negatively-charged electrons tend to be deflected in the negative z-axis direction. In this case electrons will accumulate in the bottom surface of the Hall sample, whereas positive charge will accumulate on the top surface. The electric field EH will be directed from the top surface to the bottom surface ; i. e., in the negative z-direction.

(ii) The relation between the free electron concentration, the magnetic field, current density, and Hall voltage is given as

 

n =BJnH

eVH

Substituting for B, Jn, H, VH, and e in Eq. (1) gives

(1)

 

n =

1W /m2( )10X10-3( )

1.0X10-6( ) 1.0X10-2( )A /m2

é

ë

ê ê

ù

û

ú ú

1.0X10-6 m( )

1.6X10-19 C( ) 10X10-3 V( )or

n = 0.625X1021 m-3

(iii) The electron mobility is given by

 

mn =LVH

HVB

or

mn =5X10-2 m( ) 10X10-3 V( )

1.0X10-6 m( ) 10V( ) 1W /m2( )= 50 m2V -1s-1

(2)

(3)

DRIFT AND DIFFUSION IN SEMICONDUCTORS

* In semiconductors the origins of drift current lie, as we know, in the directed motion superimposed on the random thermal motion of electrons and holes by an electric field.* The origins of diffusion current, on the other hand, lie in the random thermal motion itself.* Using Jn = envn we can write for the diffusion of electrons

* Dn is the diffusion coefficient for electrons and n is their density. * Similarly, the hole diffusion current density, Jp, can be written as

where Dp is the hole diffusion coefficient and p is the free hole density.

* If the concentration gradients are along one direction only (e. g., x-axis), then the one-dimensional current densities Jnx and Jpx for electrons and holes, respectively, take the forms

 

Jnx = eDndn

dxand

 

Jpx = -eDpdp

dx

* The total electron current density, Jn, in the presence of both drift and diffusion transport mechanisms can be obtained by combining drift and diffusion currents

* Also the total hole current density Jp is given by

* If the applied electric field and the concentration gradients are along the same direction so will the free carrier flux. In this case we may drop out the vector notations

 

Jn = emnnE + eDndn

dx and

 

Jp = emp pE - eDpdp

dx

Typical diffusion coefficient values at T = 300 K

EINSTEIN RELATIONS

Since both the mobility and the diffusion coefficient (also called the diffusivity) of a carrier capture the physics of collision processes, the question of whether they are related arises.Here we just report the relationship without going through the derivations

 

Dn =kBT

e

æ

è ç

ö

ø ÷ mn

A similar relation holds for holes in a p-type semiconductor ; i. e.,

 

Dp =kBT

e

æ

è ç

ö

ø ÷ mp

T is absolute temperature, kB is Boltzmann’s constant, ms are electron and hole mobilities and e electronic charge.

Problem (3)The hole concentration in silicon at T = 300 K is given by

where x is measured in mm. If the hole diffusion coefficient is Dp = 48 cm2s-1, determine the hole diffusion current density as a function of x.Solution

 

p x( ) =1015 exp -x

22.5

æ

è ç

ö

ø ÷ cm

-3

The hole concentration p(x) at T = 300 K is given by

 

p x( ) =1015 exp -x

22.5

æ

è ç

ö

ø ÷ cm

-3 (1)

The hole diffusion current density along the x-axis is given as

 

Jpx = -eDpdp

dx(2)

Differentiating p(x) with respect to x, and substituting for Dp and e in Eq. (2) gives

 

Jpx x( ) = - 1.6X10-19( ) 48( ) -1015

22.5exp -

x

22.5

æ

è ç

ö

ø ÷

é

ë ê

ù

û ú

or

Jpx x( ) = 3.41X10-4 exp -x

22.5

æ

è ç

ö

ø ÷ C /cm2s

(3)

where the current density is in C cm-2 s-1 and x is measured in microns.

THERMODYNAMIC EQUILIBRIUM AND THE PRINCIPLE OF DETAILED BALANCE

* In thermodynamic equilibrium the principle of detailed balance tells us that the drift component of the current density is exactly equal and opposite to the diffusion component of the current density ; i. e., in thermodynamic equilibrium Jn = 0 which implies

 

0 = emnnE + eDndn

dx

* The electric field can be written as

where V is the electric potential and eV is, hence, the potential energy of the electron.

* Since we are interested in the gradient of the potential energy, we can use any part of the band diagram that is parallel to Ec such as Ei (the intrinsic Fermi level), hence

 

eE =dE i

dx

* The electron concentration in the semiconductor in terms of Ei can be written as

 

n = ni expEF - E i

kBT

æ

è ç

ö

ø ÷

or we can write

 

dn

dx= ni exp

EF - E i

kBT

æ

è ç

ö

ø ÷

1

kBT

dEF

dx-dE i

dx

æ

è ç

ö

ø ÷

é

ë ê

ù

û ú

or

dn

dx=n

kBT

dEFdx

-dE idx

æ

è ç

ö

ø ÷

* Which yields the condition for thermodynamic equilibrium as

 

dEF

dx= 0

* Had we started by requiring Jp = 0 for thermodynamic equilibrium we would have obtained the same condition.* This condition states that the Fermi level in a semiconductor in thermal equilibrium is independent of position and it stays flat (parallel to the x-axis) throughout the entire semiconductor.