solid-state electronics chap. 5 instructor: pei-wen li dept. of e. e. ncu 1 chap 5. carrier motion ...

Solid-State Electronics Chap. 5

Instructor: Pei-Wen LiDept. of E. E. NCU

1

Chap 5. Carrier Motion

Carrier Drift Carrier Diffusion Graded Impurity Distribution Hall Effect Homework



2

Carrier Drift

When an E-field (force) applied to a semiconductor, electrons and holes will experience a net acceleration and net movement, if there are available energy states in the conduction band and valence band. The net movement of charge due to an electric field (force) is called “drift”.

Mobility: the acceleration of a hole due to an E-field is related by

If we assume the effective mass and E-field are constants, the we can obtain the drift velocity of the hole by

where vi is the initial velocity (e.g. thermal velocity) of the hole and t is the acceleration time.

qEdt

dvmF p *

Etvm

eEtv i

pd ,

*



3

Mobility

E = 0

In semiconductors, holes/electrons are involved in collisions with ionized impurity atoms and with thermally vibration lattice atoms. As the hole accelerates in a crystal due to the E-field, the velocity/kinetic energy increases. When it collides with an atom in the crystal, it lose s most of its energy. The hole will again accelerate/gain energy until is again involved in a scattering process.



4

Mobility

If the mean time between collisions is denoted by cp, then the average drift velocity between collisions is

where p (cm2/V-sec) is called the hole mobility which is an important parameter of the semiconductor since it describes how well a particle will move due to an E-field.

Two collision mechanisms dominate in a semiconductor:– Phonon or lattice scattering: related to the thermal motion of atoms; L T-3/2

– Ionized impurity scattering: coulomb interaction between the electron/hole and the ionized impurities; I T3/2/NI., : total ionized impurity conc. , I

If T, the thermal velocity of hole/electron carrier spends less time in the vicinity of the impurity. less scattering effect I

EEm

ev p

p

cpd

* *p

cpdpp m

e

E

v

adI NNN



5

Mobility

Electron mobility Hole mobility



6

Drift Current Density

If the volume charge density of holes, qp, moves at an average drift velocity vdp, the drift current density is given by

Jdrfp = (ep) vdp = eppE.

Similarly, the drift current density due to electrons is given by

Jdrfn = (-en) vdp = (-en)(-nE)=ennE

The total drift current density is given by Jdrf = e(nn+pp) E



7

Conductivity

The conductivity of a semiconductor material is defined by Jdrf E, so = e(nn+pp) in units of (ohm-cm)-1

The resistivity of a semiconductor is defined by 1/



8

Resistivity Measurement

Four-point probe measurement

factor correction: ;2 cc FFI

Vs



9

Velocity Saturation

So far we assumed that mobility is indep. of E-field, that is the drift velocity is in proportion with the E-field. This holds for low E-filed. In reality, the drift velocity saturates at ~107 cm/sec at an E-field ~30 kV/cm. So the drift current density will also saturate and becomes indep. of the applied E-field.



10

Velocity Saturation of GaAs

For GaAs, the electron drift velocity reaches a peak and then decreases as the E-field increases. negative differential mobility/resistivity, which could be used in the design of oscillators.

This could be understood by considering the E-k diagram of GaAs.



11

Velocity Saturation of GaAs

In the lower valley, the density of state effective mass of the electron mn

* = 0.067mo. The small effective mass leads to a large mobility. As the E-field increases, the energy of the electron increases and can be scattered into the upper valley, where the density of states effective mass is 0.55mo. The large effective mass yields a smaller mobility.

The intervalley transfer mechanism results in a decreasing average drift velocity of electrons with E-field, or the negative differential mobility characteristic.



12

Carrier Diffusion

Diffusion is the process whereby particles flow from a region of high concentration toward a region of low concentration. The net flow of charge would result in a diffusion current.



13

Diffusion Current Density

The electron diffusion current density is given by Jndif = eDndn/dx,

where Dn is called the electron diffusion coefficient, has units of cm2/s.

The hole diffusion current density is given by Jpdif = -eDpdp/dx,

where Dp is called the hole diffusion coefficient, has units of cm2/s.

The total current density composed of the drift and the diffusion current density.

1-D

or 3-D

dx

dpeD

dx

dneDEepEenJ pnxpxn

peDneDEepEenJ pnxpxn



14

Graded Impurity Distribution

In some cases, a semiconductors is not doped uniformly. If the semiconductor reaches thermal equilibrium, the Fermi level is constant through the crystal so the energy-band diagram may qualitatively look like:

Since the doping concentration decreases as x increases, there will be a diffusion of majority carrier electrons in the +x direction.

The flow of electrons leave behind positive donor ions. The separation of positive ions and negative electrons induces an E-field in +x direction to oppose the diffusion process.



15

Induced E-Field

The induced E-field is defined as

that is, if the intrinsic Fermi level changes as a function of distance through a semiconductor in thermal equilibrium, an E-field exists.

If we assume a quasi-neutrality condition in which the electron concentration is almost equal to the donor impurity concentration, then

So an E-field is induced due to the nonuniform doping.

dx

dE

edx

eEd

dx

dE FiFi

x

1))/((

dx

xdN

xNe

kTE

dx

xdN

xN

kT

dx

Ed

dx

EEd

n

xNkTEExN

kT

EEnn

d

dx

d

d

iiF

i

diFd

iFio

)(

)(

1

)(

)(

)()(

)(ln)(exp



16

Einstein Relation

Assuming there are no electrical connections between the nonuniformly doped semiconducotr, so that the semiconductor is in thermal equilibrium, then the individual electron and hole currents must be zero.

Assuming quasi-neutrality so that n Nd(x) and

Similarly, the hole current Jp = 0

dx

dneDEenJ nxnn 0

relation-Einstein ---

)()(

)(

1)(0

)()(0

e

kTD

dx

xdNeD

dx

xdN

xNe

kTxNe

dx

xdNeDExeNJ

n

n

dn

d

ddnn

dnxndn

e

kTD

p

p



17

Einstein Relation

Einstein relation says that the diffusion coefficient and mobility are not independent parameters.

Typical mobility and diffusion coefficient values at T=300K( = cm2/V-sec and D = cm2/sec)

n Dn p Dp

Silicon 1350 35 480 12.4

GaAs 8500 220 400 10.4

Germaium 3900 101 1900 49.2



18

Hall Effect

The hall effect is a consequence of the forces that are exerted on moving charges by electric and magnetic fields.

We can use Hall measurement to – Distinguish whether a semiconductor is n or p type– To measure the majority carrier concentration– To measure the majority carrier mobility



19

Hall Effect

A semiconductor is electrically connected to Vx and in turn a current Ix flows through. If a magnetic field Bz is applied, the electrons/holes flowing in the semiconductor will experience a force F = q vx x Bz in the (-y) direction.

If this semiconductor is p-type/n-type, there will be a buildup of positive/negative charge on the y = 0 surface. The net charge will induce an E-field EH in the +y-direction for p-type and -y-direction for n-type. EH

is called the Hall field. In steady state, the magnetic force will be exactly balanced by the indu

ced E-field force. F = q[E + v x B] = 0 EH = vx Bz and the Hall voltage across the semiconductor is VH = EHW

VH >0 p-type, VH < 0 n-type



20

Hall Effect

VH = vx W Bz, for a p-type semiconductor, the drift velocity of hole is

for a n-type,

Once the majority carrier concentration has been determined, we can calculate the low-field majority carrier mobility.

For a p-semiconductor, Jx = eppEx.

For a n-semiconductor,

H

zxzxH

xxdx edV

BIp

epd

BIV

Wdep

I

ep

Jv

H

zx

edV

BIn

WdepV

LI

x

xp

WdenV

LI

x

xn



21

Hall Effect

Hall-bar with “ear” van deer Parw configuration



22

Homework

5.14 5.20



23

solid-state electronics chap. 5 instructor: pei-wen li dept. of e. e. ncu 1 chap 5. carrier motion ...

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