Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/semiphy/16.Mobility-Hall.pdf · Log-log plot...

P.Ravindran, PHY02E – Semiconductor Physics, Autum 2013 17 December : Introduction to Semiconductors

http://folk.uio.no/ravi/semi2013

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Carrier Mobility and Hall Effect

1


calculation

Calculate the hole and electron densities in a piece of p-type silicon

that has been doped with 5 x 1016 acceptor atoms per cm3 .

ni = 1.4 x 1010 cm-3 ( at room temperature)

Undoped

n = p = ni

p-type ; p >> n

n.p = ni2 NA = 5 x 1016 p = NA = 5 x 1016 cm-3

3

316

23102

109.3105

)104.1(x

cmx

cmx

p

nn i

electrons per cm3

p >> ni and n << ni in a p-type material. The more holes you put in the less e-’s you have and vice versa.

P.Ravindran, PHY02E – Semiconductor Physics, Autum 2013 17 December : Introduction to Semiconductors3

Diffusion current

Reasons:

– concentration difference

(gradient)

– thermal movement

Proportional to the gradient

D: diffusion constant [m2/s]

nDqJ nn grad

pDqJ pp grad


where Dn (m2 /sec) is called the diffusion constant for electrons.

Repeating the same derivation for holes yields:


Total Current

It is possible for both a potential gradient and a concentration gradient to exist simultaneously within a semiconductor. In such a situation the total electron current density is:

And

The total current )( pntotal JJAI


Total currents

nDqEqnJ nnn grad

pDqEpqJ ppp grad

q

kTD Einstein's relationship

mV26V026.0[As]106.1

[K]300[VAs/K]1038.119

23

300

KT

Tq

kTU

Thermal voltage


The Einstein Relationship

Since both diffusion and mobility are statistical thermodynamic phenomena, D and μ are not independent. The relationship between them is given by the Einstein equation

where VT is the "volt equivalent of temperature” defined by

where k is the Boltzmann constant in joules per Kelvin.

T

n

n

p

pV

DD

VT

q

TkVT

11600


Current flow under equilibrium conditions

The total current under equilibrium conditions is equal to zero.

Total electron current, Jn and total hole current, Jp must also be zero.

Why?

Jn|diff = Jn|drift and Jp|diff = Jp|drift

Under equilibrium conditions, both drift and diffusion components will

vanish only if E = 0 and dn / dx = dp / dx = 0

Even under thermal equilibrium conditions, non-uniform doping will

give rise to carrier concentration gradient, a built-in E-field, and non-

zero current components.


Under equilibrium conditions, dEF / dx = 0; the Fermi

level inside a material or a group of materials in intimate

contact is invariant as a function of position.

–EF appears as a horizontal line on equilibrium energy

band diagram.

–If the Fermi level is not constant with position, charge

transfer will take place resulting in a net current flow, in

contrast to the assumption of equilibrium conditions.

Constancy of Fermi level


Constancy of Fermi level

Doping concentration varies with position.

This results in a gradient in carrier concentration. EC

EF represents the change in carrier concentration

with position.

If there is an electric field, this field causes drift

current.

The concentration gradient gives diffusion

current. These two currents exactly cancel each

other so that the net current is zero.

A non-horizontal Fermi level means there will be a

continuous movement of carriers from one side to the

other, indicating current flow (against the

assumption).


EXAMPLE: Given p = 470 cm2/V·s, what is the hole drift

velocity at E = 103 V/cm? What is tmp and what is the distance

traveled between collisions (called the mean free path) if the hole

velocity is 2.2107 cm/s? Hint: When in doubt, use the MKS

system of units.

Solution: n = pE = 470 cm2/V·s 103 V/cm = 4.7 105 cm/s

tmp = pmp/q =470 cm2/V ·s 0.39 9.110-31 kg/1.610-19 C

= 0.047 m2/V ·s 2.210-12 kg/C = 110-13s = 0.1 ps

mean free path = tmhnth ~ 1 10-13 s 2.2107 cm/s

= 2.210-6 cm = 220 Å = 22 nm

This is smaller than the typical dimensions of devices, but getting close.

Drift Velocity, Mean Free Time, Mean Free Path

P.Ravindran, PHY02E – Semiconductor Physics, Autum 2013 17 December : Introduction to SemiconductorsFrom Principles of Electronic Materials and

Devices, Third Edition, S.O. Kasap (©

McGraw-Hill, 2005)

Scattering of electrons by an ionized impurity.


From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Lattice-Scattering-Limited Mobility

L = lattice vibration scattering limited mobility, T = temperature

L T3 / 2

Ionized Impurity Scattering Limited Mobility

I = ionized impurity scattering limited mobility, NI = concentration of the ionized

impurities (all ionized impurities including donors and acceptors)

I

IN

T 2/3


Effective or Overall Mobility

e = effective drift mobility

I = ionized impurity scattering limited mobility

L = lattice vibration scattering limited mobility

1

e1

I

1

L


Log-log plot of drift mobility versus temperature for n-type Ge and n-type Si samples.

Various donor concentrations for Si are shown. Nd are in cm-3. The upper right inset is

the simple theory for lattice limited mobility, whereas the lower left inset is the simple

theory for impurity scattering limited mobility.


Drift mobility of Si at Τ= 300 K for various dopant concentration.


Conductivity (σ), resistivity(ρ) , mobility (µ) and number of carrier (n)

with temperature in semiconductors.


When we add carriers by doping, the number of additional carrers, Nd, far exceeds those in an intrinsic semiconductor, and we can treat conductivity as

s = 1/r = qdNd

Resistivity as a function of

charge mobility and number

carrier.


The Hall Effect

This phenomenon, discovered in 1879 by

American physics graduate student (!) Edwin

Hall, is important because it allows us to

measure the free-electron concentration n for

metals (and semiconductors!) and compare

to predictions of the FEG model.

The Hall effect is quite simple to

understand. Consider a B field applied

transverse to a thin metal sample carrying

a current:

I


Hall effect

Hall effect was discovered in 1879 by Edward H. Hall

Exists in all conducting materials

It is particularly pronounced and useful in semiconductors.

Hall effect sensor is one of the simplest of all magnetic sensing devices

Used extensively in sensing position and measuring magnetic fields


Hall effect

When a magnetic field is applied perpendicular to a current carrying

conductor or semiconductor, voltage is developed across the

specimen in a direction perpendicular to both the current and the

magnetic field. This phenomenon is called the Hall effect and voltage

so developed is called the Hall voltage.

Let us consider, a thin rectangular slab carrying current (i) in the x-

direction.

If we place it in a magnetic field B which is in the y-direction.

Potential difference Vpq will develop between the faces p and q which

are perpendicular to the z-direction.


i

B

P

Q

X

Y

Z

+ + ++ +VH

+++

++++

+ ++

++

+++

++ ++

+

-

P – type semiconductor


i

B

X

Y

Z

VH

+

-

__

__

__

_

__

__

__

_

_

_

_ _ P

Q

N – type semiconductor


Hall Effect in p & n-type semiconductors

Hall effect sign conventions for p-

type sample Hall effect sign conventions

for n-type sample


Hall Effect Measurements

A hypothetical charge carrier of charge

q experiences a Lorentz force in the

lateral direction:

It

w

qvBFB

As more and more carriers are

deflected, the accumulation of charge

produces a “Hall field” EH that imparts

a force opposite to the Lorentz force:

HE qEF

Equilibrium is reached when these two

opposing forces are equal in magnitude,

which allows us to determine the drift

speed:

HqEqvBB

Ev H

From this we can write the current density:B

nqEnqvJ H

And it is customary to define the Hall

coefficient in terms of the measured

quantities:

nqJB

ER H

H

1


Hall Effect Results!

In the lab we actually measure the Hall voltage VH and the current I, which gives

us a more useful way to write RH:

If we calculate RH from our

measurements and assume |q| = e

(which Hall did not know!) we can find

n. Also, the sign of VH and thus RH tells

us the sign of q!

nqIB

tV

BwtI

wV

JB

ER HHH

H

1

/

/wEV HH JwtJAI

RH (10-11 m3/As)

Metal n0 solid liquid FEG value

Na 1 -25 -25.5 -25.5

Cu 1 -5.5 -8.25 -8.25

Ag 1 -9.0 -12.0 -12.0

Au 1 -7.2 -11.8 -11.8

Be 2 +24.4 -2.6 -2.53

Zn 2 +3.3 -5 -5.1

Al 3 -3.5 -3.9 -3.9

The discrepancies between the FEG

predictions and expt. nearly vanish

when liquid metals are compared. This

reveals clearly that the source of these

discrepancies lies in the electron-lattice

interaction. But the results for Be and

Zn are puzzling. How can we have q >

0 ???

*

*

Stay tuned…..


Hall Effect

0yj →

0y c x

q qE v

m q

t t

x x

qv E

m

t

0z z

qv E

m

t

y c x

qE E

q t

x

qBE

mct

Hall coefficient:

y

H

x

ER

j B

2

x

x

qBE

mc

nqE B

m

t

t

1

nqc

electrons



The Hall Effect

Accumulation of charge leads to Hall field EH.

Hall field proportional to current density and B field

is called Hall coefficient


The Hall coefficient

carrier density form Ohm’s law?

for the steady

state we get


The Hall coefficientOhm’s law contains e

2

But for RH the sign of e

is important.


What would happen for positively charged carriers?


Hall effect and magnetoresistance

Edwin Herbert Hall (1879): discovery of the Hall effect

HvF c

eL

the Lorentz force

in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges

balances the Lorentz force

quantities of interest:

magnetoresistance (transverse magnetoresistance)

Hall (off-diagonal) resistance

( ) xxx

x

y

yx

x

y

H

x

EH

j

E

j

ER

j H

r r

r

RH → measurement of the sign of the carrier charge

RH is positive for positive charges and negative for negative charges

the Hall effect is the electric

field developed across two

faces of a conductor

in the direction j×H

when a current j flows across

a magnetic field H

( ) xxx

x

y

yx

x

VR H R

I

VR

I

resistivity

Hall resistivity

the Hall coefficient


mc

eH

ppeE

ppeE

mce

dt

d

ce

c

y

xcy

xycx

t

t

t

0

0

1

1

pHpE

p

HvEfforce acting on electron

equation of motion

for the momentum per electron

in the steady state px and py

satisfy

cyclotron frequency

frequency of revolution

of a free electron in the

magnetic field H

at H = 0.1 T

0yj xxc

y jnec

HjE

0s

t

multiply by

yxcy

xycx

jjE

jjE

ts

ts

0

0

m

ne

m

pnej

mne

ts

t

2

0

/

the Drude

model DC

conductivity

at H=0

Hrc

erm cc

2

weak magnetic fields – electrons can complete only a small part of revolution between collisions

strong magnetic fields – electrons can complete many revolutions between collisions1

1

t

t

c

c

1tcj is at a small angle f to E f is the Hall angle tan f ct

RH → measurement of the density

necRH

1xx Ej 0s

the resistance does not

depend on H

GHzcc 1~

2

n




measurable quantity – Hall resistance HRHH r

ecn

H

I

V

Dx

y

H

2

r zD nLn 2for 3D systems

for 2D systems n2D=n

jE r Ej sin the presence of magnetic field the

resistivity and conductivity becomes tensors

yxcy

xycx

jjE

jjE

ts

ts

0

0

yyyx

xyxx

rr

rrr

0 0

0 0

1

1

cx x y

cy x y

E j j

E j j

t

s s

t

s s

for 2D:

00

00

1

1

sst

stsr

c

c

nec

H

ne

m

cxy

xx

0

2

0

1

s

tr

tsr

22

22

xyxx

xy

xy

xyxx

xxxx

rr

rs

rr

rs

2

0

2

0

)(1

)(1

t

tsss

t

sss

c

cyxxy

c

yyxx

1

yyyx

xyxx

yyyx

xyxx

rr

rr

ss

sss

x xx xy x

y yx yy y

E j

E j

r r

r r

0 0

0 0

1

1

x xc

y yc

E j

E j

s t s

t s s

x xx xy x

y yx yy y

j E

j E

s s

s s

P.Ravindran, PHY02E – Semiconductor Physics, Autum 2013 17 December : Introduction to Semiconductorsfrom D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999)

nec

H

ne

m

cxy

xx

0

2

0

1

s

tr

tsr

nec

HHRHH rHall resistance

strong magnetic fields

quantization of Hall resistance

at integer and fractional

the integer quantum Hall effect

and the fractional quantum Hall effect

2e

hxy

nr

hc

eHnn

the Drude

model

the classical

Hall effect1tc

1tc

weak

magnetic

fields

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