Lecture contents - University at Albany, SUNYsoktyabr/NNSE618/NNSE618-L18-MS-junction.pdf · Schottky effect = Image force barrier lowering 23 Barrier height slightly depends on current

NNSE 618 Lecture #18

1

Lecture contents

• Metal-semiconductor contact

– Electrostatics: Full depletion approximation

– Electrostatics: Exact electrostatic solution

– Current

– Methods for barrier measurement


2

• metal-semiconductor contact

• p-n homojunctions

• heterojunctions

Formalism includes the following phenomena:

• Electrostatics (Gauss law)

• Continuity equations:

• Current equations

– Drift and diffusion currents

• Einstein relation (in non-degenerate

semiconductor)

– Thermionic current

– Tunneling current

Junctions: general approaches, conventions

q

TkD B nTknqJ Bnn

nqDnqJ nnn

nnn Jq

RGt

n

1ppp J

qRG

t

p

1

pqDnqJ ppp

2 4

2

0

replace

0

14

0 8.85 10F

cm

=> - Formation of potential barriers

- Different from bulk material

CGS

SI

Poisson equation


3

eVq M 75.4

: The Vacuum level. It represents the energy of a free electron

: work function of the metal (constant of the material)

: work function of the semiconductor (depends on doping)

: electron affinity (constant for semiconductor)

0

M

S

E

SM Consider the case where , and the two materials come in contact. On

average, the electrons in the metal will tipically have lower energy than in the

semiconductor (lower EF). Thus there will be a transfer of electrons from the

semiconductor into the metal (holes are ignored).

Metal and semiconductor: Schottky approach

Two systems are isolated

from Muller, Kamins 2003


4

Formation of metal/semiconductor interface

Band diagram before contact established

Charge redistribution at contact

Band diagram of M-S contact

(Schottky)

Barrier height

Built-in potential

From Van Zeghbroeck, 1996


Band structure after contact:

• If trap density is very high the alignment of the

Fermi levels will be accomplished by the transfer

of electrons from the traps into the metal, instead

of from the semiconductor into the metal:

(1)

• The trap density is finite, therefore, height of the

potential barrier is somewhere between the (1)

and Schottky model:

5 Formation of metal/semiconductor interface: Fermi-level

pinning (Cowley-Sze)

• M-S contact properties are determined by potential variation in the semiconductor (not metal)

• Usually difference in work-functions does not determine the contact barrier due to existence of

interface states (Fermi level pinning) – this behaviour is technology dependent!

• However, the basic formalism for electrostatics currents is still valid

• Band structure before contact: electrons

trapped in the interface states create depletion

zone and band bending

From Colinge & Colinge 2005 ( )B Mq q

0B Gq E q


6

Metal-semiconductor contact

• In GaAs there is hardly a

dependence of M-S contact barrier

properties on metal

• Most of the barrier are within 0.2

eV though metal work functions are

within 0.8 eV

From Murakami, 1993

M-S barrier height of n-GaAs vs. work function of metal


7

Schottky-barrier heights

From Sze, 1981


8 Electrostatics: Full depletion approximation

Charge density in MS contact

dqN

We consider semiconductor

fully depleted up to xd

(donors are ionized, no

electrons)

Integrate charge density => field

Integrate field => potential (built-in + applied) 2

2

s

d d

dx dx


( ) dd

s

qNx xx

2

( )2

di a d

s

qNx V x x

Integrating from 0 to xd :


9

Applying potential

Capacitance (per unit area)

Potential (built-in + applied) Band diagram


21

q

kT

qnLD

Compare to the Debye length:

Depletion width


10

Charge density in general case

In n-type semiconductors without acceptors and

holes, considering zero potential deep into

nondegenerate and fully ionized semiconductor :

Poisson equation yields:

Solution (field vs. potential):

Compare to full depletion approximation:

Capacitance:

Electrostatics: exact solution


11 Electrostatics: exact solution


Capacitance

Depletion width (effective):

Field and potential may be also taken from full

depletion approximation

tai

d

d VVqN

x 2 ai

d

d VqN

x 2

Full depletion approximation:

dd xx

qN

22

dd

a xxqN

V

Numerical solution needed for (x) calculation


12 Electrostatics: exact solution


Results of numerical solution


13

Applicability: Diffusion and drift are valid if concentration is not changing

at a mean-free path:

This requirement is stronger than

Drift-diffusion dominates in low-doped low-mobility semiconductors

Current: diffusion theory: 1

mmthd vx

driftJ in equilibriumdiffJ

Electron fluxes – currents are opposite

Applying drift-diffusion equation in

the semiconductor

The current:

Forward bias Reverse bias


nn

1i

B d

e

k T x

;i Bk T n

e ndx


14

The drift/diffusion current:

Multiplying both sides by , and integrating from 0 to xd :

Boundary conditions for electron density and potential:


exp kT

0 0 0

( )

dd d

q q q

kT kT kTx n n

xx xq q q

kT kT kTx n n

d dnJ e qn e qD e

dx dx

dJ e dx qD ne dx qD ne

dx

KTq

CBeNn

/)0(

KTq

CddneNNxn

/)(

KTEE

CfCeNn

/)(

anBaidVVx

)(

0)0(

( )xx

0

xd


Using also:

kTD

e

2

( )2

di a d

s

qNx V x x


15 Current: diffusion theory: 3

Integrating:

Estimating denominator:

From electrostatic solution:

Assuming

Finally for the current density: with saturation current

1exp

kT

qVJJ a

SD

2 2exp

d i an c BSD

s

qN Vq D N qJ

kT kT

( )

0 0

( ) ( 1)n B n a aB B

d d

q q V qVq q

kT kT kT kT kTn C n C

x x xq q

kT kT

qD N e e e qD N e eJ

e dx e dx

2

( ) 22 2

d d di a d d d i a

s s s d

qN qN qNx xx V x x x x xx V

x

2 2 sd i a

d

x VqN

Leaving the linear

dominant term

i aV kT

2( )2( )

0 0

( 1)2 ( ) 2 ( )

i ad di a

d

V xqx xq qV

kT x d dkT kT

i a i a

kTx kTxe dx e dx e

q V q V


Schottky diode drift-diffusion current can be written

slightly differently, given that JSD depends on Va

The saturation current can be written as a drift current at the metal-

semiconductor interface :

maxd d

s

qN x

max max( 1) ( 1)

a aB qV qVq

kT kT kTCJ q N e e q n e

2'2 ( )

( 1) ( 1)a aB qV qVq

n C d i a kT kT nkTS

s

q D N qN VJ e e J e

kT


2 2 sd i a

d

x VqN

Using also:

Log-Linear Plot for Al/Si diode

where n – ideality factor is a fitting

parameter (don’t mix it up with

concentration!)


17

Thermionic emission dominates in semiconductors with high mobility and

high doping.

• Current is due to electrons with energy higher than the barrier. Although

the potential barrier is larger than kT/q at room temperature there exists a

non-zero probability that some electrons gather enough energy to

overcome the barrier.

• Current from semiconductor to metal:

• After substituting non-degenerate 3D-DOS and averaging, current from

semiconductor to metal:

with Richardson constant

Current: thermionic theory: 1

E

Electron density

vx

kT

qV

kT

qTAJ aB

ms expexp* 2

2

3 2 2

0

4 * ** 120

qm k m AA

h m cm K


18

For non-degenerate semiconductor density

of states times distribution function is:


minimal velocity of an electron in the quasi-neutral

n-type region to cross the barrier



19

Current from metal to semiconductor does

not depend on applied voltage and at

zero bias equals to

Total current:

with

Richardson constant


20 * exp Bm s s m a

qJ J V A T

kT

1exp

kT

qVJJ a

ST

kT

qTAJ B

ST

exp* 2

2

3 2 2

0

4 * ** 120

qm k m AA

h m cm K


• Quantum mechanics, i.e. An electron with E > b may be QM reflected or

an electron with E < b may tunnel through the barrier when biased

• Injection of minority carriers (holes from n-type semicomnductor) at

high reverse bias

• Often phenomenological equation is used:

Log-Linear Plot for Al/Si diode

)1(' nKT

qV

Sx

a

eJJ

J

V

Linear-Linear Plot

Forward Bias Reverse Bias


Additional factors affecting the current


21

Dominates in highly-doped semiconductors and at low

temperatures

Quantum mechanic tunneling is described by wave

function

Transmission coefficient (triangular barrier):

Current is calculated by integrating over density of

states*distribution function, similarly to the thermionic

current but with tunneling probability

Current: Tunneling

Electron density

E

Jthermionic

Jtunnel

t x

dnJ q v dE

dE


22

Tunneling dominates in highly-

doped semiconductors and at low

temperatures

Current: tunneling vs. thermionic

From Sze, 1981


23 Schottky effect = Image force barrier lowering

Barrier height slightly depends on current

Charge near a metal surface is attracted to the

surface with force

Maximum of the barrier occurs at

Reduction of potential barrier:

224 x

qF

mxq

24

From Sze, 1981

16

qxm


24

Measuring the barrier height: internal photoemission

• Internal photoemisison: current through

Schottky diode is measured as a function

of incident photon energy

• Illumination through a thin metal is

usually used

• Photorsponce is given by Fowler theory

(thermionic), where quadratic

dependence is asymptotic at

• Can be used to study image-force

lowering of barrier

kTh b 3

From Sze, 1981


25

Measuring the barrier height: C-V

From Sze, 1981

• Slope gives carrier density in

semiconductor

• Deep levels can be probed (or even

identified by DLTS)


26

Measuring the barrier height: I-V activation energy

From Sze, 1981

• Slope of log(Is/T2) vs. 1/T (for

thermionic contact) gives activation

energy corresponding to the barrier

height

• Does not need knowledge of contact

geometry

• 0.71 - 0.81 eV for Al-n-Si is measured

Arrenius plots of Al - n-Si diode current of

current at a fixed forward voltage


27

Ohmic contacts

• Ohmic contacts should have negligible

resistance relative to bulk or spreading

resistance of semiconductor device

• Since M-S contact usually has a potential

barrier due to interface states there are

two ways to obtain low resistance

junctions:

• Reduce barrier height by choice of

metal and processing techniques

• Reduce the barrier width by doping

From Sze, 1981


28

Ohmic contacts

• Figure of merit for Ohmic contacts = Specific contact resistivity

• When thermionic current dominates:

• When tunneling current dominates

(high doping level)

From Sze, 2002

12

0

C

V

JR cm

V

*exp B

C

qkR

kTqA T

4 *exp B

C

D

mR

N

thermionic tunneling

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Lecture contents - University at Albany, SUNYsoktyabr/NNSE618/NNSE618-L18-MS-junction.pdf · Schottky effect = Image force barrier lowering 23 Barrier height slightly depends on current