Lect_02.pdf

1Jose E. Schutt‐Aine ‐ ECE 442

ECE 442Spring 2007

2. Semiconductor Physics

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


8A1A

2A

3B 4B 5B 6B 7B 8B 11B 12B

3A 4A 5A 6A 7APeriodic Table of the Elements

Los Alamos National Laboratory Chemistry Division

11

1

3 4

12

19 20 21 22 23 24 25 26 27 28 29 30

37 38 39 40 41 42 43 44 45 46 47 48

55 56 57

58 59 60

72 73 74 75 76 77 78 79 80

87 88 89

90 91 92 93 94 95 96

104 105 106 107 108 109 110 111 112

61 62 63 64 65 66 67

97 98 99

68 69 70 71

100 101 102 103

114 116 118

31

13 14 15 16 17 18

32 33 34 35 36

49 50 51 52 53 54

81 82 83 84 85 86

5 6 7 8 9 10

2H

Li

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc Ti V Cr Mn Fe Co Ni Cu Zn

Y Zr Nb Mo Tc Ru Rh Pd Ag Cd

La* Hf Ta W Re Os Ir Pt Au Hg

Ac~ Rf Db Sg Bh Hs Mt Ds Uuu Uub Uuq Uuh Uuo

B C N O F

Al Si P S Cl

Ga Ge As Se Br

In Sn Sb Te I

Tl Pb Bi Po At

He

Ne

Ar

Kr

Xe

Rn

39.10

85.47

132.9

(223)

9.012

24.31

40.08

87.62

137.3

(226)

44.96

88.91

138.9

(227)

47.88

91.22

178.5

(257) (260) (263) (262) (265) (266) (271) (272) (277) (296) (298) (?)

50.94

92.91

180.9

52.00

95.94

183.9

54.94

(98)

186.2

55.85

101.1

190.2 190.2

102.9

58.93 58.69

106.4

195.1 197.0

107.9

63.55 65.39

112.4

200.5

10.81

26.98

12.01

28.09

14.01

69.72 72.58

114.8 118.7

204.4 207.2

30.97

74.92

121.8

208.9 (209) (210) (222)

16.00 19.00 20.18

4.003

32.07 35.45 39.95

78.96 79.90 83.80

127.6 126.9 131.3

140.1 140.9 144.2 (147) (150.4) 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0

232.0 (231) (238) (237) (242) (243) (247) (247) (249) (254) (253) (256) (254) (257)

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

barium

francium radium

vanadium

cesium

helium

boron carbon nitrogen oxygen fluorine neon

aluminum silicon phosphorus sulfur chlorine argon

scandium titanium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton

yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon

lanthanum hafnium

cerium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium

tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon

actinium

thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium nobelium lawrencium

dubnium seaborgium bohrium hassium meitnerium darmstadtium

1.008

6.941

22.99

Lanthanide Series*

Actinide Series~

1s1

[Ar]4s23d104p3[Ar]4s23d3[Ar]4s13d10

[Ne]3s23p6[Ne]3s23p4

[Ar]4s1 [Ar]4s23d10

1s2

[He]2s1 [He]2s2

[Ar]4s23d7

[Ne]3s23p5

[He]2s22p1 [He]2s22p2 [He]2s22p3

[Ar]4s23d5

[He]2s22p4 [He]2s22p5 [He]2s22p6

[Ar]4s23d104p5

[Ne]3s1 [Ne]3s23p1 [Ne]3s23p3[Ne]3s23p2

[Rn]7s25f146d2

[Ne]3s2

[Ar]4s2 [Ar]4s23d1 [Ar]4s23d2 [Ar]4s13d5 [Ar]4s23d6[Ar]4s23d8 [Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p4 [Ar]4s23d104p6

[Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s24d2 [Kr]5s14d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d7 [Kr]5s14d8 [Kr]4d10 [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 [Kr]5s24d105p2 [Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6

[Xe]6s1 [Xe]6s2 [Xe]6s25d1

[Xe]6s24f15d1 [Xe]6s24f3 [Xe]6s24f4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f75d1 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13 [Xe]6s24f14 [Xe]6s24f145d1

[Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10

[Rn]7s1 [Rn]7s2 [Rn]7s26d1

[Rn]7s26d2 [Rn]7s25f26d1 [Rn]7s25f36d1 [Rn]7s25f46d1 [Rn]7s25f6 [Rn]7s25f7 [Rn]7s25f76d1 [Rn]7s25f9 [Rn]7s25f10 [Rn]7s25f11 [Rn]7s25f12 [Rn]7s25f13 [Rn]7s25f14 [Rn]7s25f146d1

[Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9


The Periodic TableThe periodic table of elements is organized based on the periodicity of the electronic structure in atoms

– All the elements in the same row make up a period– All the elements in the same column make up a group– Elements in a group have the same valence shell configuration


Semiconductor – Silicon (Si)Si: 4th column in periodic table - 4 valence electrons shared through covalent bonds

+4 - - +4 +4- -

-

-

-

-

-

-

+4 +4+4

+4 - - +4 +4- -

-

-

-

-

-

-

- - - -

siliconatom

covalentbonds

electron

Intrinsic Semiconductor Si material


• Free Electrons and Holes– At low temperatures, all covalent bonds are intact and very few electrons are

available to conduct electric current– At room temperature, some of the bonds are broken by thermal ionization and

some electrons are freed. This creates a hole.– Free electrons and holes move randomly across silicon crystal structure

Recombination: electron filling a hole

Intrinsic Semiconductor – Silicon (Si)

n: concentration of free electronsp: concentration of holes

In thermal equilibrium: n = p = ni

ni: concentration of free electrons in intrinsic silicon at room temperature


Electrons and holes move in Si via drift or diffusion

Intrinsic Semiconductor – Silicon (Si)

diffusion: associated with random motion due to thermal agitation or gradient

Hole diffusion current:

/2 3 GE kTin BT e=

B: material dependent parameter = 5.4 × 1031 for SiEG: Bandgap energy = 1.12 eVk: Boltzmann constant=8.62×1015 ev/KAt T = 300 K, ni = 1.5 × 1010 carriers/cm3

p p

dpJ qD

dx= −

Jp: current density A/m2

q: electron chargeDp: Diffusion constant (diffusivity) of holes

electron diffusion current: n n

dnJ qD

dx=


Drift in SemiconductorsDrift: occurs when electric field is present and is applied across the Si. Free electrons & holes are accelerated by Efield and acquire velocity

for holesdrift pv Eµ=

µp: mobility for holes = 480 cm2 /V sec µn: mobility for electrons = 1350 cm2 /V sec

Associated currents are:

p drift pJ qp Eµ− =

mobility is 2.5 times greater for electrons than holes

n drift nJ qn Eµ− =

for holes

for electrons

The total drift current is:

( )drift p nJ q p n Eµ µ= +


Drift in Semiconductorswhich gives the conductivity

Resistivity is:

driftJ Eσ=

From thermodynamics, we have the Einstein relation:

( )p nq p nσ µ µ= +and

1/ρ σ=

pnT

n p

DD kTV

qµ µ= = =


N-Type Doped Semiconductor

Introducing atoms of a pentavalent element (e.g. phosphorus) results in n-type silicon phosphorus is donor

+4 - - +4 +4- -

-

-

-

-

-

-

+4+4

+4 - - +4 +4- -

-

-

-

-

-

-

- - - -siliconatom

covalentbonds

electron

+5

-freeelectronphosphorus

atom

ND: concentration of donor atoms nno: concentration of free electrons at thermal equilibrium

2no no in p n= no Dn N≅

2i

noD

np

Nthe concentration of holes is:

n-type Si

- Electrons are majority carriers- Holes are minority carriers


P-Type Doped Semiconductor

Introducing atoms of a trivalent element (e.g. boron) results in p-type silicon boron is donor

NA: concentration of acceptor atoms ppo: concentration of holes at thermal equilibrium

2po po in p n= po Ap N≅

2i

poA

nnNthe concentration of free electrons is:

p-type Si

- Holes are majority carriers- Electrons are minority carriers

+4 - - +4 +4- -

-

-

-

-

-

-

+4+4

+4 - - +4 +4- -

-

-

-

-

-

-

- - -siliconatom

covalentbonds

electron

+3 +

hole

boronatom


A bar of silicon (n-type) has 1cm2 cross-sectional area and is 10 cm long. A potential of 1V is applied across its length leading to a current of 100 mA. Calculate the concentration of free electrons in this material.

3

1 10100 10

VRI −= = = Ω

×I = 100 mA

10 1 1 10

l RAR cmA l

ρ ρ ρ ×= ⇒ = ⇒ = = Ω−

-11 1 / cmσρ

= = Ω

191.6 10 coulq −= ×21300 cm / secn voltµ = −

194 19 3

19

1 10 4.8 10 10 /1.6 10 1300 1.6 1300

nn n D D

n

q N N cmqσσ µµ

−−= ⇒ = = = = × ×

× × ×

15 34.8 10 carriers /DN cm= ×

Si Doping - Example


What is the hole diffusion flow for the carrier distribution shown below. Assume unit cross-section area.

I = 100 mA

p pdpI qD Adx

= −

5332.8 10 A=3.328 pI mA−= ×

Diffusion - Example

0.1 cm

1014/cm3

4 x 1013 /cm3

Concentration of holesConstant slope

13 3 13 320 4 10 160 100.1

dp cm cmdx

− −−= × = ×

19 14 51.6 10 13 16 10 1.6 13 16 10 1pI A Amps− −= − × × × × = − × × × ×

191.6 10 coulq −= ×213 cm /secpD =

2 ×

p pI J A=


– At room temperature, the intrinsic carrier density is small compared to device doping levels.

– ni increases rapidly with temperature (doubles for every 11o C in Si).

– At high temperature, thermal generation can be the dominant process.

• Donors and Acceptors– When semiconductor is doped, impurity energy levels are introduced.

– Donor level is neutral if filled by electron and positive if empty.

– Acceptor level is neutral if empty and negative if filled by electrons.

– Impurity levels are determined by ionization energy (for donors and acceptors).

Doped Semiconductor


Ec

Ev

-

Eg

-

+ +

CB

VB

Ec

Ev

- -

+

- - -EDEF

CB

VB

Ec

Ev +

-

EA

EF

CB

VB

+ + + +

Band Diagrams

intrinsic n type p type


( ) ( )2 21 42no D A D A in N N N N n⎡ ⎤= ≥ − + − +⎢ ⎥⎣ ⎦

2 2i i

nono D

n npn N

=

no D D A i D An N if N N n and N N−

ln CC F

D

NE E kTN

⎛ ⎞− = ⎜ ⎟

⎝ ⎠

ln noF i

i

nE E kTn

⎛ ⎞− = ⎜ ⎟

⎝ ⎠

( ) ( )2 21 42po A D A D ip N N N N n⎡ ⎤= ≥ − + − +⎢ ⎥⎣ ⎦

2 2i i

popo A

n nnp N

=

po A A D i A Dp N if N N n and N N−

ln VF V

A

NE E kTN

⎛ ⎞− = ⎜ ⎟

⎝ ⎠

ln poi F

i

pE E kT

n⎛ ⎞

− = ⎜ ⎟⎝ ⎠

n-type and p-type semiconductors

• For n-type semiconductors– Electrons are majority carriers.– Holes are minority carriers.

• For p-type semiconductors– holes are majority carriers.– electrons are minority carriers.


Mobility: At low electric field, the drift velocity is proportional to the electric field strength E. The proportionality constant is defined as the mobility µ(cm2/V-s)

dv Eµ=

– Acoustic phonons and ionized impurities result in carrier scattering – These affect the mobility– Temperature affects mobility

Diffusion: Carrier diffusion coefficient Dn or Dp is another important parameter associated with mobility. In thermal equilibrium, we get

Einstein Relationpn

n p

DD kT kTandq qµ µ

= =

Dn: diffusion constant for electronsµn: mobility for electronsDp: diffusion constant for holesµp: mobility for holes

Drift and Diffusion


ExVH

Va

+-

+E

yEy

W

x

yzBz

I

Hall Effect

,J E where is conductivityσ σ=

( )n pq n pσ µ µ= +

nn p q nσ µ⇒ =

If material is n-type

• Consider a p-type sample– E field is applied along x axis– B field is along z axis– The Lorentz force qvx×Bz exerts an average downward force on the holes– Downward directed current (of holes) causes a piling up of holes at the

bottom side of the sample– This gives rise to an electric field Ey– No net current in y-direction Ey exactly balances the Lorentz force


yy H x z

VE R J B

W= =

RH is the Hall coefficient

The carrier concentration and the carrier type (electron or hole) can be obtained from the Hall measurement, provided that one type of carrier dominates

Hall Effect


Recombination Processes

Whenever the thermal equilibrium condition of a physical system is disturbed pn≠ni

2, processes exist to restore the system to equilibrium pn=ni

2

Recombination: Transition of an electron from the conduction band to the valence band. It is made possible by:-The emission of a photon (radiative process) optical-Transfer of energy to another free electron or hole (Auger process)-Trapping energy in bandgap


Under low injection condition ((Dn, Dp)<< majority carriers) recombination process is given by:

n no

p

p pUτ−

=

U: recombination rateτn: minority carrier lifetimevth: carrier thermal velocityσp: conductivity for holesNt: trap density

For n-type, can show that:

1p

p th tv Nτ

σ=

The minority carrier lifetime is measured using the photoconductive effect

Recombination

For p-type, 1n

n th tv Nτ

σ=


Faraday’s Law of Induction

Ampère’s Law

Gauss’ Law for electric field

Gauss’ Law for magnetic field

Constitutive Relations

BEt

∂∇× = −

∂

H J∇× = Dt

∂+∂

D ρ∇⋅ =

0B∇⋅ =

B Hµ= D Eε=

MAXWELL’S EQUATIONS


Current Density Equations

n n nJ q nE qD nµ= + ∇

p p pJ q pE qD pµ= − ∇

cond n pJ J J= +

One dimensional

n n n nn kT nJ q nE qD q nEx q x

µ µ⎛ ⎞∂ ∂

= + = +⎜ ⎟∂ ∂⎝ ⎠

p p p pp kT pJ q pE qD q pEx q x

µ µ⎛ ⎞∂ ∂

= − = +⎜ ⎟∂ ∂⎝ ⎠


Continuity Equations

1n n n

n G U Jt q

∂= − + ∇ ⋅

∂

Gn and Gp are the electron and hole generation rates (cm-3/s) respectively caused by internal influence (optical, Auger). Un is the electron recombination rate in p-type semiconductor and can be approximated by (np-npo)/τn

1p p p

p G U Jt q

∂= − − ∇ ⋅

∂

One dimensional:

( ) 2

2p pop p p

n p n n nn

n nn n nEG n E Dt x x x

µ µτ−∂ ∂ ∂∂

= − + + +∂ ∂ ∂ ∂

( ) 2

2n non n n

p n p p pp

p pp p pEG p E Dt x x x

µ µτ−∂ ∂ ∂∂

= − − − +∂ ∂ ∂ ∂


Decay of Photoexcited Carriers

N-type sample illuminated by light uniformly generating electron-hole pairs at a rate G

0 constantτ∂= ⇒ = + =

∂n

n no pp p p Gt

The boundary conditions are:

( )τ−∂

= −∂

n non

p

p pp Gt

At steady state,

If at t=0, the light is turned off

(0)n no pp p Gτ= + ( )n nop t p→∞ =

0, 0∂= =

∂npE

x


( )τ−∂

= −∂

n non

p

p ppt

/: ( ) ptn no pand the solution is p t p Ge ττ −= +

Decay of Photoexcited Carriers

Decay of this photoconductivity can be observed in an oscilloscope and is a measure of the lifetime. (The pulse width must be much less than the lifetime)


Steady State Injection from One Side

High energy photons that create electron-hole pairs at the surface only

( ) 2

20τ−∂ ∂

= = − +∂ ∂

n non np

p

p pp pDt x

Boundary conditions are pn(x=0)=pn(0)=constant

[ ] /: ( ) (0) px Ln no n noSolution p x p p p e−= + −

,p p pL D the diffusion lengthτ=

n n nL D τ=


If the second boundary condition is changed so that all excess carriers at x=W are extracted

( )n nop W p=

[ ]sinh

( ) (0)sinh

pn no n no

p

W xL

p x p p pWL

⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠= + − ⎢ ⎥⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Current density at x=W is given by:

[ ] 1(0)sinh( / )

np p n no

W p p

DpJ qD q p px L W L∂

= = −∂

Steady State Injection from One Side


Transient & Steady-State Diffusion

Localized light pulses generate excess carriers

0 & 0dEGdx

= =

( ) 2

2n non n n

pp

p pp p pE Dt x x

µτ−∂ ∂ ∂

= − − +∂ ∂ ∂

If no field is applied, E=0

2

( , ) exp44n no

p pp

N x tp x t pD tD t τπ

⎛ ⎞= − − +⎜ ⎟⎜ ⎟

⎝ ⎠


Surface Recombination

The boundary condition at x=0 is:

[ ]0

(0)nn p n no

x

pqD qS p px =

∂= −

∂

Sp: Surface recombination velocity (cm/s)

( ) 2

2n non n

pp

p pp pG Dt xτ

−∂ ∂= − +

∂ ∂

exp( / )( ) 1 p p p

n no pp p p

S x Lp x p G

L Sτ

ττ

⎡ ⎤−= + −⎢ ⎥

+⎢ ⎥⎣ ⎦

0, ( )p n no pWhen S p x p Gτ→ → +

, ( ) 1 exp( / )p n no p pWhen S p x p G x Lτ ⎡ ⎤→∞ → + − −⎣ ⎦

surfacerecombination n-type

pn(0)τpG

pno

τp x0

pn(x)

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