the semiconductor in equilibrium

Chapter 4. The Semiconductor in Equilibrium

Charge Carriers in SemiconductorsDopant Atoms and Energy LevelsThe Extrinsic SemiconductorStatistics of Donors and AcceptorsCharge NeutralityPosition of Fermi Energy Level

Young-Hwan Leehttp://cafe.daum.net/lyh201circuitE-mail : [email protected] : 010-7178-1884

Hanyang University fall – 2007 2

4.1 Charge Carriers in SemiconductorsThermal equilibrium ; no external forces (electric fields, magnetic fields, or temperature gradients)Dopant atoms : the electrical properties of a semiconductor can be altered in desirable ways by adding controlled amounts of specific impurity atoms. Current density in semiconductorJ = (drift current density: Jdri) + (diffusion current density: Jdif)

Electrical properties (current) 중요: electron and hole 의band 구조Density of states function and Fermi-Dirac distribution function 중요

4.1.1 Equilibrium Distribution of Electrons and Holes


4.1 Charge Carriers in Semiconductors

)()()( EfEgEn Fc=

The distribution of holes as a function of energy in V.B

functiony probabilit Dirac-Fermi :)(EfF

band conduction in the states quantum ofdensity : )(Egc

The distribution of electrons as a function of energy in C.B.

[ ])(1)()( EfEgEp Fv −=band valencein the states quantum ofdensity : )(Egv

An ideal intrinsic semiconductor is a pure semiconductor with no impurity atoms and no lattice defects in the crystal.At T = 0 K,

all energy states in the valence band : filled with electronsall energy states in the conduction band : empty of electrons


4.1 Charge Carriers in Semiconductors

Density of states functions, Fermi-Dirac probability function, and areas representing electron and hole concentrations for the case when EF is near the midgap energy


4.1.2 The n0 and p0 EquationsIf E – EF >> kT, then Fermi-Dirac function을 Boltzmann approximation로

Nc: 전도대의 유효 상태 밀도함수(effective density of states function in the conduction band)

⎥⎦

⎤⎢⎣

⎡ −−=

TkEENn

B

Fcc

)(exp0

The thermal-equilibrium concentration of electrons in the conduction band

⎥⎦

⎤⎢⎣

⎡ −−≈

−+

=TkEE

TkEEEf

B

F

B

FF

)(exp)(exp1

1)(

2/3

2

*22 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

hTkmN Bn

cπ

dETkEEEE

hm

B

FE c

n

c⎥⎦

⎤⎢⎣

⎡ −−−≈ ∫

∞ )(exp)2(43

23*π∫= dEEfEgn Fc )()(0

kB : 1.380662×10–23 [J/K] Boltzmann constant


4.1.2 The n0 and p0 EquationsExample 4.1

Calculate the probability that a state in the conduction band is occupied by an electron and the thermal-equilibrium electron concentration in silicon at T = 300 K. Confer table 4.1. Ec – EF = 0.25 eV.

TkEE

TkEEEf

B

F

B

FcF

)]([exp)(exp1

1)( −−≈

−+

=

51043.60259.0

25.0exp)( −×=⎟⎠⎞

⎜⎝⎛ −=cF Ef

( )5190 1043.6)108.2()(exp −××=⎥⎦

⎤⎢⎣⎡ −−=

kTEENn Fc

c

3150 cm 108.1 −×=n


4.1.2 The n0 and p0 EquationsThermal-equilibrium concentration of holes in the valence band

⎥⎦⎤

⎢⎣⎡ −−=

kTEENp vF

v)(exp0

Nv: 가전자대의 유효 상태 밀도함수(effective density of states function in the valence band)

Table 4.1 Effective density of states function and effective mass values (at T =300 K)

0.370.556.0 x 10 181.04 x 10 19Ge0.480.0677.0 x 10 184.7 x 10 17GaAs0.561.081.04 x 10 192.8 x 10 19Simp

*/m0mn*/m0Nv [cm–3]Nc [cm–3]

2/3

2

*22 ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

hkTm

N pv

π

k : 1.380662×10–23 [J/K] Boltzmann constant


4.1.2 The n0 and p0 EquationsExample 4.2

Calculate the thermal-equilibrium hole concentration in silicon at T = 400 K. Fermi energy is 0.27 eV above the valence band energy. Confer table 4.1.

3192/3

19 cm 1060.1300400)1004.1( −×=⎟

⎠⎞

⎜⎝⎛×=vN

eV 03453.03004000259.0 =⎟

⎠⎞

⎜⎝⎛=kT

⎟⎠⎞

⎜⎝⎛ −

×=⎥⎦⎤

⎢⎣⎡ −−=

03453.027.0exp)106.1()(exp 19

0 kTEENp vF

v

3150 cm 1043.6 −×=p


4.1.3 Intrinsic Carrier ConcentrationFor an intrinsic semiconductor with no impurity atoms p0 = n0 = ni, p0n0 = ni

2

⎥⎦

⎤⎢⎣

⎡ −−==

TkEENnn

B

Ficci

)(exp0

⎥⎦

⎤⎢⎣

⎡ −−===

TkEENnpp

B

vFivii

)(exp0

⎥⎦

⎤⎢⎣

⎡ −−⋅⎥

⎦

⎤⎢⎣

⎡ −−=

TkEE

TkEENNn

B

vFi

B

Ficvci

)(exp)(exp2

⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡ −−=

TkE

NNTkEENNn

B

gvc

B

vcvci expexp2


4.1.3 Intrinsic Carrier Concentration

ni =2.4x1013 cm–3Geni =1.8x106 cm–3GaAsni =1.5x1010 cm–3Si

주어진반도체물질과온도에서 ni는상수(Fermi energy 와무관)

Table 4.2 Commonly accepted values of ni at T =300 K

The intrinsic carrier concentration of Ge, Si, and GaAs as a function of temperature.


Example 4.3Calculate the intrinsic carrier concentration in gallium arsenide at T = 300 K and T = 450 K. The bandgap energy of gallium arsenide is 1.42 eV and does not vary with temperature over this range. Confer table 4.1.

eV 03885.03004500259.0 =⎟

⎠⎞

⎜⎝⎛=kT

1218172 1009.50259.0

42.1exp)100.7)(107.4( ×=⎟⎠⎞

⎜⎝⎛ −××=in

36 cm 1026.2 −×=in

at T = 300 K

at T = 450 K

213

18172 1048.103885.0

42.1exp300450)100.7)(107.4( ×=⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛××=in

310 cm 1085.3 −×=in

4.1.3 Intrinsic Carrier Concentration


4.1.4 The Intrinsic Fermi-level Position

: Emidgap= (Ec+Ev)/2

⇐ Intrinsic Semi.

EFi는 density of states function이큰 band 쪽에서멀어진다.If mp

* = mn*, then EFi : the center of the bandgap

mp* < mn

*, then EFi : slightly below the center of the bandgapmp

* > mn*, then EFi : slightly above the center of the bandgap

⎥⎦⎤

⎢⎣⎡ −−

=⎥⎦⎤

⎢⎣⎡ −−

kTEEN

kTEEN vFi

vFic

c)(exp)(exp

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

c

vvcFi N

NkTEEE ln21)(

21

⎟⎟⎠

⎞⎜⎜⎝

⎛++= *

*

ln43)(

21

n

pvcFi m

mkTEEE

**when pn mm =**when pn mm ≠


4.2 Dopant Atoms and Energy LevelsPractical semiconductor devices는주로 specific dopant(or impurity) atoms을적당량첨가한 extrinsic(or impurity) semiconductor를사용하여제작.→impurity atoms<substitutional impurity>: dopant atoms

for Si, n-type: P, As, Sb etc. → group V elements (donor atom)p-type: B, Al, Ga, In etc. → group III elements (acceptor atom)

extrinsic semiconductor: impurity atoms의 ionization에의해carriers 발생.

intrinsic semiconductor: electrons (in valence band)이conduction band로 transition 하므로electron-hole pair 형태로 carriers 발생. p0 = n0

Carrier generation in semiconductor


4.2.1 Qualitative Descriptionfor n-type semiconductor (n0 > p0)

The intrinsic silicon lattice. The silicon lattice doped with a phosphorus atom.

The discrete donor energy state. The effect of a donor state being ionized.


4.2.1 Qualitative Descriptionfor p-type semiconductor (n0 < p0)

The ionization of the boron atom.

The discrete acceptor energy state. The effect of an acceptor state being ionized.

The silicon lattice doped with a boron atom.


4.2.2 Ionization Energy

근사적 Bohr Model을이용 donor 전자의반경계산

Bohr 반경 a0와의비교

for Si, ε = 11.7, m*/m0 = 0.26, r1/a0 = 45, r1 = 23.9 Å

결론: Donor 전자는 donor atom에 bound 되어있지않다.

nn rvm

πεre 2*

2

2

4=

oh A53.042

0

20

0 ==em

a πε

hnvrm n =*2*

22 4em

nrnπεh

=

⎟⎠⎞

⎜⎝⎛= *

02

0 mmn

ar

rn ε

ionization energy : extrinsic semiconductor에서 impurity atoms을ionization 하는데필요한 energy


4.2.2 Ionization Energy

Boron

Phosphrous

0.01020.06Aluminum

GeSi

0.01040.045Acceptors

0.01270.05Arsenic

0.0120.045Donors

Ionization energy [eV] Impurity

Table 4.3 Impurity ionization energies in silicon and germanium

Ionization Energy of hydrogen : E = – 13.6 eV, Si : –25.8 meV << Eg

VTE +=Total energy of orbital electron

Kinetic energy 22

4*2*

)4()(221

πεhnemvmT ==

Potential energy 22

4*2

)4()(4 πεπε hnem

reV

n

−=

−= 22

4*

)4()(2 πεhnemE −

=


4.2.3 Group III-V Semiconductors

0.0404Germanium0.0345Silicon

0.0307Zinc

0.0061Germanium

0.058Tellurium

Beryllium

Selenium

0.0347Cadium

0.028Acceptors

0.058Silicon

0.0059Donors

Ionization energy [eV] Impurity

Table 4.4 Impurity ionization energies in gallium arsenide

Amphoteric : GaAs에서 Si, Ge


4.3 The Extrinsic SemiconductorExtrinsic semi. : 특정량의 불순물을첨가하여 electron, hole 농도를intrinsic semi.에서의 농도와 다르게 만든 반도체. 한가지 운반자(carrier)가우세

For n-type semi. EF>EFi then n0 > ni, p0 < ni→ n0 > p0

For p-type semi. EF<EFi then p0 > ni, n0 < ni→ p0 > n0

⎥⎦⎤

⎢⎣⎡ −−=

kTEENn Fc

c)(exp0

⎥⎦

⎤⎢⎣

⎡ −−=

TkEENp

B

vFv

)(exp0

⎥⎦

⎤⎢⎣

⎡ −=

TkEEnn

B

FiFi exp0 ⎥

⎦

⎤⎢⎣

⎡ −−=

TkEEnp

B

FiFi

)(exp0


4.3.2 The n0p0 Product

The n0p0 product is always a constant for a given semiconductor material at a given temperature.The values of n0 and p0 are not necessarily equal.즉, extrinsic semiconductor에대해서도성립가정: The Boltzmann approximation is valid.

n0p0 product2

00 inpn =

⎥⎦⎤

⎢⎣⎡ −

=kT

EEnn FiFi exp0 ⎥

⎦

⎤⎢⎣

⎡ −−=

TkEEnp

B

FiFi

)(exp0

4.3.3 Fermi-Dirac Integral

⎥⎦

⎤⎢⎣

⎡−=

TkE

NNpnB

gvc exp00


4.3.4 Degenerate and Nondegenerage Semiconductors

at low impurity concentration : local energy states는 Eg내에형성, impurity atoms간의거리는크므로그들간의상호작용(간섭)이없고, carrier는Maxwell-Boltzmann statistics에따름.

at higher impurity concentration : impurity atom간의거리가작으므로상호작용무시할수없음.

electron의 wave functions이overlap. (neighboring impurity center의) local energy level은원위치에서상하로대칭적으로넓어져 impurity bands 형성.

→ impurity atoms의 ionization energy를감소시킴.


4.3.4 Degenerate and Nondegenerage Semiconductors

high doping 에의한 three major effects (degenerate semiconductors)(1) The Fermi level is located within the allowed bands(C.B. or V.B.) themselves. → impurity atoms의 ionization energy는감소하여insignificant하게됨.(2) The impurity states broaden into bands.(3) The intrinsic band gap is reduced. (so-called band-edge tailing)

impurity atoms간의상호작용무시할수없고(atomic spacing의감소때문), carriers는 Fermi-Dirac statistics에따름.


4.4 Statistics of Donors and Acceptors

Ni개의전자와 gi개의양자상태

½는전자의 spin 때문½이일반적으로는 1/gi로바뀜

nd는 donor level에있는전자의농도Nd

+는이온화된 donor의농도

pa 는 acceptor level에있는홀의농도Na

–는이온화된 acceptor 의농도

4.4.1 Probability Function

−−=⎟⎠⎞

⎜⎝⎛ −

+= aa

aF

aa NN

kTEE

g

Npexp11

⎟⎠⎞

⎜⎝⎛ −

+=

kTEE

NnFd

dd

exp211

+−= ddd NNn

−−= aaa NNp


4.4.2 Complete Ionization and Freeze-OutIf (Ed –EF) >>kBT, then the Boltzmann approximation is also valid.

⎥⎦⎤

⎢⎣⎡ −−=

kTEENn Fc

c)(exp0

⎥⎦⎤

⎢⎣⎡ −−

+=

+kT

EENNnn

ndc

d

cd

d

)(exp2

1

1

0

⎟⎠⎞

⎜⎝⎛ −−=

⎟⎠⎞

⎜⎝⎛ −

≈kT

EEN

kTEE

Nn Fdd

Fd

dd exp2

exp21

Ec – Ed : donor electron의 ionization energy

The ratio of electrons in the donor state to the total number of electrons in the conduction band plus donor state


4.4.2 Complete Ionization and Freeze-OutAt T = 300 K : Complete ionization

1016 cm–3 doping

At T = 0 K : Freeze-Out

0== ad pn

0or == +ddd NNn

0exp =⎟⎠⎞

⎜⎝⎛ −

kTEE Fd

⎟⎠⎞

⎜⎝⎛ −

+=

kTEE

NnFd

dd

exp211

0=−∞e∴ EF > Ed→ fig 4.13

Energy-band diagrams showing complete ionization (a) donor states and (b) acceptor states


4.4.2 Complete Ionization and Freeze-OutExample 4.7Determine the fraction of total electrons still in the donor states at T = 300 K. Phosphorus doping in silicon, for T = 300 K, at a concentration of Nd = 1016 cm–3.

⎟⎠⎞

⎜⎝⎛ −×

+=

+0259.0

045.0exp)10(2

108.21

1

16

190 d

d

nnn

%41.00041.0 ==


4.5 Charge Neutrality

Donor와 Acceptor impurity atoms 모두를가지고있음. Device 제작과정에자연스럽게생김.electron와 hole농도의평형

In thermal equilibrium, the semiconductor crystal is electrically neutral. The net charge density is zero. The charge-neutrality condition is used to determine the thermal-equilibrium electron and hole concentration as a function of theimpurity doping concentration.

An n-type compensated semiconductor ; Nd > Na

A p-type compensated semiconductor ; Na > Nd

A completely compensated semiconductor ; Na = Nd

4.5.1 Compensated Semiconductors


4.5.2 Equilibrium Electron and Hole Concentrations

n0 + Na– = p0 + Nd

+

n0 + (Na – pa) = p0 + (Nd – nd)

Complete ionization : nd = pa = 0n0 + Na = p0 + Nd


22

0 22 idada nNNNNp +⎟⎠⎞

⎜⎝⎛ −

+−

=

02

02

00 / nnpnpn ii =→=

22

0 22 iadad nNNNNn +⎟⎠⎞

⎜⎝⎛ −

+−

=

Charge Neutrality Condition

density of negative charge = density of positive charge (fig. 4.14)

n-type (Nd > Na) p-type (Nd < Na)



Determine the thermal equilibrium electron and hole concentration for a given doping concentration. Nd=1016 cm-3, Na = 0, ni = 1.5 x 1010 cm-3.

Example 4.9

n0 ≅1016 cm –3, p0 ≅ 2.25 x 104 cm –3

3416

210

0

2

0 cm 1025.210

)105.1( −×=×

==nnp i

22


⎜⎝⎛ −

+−

=

31621021616

cm 10)105.1(2

0102

010 −≈×+⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−=


EV

EC

EFi

Intrinsic Semi. Extrinsic Semi.

Ed

EV

EFi

EC


A few donor electrons annihilate some intrinsic holes.

Cf) Fig 4.16 Electron concentration versus temperature showing the three regions.


Example 4.10Calculate the thermal-equilibrium electron and hole concentration in a germanium sample for a given doping density. Nd= 5 x 1013 cm-3, Na = 0, ni = 2.4 x 1013 cm-3.

22


⎜⎝⎛ −

+−

=


n0 ≅ 6 x1013 cm-3, p0 ≅ 9.6 x 1012 cm-3

31213

213

0

2

0 cm 106.9106

)104.2( −×=××

==nnp i

31321321313

cm 106)104.2(2

01052

0105 −×≈×+⎟⎟⎠

⎞⎜⎜⎝

⎛ −×+

−×=


4.6 Position of Fermi Energy Level

for n-type semi. Nd >> ni , then n0≈ Nd

Compensated semi. : Nd⇒ Nd – Na

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

iBFiF n

nTkEE 0ln

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

d

CBFC N

NTkEE ln

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

0

lnnNTkEE C

BFC

The position of Fermi level as a function of the doping concentrations and as a function of temperature.4.6.1 Mathematical Derivation

⎥⎦⎤

⎢⎣⎡ −−=

kTEENn Fc

c)(exp0

⎥⎦

⎤⎢⎣

⎡ −=

TkEEnn

B

FiFi exp0


4.6 Position of Fermi Energy Level

Determine the required donor impurity concentration to obtain a specified Fermi energy. Silicon at T = 300 K contains an acceptor impurity concentration of Na=1016 cm–3. Silicon is n-type and the Fermi energy is 0.2 eV below the conduction band edge.

Example 4.13

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=−ad

CBFC NN

NTkEE ln

⎥⎦

⎤⎢⎣

⎡ −−=−

TkEENNN

B

FCCad

)(exp

31619 cm 1024.10259.0

2.0exp108.2 −×=⎥⎦⎤

⎢⎣⎡ −

×=− ad NN

31616 cm 1024.21024.1 −×=+×= ad NN


4.6.1 Mathematical DerivationFor a p-type semiconductor,

Na >> ni, then p0 ≈ Na

Compensated semiconductor : Na⇒ Na – Nd

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

iBFFi n

pTkEE 0ln

⎥⎦⎤

⎢⎣⎡ −−=

kTEENp vF

v)(exp0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=−

0

lnpNTkEE v

BvF

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

a

vBvF N

NTkEE ln

⎥⎦

⎤⎢⎣

⎡ −−=

TkEEnp

B

FiFi

)(exp0

n-type

EF

EV

EC

EFi

p-type

EF

EV

EC

EFi


4.6.2 Variation of EF with Doping Concentration & Temp.

ni (intrinsic carrier concentration) : strong function of temperatureEF (Fermi energy level): function of temperatureAt high temperature : EF ⇒ EFi불순물의영향이줄어들고, 점차 intrinsic semiconductor처럼행동.

At the very low temperature (Freeze-out): 양자역학적효과우세.

Boltzmann approximation is no longer valid.n-type : EF > Edp-type : EF < Ea



Position of Fermi level as a function of temperature for various doping concentrations.

Position of Fermi level as a function of donor concentration (n type) and acceptor concentration (p type).



Example 4.14

Table 4.3 Boron의 ionization energy : 0.045 eV

Si의 EFi가 band gap 의중간이라고가정

식 4.68 에서 EFi – EF = Eg/2 + Ev – EF = Eg/2 – (Ea – Ev) – (EF –Ea)

= 1.12 eV/2 – 0.045 eV – 3*(1/0.0259 eV)

= kBT ln (Na/ni) = 0.437 eV

Na = ni exp (0.437/0.0259)

= 1.5 x 1010 exp (0.437/0.0259) ≈ 3.2 x 1017 cm-3

Determine the Fermi level and the maximum doping at which the Boltzmann approximation is still valid. Consider p-type Si, at T = 300 K, doped with boron. 단, EF–Ea = 3kBT일때까지 Boltzmann분포가타당하다고하자.


4.6.3 Relevance of the Fermi Energy

두물질이접합했을때:In thermal-equilibrium, the Fermi energy level is a constant throughout a system.

(a) Electrons are more energetic in Mo, so they tunnel to the surface of Pt.

(b) Equilibrium is reached when the Fermi levels are lined up.When two metals are brought together, there is a contact potential ΔV.

the semiconductor in equilibrium

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