Chp2. Carrier Modeling

013

2. Carrier Modeling

carriers : electric charge transporters

electrons and holes

2.1 Quantization Concept

“ Particle – in – a – box ” Model (electron)

The energy of the electron is not

continuous but quantized.

014

eV = 1.6 * 10-19 J

Si은 14개의 전자를 갖고 있으므로 좀 더 복잡하다.

+

proton n = 1 n = 2 n =

. . . . .

EN = - 13.6/n2 eV n = 1, 2, 3, …

015

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.2 Semiconductor Models

Bonding Model / Band Model

2.2.1 Bonding Model

4 nearest neighbors

Figure 2.3 The bonding model

Line represents a shared valence electron

Circle represents the core of a semiconductor (e.g. Si) atom

016

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.2.2 Energy Band Model

Energy – related aspects of events

( Bonding Model은 실제 공갂에서 일어나는 일을 예상)

Figure 2.5 Conceptual development of the energy band model starting with N isolated Si atoms on the top left and concluding with a “dressed-up” version of the energy band model on the top right.

017

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.2.3 Carriers (운반자)

전류의 흐름을 맡는 졲재

Figure 2.7 Visualization of carriers using the bonding model (left) and the energy band model (right).

(a) No-carrier situation; (b) visualization of an electron; (c) visualization of a hole.

(a) No Carriers

(b) The electron

(c) The hole

018

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.2.4 Band Gap and Material Classification

Eg의 크기에 의해서 순수한 상태의 전기적

특성이 결정된다.

부도체 SiO2 Eg 8 eV

Diamond Eg 5 eV

금 속 very small band gap or no band gap

반도체 Eg = 1.12 eV (Si)

0.67 eV (Ge)

1.42 eV (GaAs)

019

Figure 2.8 Explanation of the distinction between (a) insulators, (b) semiconductors, and (c) metals using the energy band model

020

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.3 Carrier Properties

2.3.1 Charge

electrons and holes have same magnitude of

the carrier charge q = 1.6 * 10-19 Coul.

2.3.2 Effective Mass m*

different for electrons and holes

different in different materials

021

Figure 2.9 An electron moving in response to an applied electric field

(a) within a vacuum, and (b) within a semiconductor crystal.

within a vacuum within a semiconductor

022

dt

vdmqF

0

dt

vdmqF e

*

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.3.3 Carrier Numbers in Intrinsic Materials

“ intrinsic semiconductor ” = pure semiconductor

물질 고유의 성질 (ex. 전기전도도)을 보유하고 있다.

Let’s define. n = number of electrons, cm-3

p = number of holes, cm-3

For intrinsic semiconductors at equilibrium

n = p = ni

intrinsic semiconductor의 정의

023

For Si ni 1010cm-3, For Ge ni 1013cm-3 (암기할 것)

2.3.4 Manipulation of Carrier Numbers – Doping

Figure 2.10 Visualization of (a) donor and (b) acceptor action using the bonding model. In (a) the Column Ⅴ element P is substituted for a Si atom; in (b) the Column Ⅲ element B is substituted for a Si atom.

Doping : 특정한 불순물을 첨가하여 반도체의 전기전도도

를 제어하는 조작. 이 때 사용되는 불순물을 dopant

(impurity)라고 부른다.

024

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

Figure 2.11 Pseudo-hydrogen atom model for the donor-site bond

Dopant (impurity)는 일반적으로 matrix atom (e.g. Si)

보다 가전자 수가 하나 많거나 (e.g. Ⅴ족 원소) 하나 작은

원소 (e.g. Ⅲ족 원소)를 선택한다.

Si내에 Ⅴ족 원소를 첨가하였을 때의 그림이 Fig2.11에

나타나있다. 이는 수소원자와 비슷한 형상이므로, 최외각

전자가 취할 수 있는 에너지도 수소원자 모델로부터 구할

수 있다.

025

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

(보통 반도체의 ε r 는 거의 10)

EB의 의미?

Donor and Acceptor ?

대체로 donor level은 CB 가까이

acceptor level은 VB 가까이 위치한다.

정의 : negatively charged when ionized - acceptor

positively charged when ionized - donor

026

eVeVKm

mE

S

eB 1.06.13

12

0

*

실제 energy level은 어떻게 측정하는가 ? 온도를 올리면서 conductivity 측정

Figure 2.13 Visualization of (a) donor and (b) acceptor action using the energy band model.

027

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

2.3.5 Carrier Related Terminology

• Dopants ( Dopant impurities )

• Intrinsic semiconductor

• Extrinsic semiconductor

• Donor

• Acceptor

• n-type materials

• p-type materials

• Majority carriers

• Minority carriers

028

2.4 States and Carrier Distributions

2.4.1 Density of States

distribution of energy states density of states

QM calculations

gc(E)dE : number of conduction band states/cm3 between E and E+dE (E ≧ Ec)

gv(E)dE : number of valence band states/cm3 between E and E+dE (E ≦ Ev)

029

C32

**

EE 2

Cee

c

EEmmEg

V32

**

EE 2

EEmmEg

Vpp

c

2.4.2 Fermi Function ( Fermi-Dirac Distribution Function)

전자의 state가 전자로 채워져 있을 확률 ( at thermal equilibrium )

EF : Fermi level

Figure 2.14 General energy dependence of gc(E) and gv(E) near the band edges.gc(E) and gv(E) are the density of states in the conduction and valence bands, respectively.

030

kTEE FeEf

/1

1

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

정의 : ① f(EF) = 0.5 ② Electrochemical potential energy of electrons

When T > 0K

⑴ E = EF f(EF) = 1/2

⑵ E (EF + 3kT) states mostly empty

Boltzmann Distribution

Figure 6.1 (a) The Fermi-Dirac distribution function for a Fermi energy of 2.5 eV and for temperatures of0K, 600K, and 6000K. (b) The classical Maxwell-Boltzmann distribution function of energies for the same temperatures.

031

kTEE FeEf/

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

⑶ E (EF – 3kT) states mostly filled

exp [(E- EF)/kT] << 1

f(E) 1 – exp [(E- EF)/kT]

1 – f(E) exp [(E- EF)/kT]

exp [-(EF-E)/kT]

⑷ At R.T. T=300ok

kT = 0.026 eV

3kT = 0.078 eV << Eg

Fermi level equilibrium에서 적용

032

2.4.3 Equilibrium Distribution of Carriers

g(E)*f(E) Available states * Occupation Probability

f(E)*gC(E):electrons in CB, [1-f(E)]*gV(E):holes in VB

Figure 7.17 (a) The two highest bands at T=0K. (b) The two highest bands at T>>0K. There are electrons at the bottom of the conduction band, and holes at the top of the valence band.

033

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

교과서 p47 Fig. 2.16 참조

① electron과 hole이 band edge에 가까이 분포되어 있다.

② EF의 위치에 따라 CB내의 전자 분포와 VB내의 hole

분포가 결정된다. majority carriers, minority carriers

③Thermal equil. 상태에서는 n이 증가하면 p는 감소한다.

Figure 2.18 “At a glance” representation of intrinsic (left), n-type (middle), and p-type (right)Semiconductor materials using energy band diagram.

034

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company

Figure 4.8 Density of states function, Fermi-Dirac Probability function, and areas representing electron and hole concentrations for the case when EF is above the midgap energy.

Figure 4.9 Density of states function, Fermi-Dirac Probability function, and areas representing electron and hole concentrations for the case when EF is below the midgap energy.

035

Figure reference: “Semiconductor Device Fundamentals”

Robert F. Pierret, Addison-Wesley Publiching Company