Session 4: Solid State Physics Dopants and Carrier Concentration

1 1. Introduction 2. Crystal 3. Cubic Lattices Outline 4. Other 5. Miller Indices

A  B  C  D  E
F  G
H
I
J

2 1. Introduction 2. Crystal 3. Cubic Lattices Intrinsic Material 4. Other 5. Miller Indices

Conduction band Free electron Intrinsic = pure 14P electron/hole pair Thermal is generated excitation hole 14P 14P 14P Valance band

= # of e-/cm 3 14P = # of h +/cm 3 = intrinsic carrier concentration

note that totally there are electrons 1.5 10 ° 2 10

1 out of bond is broken! 10

3 1. Introduction 2. Crystal 3. Cubic Lattices Intrinsic Material 4. Other 5. Miller Indices

10 22 Conduction band Ge Intrinsic = pure Si 10 18 GaAs Thermal excitation 10 14

10 Valance band 10 10 6

10 2 1 0 400 800 1200 1600

Energy Band Gap determines the intrinsic carrier concentration. ni EgGe< EgSi< EgGaAs

4 1. Introduction 2. Crystal 3. Cubic Lattices Doping 4. Other 5. Miller Indices Doping means mixing a pure semiconductor with impurities to increase its electrical conductivity Can be done in two ways:

Increasing the number of electrons by Increasing the number of holes by mixing pentavalent elements such as mixing trivalent elements such as phosphorous, arsenic, antimony aluminum, boron, gallium (means (means adding donor impurities) adding acceptor impurities)

Donors and acceptors are known as dopants. Dopant ionization energy ~50meV (very low).

Possible dopant deactivation & defect formation -- -- 14 -3 N or P : N D or N A < 10 cm - - 14 -3 16 -3 N or P : 10 cm < ND or N A < 10 cm 16 -3 18 -3 N or P : 10 cm < N D or N A < 10 cm + + 18 -3 20 -3 N or P : 10 cm < N D or N A < 10 cm ++ ++ 20 -3 N or P : N D or N A > 10 cm 5 1. Introduction 2. Crystal 3. Cubic Lattices Donor and Acceptor in the Band Model 4. Other 5. Miller Indices

Conduction band Donor ionization Donor level energy

Acceptor Acceptor level Valance band ionization energy

@ 300° 26 Ionization energy of selected donors and acceptors in silicon

Donors Acceptors Dopant Sb P As B Al In Ionization Eng Ec-Ed or 39 44 54 45 57 160 Ea-Ev (meV) 6 1. Introduction 2. Crystal 3. Cubic Lattices Donor and Acceptor in the Band Model 4. Other 5. Miller Indices

Si Ionization energies of shallow DONORS(eV) As = 0.054; P = 0.045; Sb = 0.043 Ionization energies of shallow ACCEPTORS(eV) Al = 0.072; B = 0.045; Ga = 0.074; In = 0.157

Ionization energy of various impurities in Ge, Si, and GaAs at 300K.

7 1. Introduction 2. Crystal 3. Cubic Lattices Density of States 4. Other 5. Miller Indices

= number of states per in the energy range between and near the band edges:

number of states in ∆ ( ) ≡ ∆ volume .

2

2 8 1. Introduction 2. Crystal 3. Cubic Lattices Thermal Equilibrium and Fermi Function 4. Other 5. Miller Indices An Analogy for Thermal Equilibrium

Sand particles

Dish

Vibrating Table

There is a certain probability for the electrons in the conduction band to occupy high-energy states under the agitation of thermal energy.

9 1. Introduction 2. Crystal 3. Cubic Lattices Probability of a State at E being Occupied 4. Other 5. Miller Indices •There are g1 states at E1, g2 states at E2… There are N electrons, which constantly shift ••• among all the states but the average electron 1 2 3 … gk energy is fixed at 3kT/2.

•There are many ways to distribute N among n1, n2, n3….and satisfy the 3kT/2 condition. ••• ••• •The equilibrium distribution is the 1 2 3 … g2 distribution that maximizes the number of ••• •• ••• combinations of placing n1 in g1 slots, n2 in g2 1 2 3 … g slots…. : 1

1 ⁄ 1 is a constant determined by the condition 10 1. Introduction 2. Crystal 3. Cubic Lattices Fermi Function 4. Other 5. Miller Indices

Particles can be classified into 3 categories: 1. Classical particles (ball) Maxwell-Boltzmann dist. 2. Bosons (photons) Bose-Einstein dist. 3. Fermions(undist + Pauli Exclution) (electrons) Fermi-Dirac dist. Probability that an available state at energy E is occupied:

1 Ef is called the Fermi energy or the Fermi level. There is only one Fermi level in a system at equilibrium. 1 ⁄ 1 Boltzmann constant, 0.9 0° k = 1.38 × 10 −23 J/K 0.8 100° k = 8.62×10 -5 eV/K 0.7 300° 0.6 500° 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 (ev) 11 1. Introduction Fermi Function 2. Crystal 3. Cubic Lattices 4. Other Probability of Electron Distribution 5. Miller Indices

1 Ef is called the Fermi energy or the Fermi level. 1 ⁄ If we are 3kT away from the Fermi energy then we might use Boltzmann approximation: if ⁄ ≫

⁄ if 1 ≪ 3 2 ⁄ 2 3

0.5 1 12 1. Introduction 2. Crystal 3. Cubic Lattices Electron / Hole Concentrations 4. Other 5. Miller Indices Derivation of n and p from D(E) and f(E) Integrate n(E) over all the energies in the conduction band to obtain n Top Of Conduction Band

2 ⁄

2 ⁄ ⁄ ⁄ Nc is called the effective density 2 ⁄ ≡ 2 of states of the conduction band.

1 Top Of Conduction Band ⁄ Nv is called the effective density ⁄ 2 of states of the valence band. ≡ 2 Closer Ef to Ec the larger n , closer Ef to Ev the larger p For Si: Nc =2.8×1019 cm-3 , Nv =1.04×1019 cm-3 13 1. Introduction 2. Crystal 3. Cubic Lattices Fermi Level and Carrier Concentration 4. Other 5. Miller Indices

⁄

ln

18 20 10 13 10 14 10 15 10 16 10 17 10 10 19 10 or

14 1. Introduction 2. Crystal 3. Cubic Lattices Carrier Concentrations 4. Other 5. Miller Indices

Multiply ⁄ and ⁄

⁄ ⁄

Law of Mass Action ⁄ ≡ In an intrinsic (undopped) semiconductor, ⁄ ⁄ 2 2 ≡ 2 ≡ 2 ⋯

≡ ∗ 3 → ln ln ∗ 2 2 2 4 2

⁄ ⁄ ⁄ ⁄ 15 1. Introduction 2. Crystal 3. Cubic Lattices Intrinsic Material 4. Other 5. Miller Indices 500 400 300 250 10 16

Ge 10 14 2.5 10 ⁄ ≡ Si 10 12

⁄ 2 GaAs 2 10 10 1.5 10

/ ∗ ∗ ⁄ 10 8

2 10 10 6 1000 / T K -1 2 3 4 16 1. Introduction 2. Crystal 3. Cubic Lattices Density of States 4. Other 5. Miller Indices . intrinsic

n-type

p-type

17 1. Introduction 2. Crystal 3. Cubic Lattices Density of States 4. Other 5. Miller Indices Donor

0° 50° 300°

Acceptor

0° 50° 300°

18 1. Introduction 2. Crystal 3. Cubic Lattices Carrier Concentration vs. Temperature 4. Other 5. Miller Indices At room temperature, all the shallow dopants are ionized. (Extrinsic region) 2 When the temperature is Freeze decreased sufficiently (~100 K), out some of the dopants are not Extrinsic region ionized. (Freeze out region) 1 Intrinsic region When the temperature is increased so high that the intrinsic carrier conc. approaches the active dopant conc. (T
Ti, > 450K for Si), the semiconductor is 100 200 300 400 500 said to enter the intrinsic region.

0

0° low moderate high 19 1. Introduction 2. Crystal 3. Cubic Lattices Fermi Level vs. Temperature 4. Other 5. Miller Indices

When the temperature is decreased, the Fermi level rises towards the donor level (N-type) and eventually gets above it.

When the temperature is increased, the Fermi level moves towards the intrinsic level.

20 1. Introduction 2. Crystal 3. Cubic Lattices Carrier Concentrations 4. Other 5. Miller Indices

Q: What is the hole concentration in an n-type semiconductor with of donors? 10 how much hole concentration will change if T increase by 60 Sol: 10

After increasing T by 60°C, n remains the same at while p increases by about a 10 factor of 2300 because / ∝

Q: What is n if in a p-type silicon wafer? 10 Sol: 10 10 10

21 1. Introduction 2. Crystal 3. Cubic Lattices Dopant Ionization 4. Other 5. Miller Indices

Consider a phosphorus-doped Si sample at 300K with . What fraction 10 of the donors are not ionized?

Sol: Suppose all of the donor atoms are ionized.

then ln 150 1 1 ⁄ ⁄ 1 1

2 4

Probability of non-ionization 1 ~ ⁄ 1 1 ⁄ 0.034 1

22 1. Introduction 2. Crystal 3. Cubic Lattices Most General Eq. 4. Other 5. Miller Indices

Charge neutrality

Mass law

1 1 ⁄ ⁄ 1 1

23 1. Introduction 2. Crystal 3. Cubic Lattices Nondegenerately Doped Semiconductor 4. Other 5. Miller Indices

Recall that the expressions for n and p were derived using the Boltzmann approximation, i.e. we assumed

3 3

Conduction band 3

in this range

Valance band 3

The semiconductor is said to be nondegenerately doped in this case.

24 1. Introduction 2. Crystal 3. Cubic Lattices Degenerately Doped Semiconductor 4. Other 5. Miller Indices

If a semiconductor is very heavily doped, the Boltzmann approximation is not valid.

In Si at if 300: 3 1.6 10 3 if 9.1 10 The semiconductor is said to be degenerately doped in this case.

Terminology:

“n+” degenerately n-type doped. ~ “p+” degenerately p-type doped. ~

25 1. Introduction 2. Crystal 3. Cubic Lattices Band Gap Narrowing 4. Other 5. Miller Indices

If the dopant concentration is a significant fraction of the silicon atomic density, the energy-band structure is perturbed
the band gap is reduced by : ∆

/ 300 ∆~3.5 10

10 → ∆~35 10 → ∆~75

26