ENE 311

ENE 311 Lecture 5

The band theory of solids

• Band theories help explain the properties of materials.

• There are three popular models for band the ory:

- Kronig-Penney model

- Ziman model

- Feynman model

Kronig-Penney Model

• Band theory uses V 0 . The potential is peri odic in space due to the presence of immobi

le lattice ions.

Kronig-Penney modelIdeal

w

Kronig-Penney Model

• Ions are located at x 0= , a 2, a, and so on. The potential wells are separated from each other by barriers of height U0 and width w.

• - From time independent Schrödinger equation in1- dimension (x- only), we have

(1)

Kronig-Penney Model

• For this equation to have solution, the following must be satisfied

(2)

(3)

(4)

Kronig-Penney Model

Allowed band Forbidden band

Kronig-Penney Model

• We plot the right-hand side of (2) as a function of a and since the left-hand side of the same equation is always between -1 and +1, a solution exists only for the shaded region and no solution outside the shaded region.

• These regions are called “allowed and forbidden bands of energy” due to the relation between and E.

Kronig-Penney Model From equation (2), we have

• If P increases, allowed bands get narrower and the forbidden bands get wider.

• If P decreases, allowed bands get wider an d forbidden bands get narrower.

• If P = 0 , then cos( a) = cos(ka)

Kronig-Penney Model

• If P , then sin( a) = 0

• At the boundary of an allowed band cos(ka) = 1 , this implies k = n /a for n = 1 ,2 ,3 , …

Kronig-Penney Model - How to plot E k diagram

• Choose values between -1 to +1 , then find - argument of right hand side ( a) which sat

isfies chosen values.

• - Likewise to left hand side (ka).

a = (any number in radian)

Kronig-Penney Model

• ka = (any number in radian)

• - Plot E k diagram

Brillouin Zone

Reduced Brillouin Zone

Number of electrons per unit volume• The total number of electrons per unit volume i

n the range dE (between E and E + dE) is givenby

where N(E) = density of states (number of energy levels p

er energy range per unit volume)

F(E) = a distribution function that specifies expecta ncy of occupation of state or called “probability of o

ccupation”.

Number of electrons per unit volume

• The density of states per unit volume in thr ee dimensions can be expressed as


• The probability of occupancy is given by the Fermi-Dirac- distribution as

whereEF = Fermi energy level ( the energy at F(E) = 0.5)

k = Boltzmann’s constant

T = absolute temperature (K)


• For T =0 K:

If E > EF, F(E) =0 F(E) =1/(e +1 ) = 0

If E < EF, F(E) =1 F(E) =1 /( e- + 1) =

1

• For T >0 , F(EF ) = 0.5


• From equation (5),


• For T > 0

Fermi levels of various materials

Li 4.72 eV

Na 3.12 eV

K 2.14 eV

Cu 7.04 eV

Ag 5.51 eV

Al 11.70 eV


Characteristics of F(E)

1 . F(E), at E = EF , equals to 0.5.

2 . For (E – EF ) > 3 kT

This is called “Maxwell – Boltzmann distribution”.


Characteristics of F(E)

3. For (E – EF) < 3kT

4. 3F(E) may be distinguished into regions for T> 0 aa− E =0 to (E = EF – 2.2 aaaaa aa aaaaaa): () .− (E = EF – 2.2 kT) to (E = EF + 2.2 kT): F(E)

aaaaaaa aaaa aaaaaa 1 aa aaaaaa 0.− (E = EF + 2.2 kT) to E = aa aaaaa aa aa: ()

.

Intrinsic carrier concentration

• Free charge carrier density or the number of electrons per unit volume

• For electrons: E1/2 - = (E EC)1/2 and

• For holes: E1/2 = (EV - E)1/2 and


• At room temperature, kT = 0.0259 eV and (E – EF) >> kT, so Fermi function -can be reduced to Maxwell Boltzman

n distribution.




• Therefore, the electron density in the condu ction band at room temperature can be exp

ressed by

(6)

= effective density of states in the con duction band.


• Similarly, we can obtain the hole density p in the valence band as

(7)

= effective density of states in the valence band


(a) Schematic band diagram. (b) Density of states. (c)Fermi distribution function.(d) Carrier concentration


• Forintrinsicsemi conduct or s, t he number of el ect r ons peruni t vol ume i n t he conduct i on band equal s tot he number of hol es per uni t e vol ume i n t h eval ence band.

(8)

wher e n i = intrinsic carrier density

in p n

2. in p n


• From (8);


• The Fermi level of an intrinsic semiconducto r can be found by equating (6) = (7) as



Ex. Calculate effective density of states NC andNV for GaAs at room temperature if GaAs ha s and .


Soln

We clearly see that the only difference betwee n NC and NV is the values of effective electro n and hole mass.


Soln


Ex. From previous example, calculate intrinsic carrier density ni for GaAs at room temperat

ure where energy gap of GaAs is 1.4 eV.


Soln


We may have a conclusion that

• As EF EC , then n increases.

• As EF EV , then p increases.

• As T = 0 K, then EF is at Eg/2

• If EF > EC or EF < EV , then that semiconductor is said to be “degenerate”.

Donors and Acceptors

• When a semiconductor is doped with some i mpurities, it becomes an extrinsic semicon

ductor.

• Also, its energy levels are changed.


The figure shows schematic bond pictur - -es for n type and p type.


• - For n type, atoms from group V impurity rel ease electron for conduction as free charge

carrier.

• An electron belonging to the impurity atom clearly needs far less energy to become ava

ilable for conduction (or to be ionized).

• The impurity atom is called “a donor”.

• The donor ionization energy is EC – ED whereED is donor level energy.


• - For p type, atoms from group III capture ele ctron from semiconductor valence band and

produce hole as free charge carrier.

• EA is called “acceptor level” and EA – EV is c alled “acceptor ionization level energy”.

• This acceptor ionization level energy is smal l since an acceptor impurity can readily acc

ept an electron.


• The ionization energy or binding energy, pro ducing a free charge carrier in semiconduct

or, can be approximately expressed by



Ex. Calculate approximate binding energy for donors in Ge, given that r = 16 and = 0.12

m0.


Soln


(a ) donor ions and (b ) acceptor ions.


• - Consider an n type semiconductor, if ND is t he number of donor electrons at the energy

level ED , then we define to be the number of free electron carrier (number of N D that hav

e gone for conduction). or ionized donor ato m density can be written as


• - For a p type, the argument is similar. Theref ore, NA

- - or free hole density or ionized accep tor atom density is written as


We can obtain the Fermi level dependence on temperature for three cases:

• very low temperature

• intermediate temperature

• very high temperature.


1 . Very low temperature



2. Intermediate temperature



3. Very high temperature

• In this case, all donors are ionized and electr ons are excited from valence band to conduc tion band.

• This is acting like an intrinsic semiconductor or EF = Ei.

• It may be useful to express electron and hole densities in terms of intrinsic concentration ni and the intrinsic Fermi level Ei .


• From (6), we have


• - Similarly to p type, we have

• This is valid for both intrinsic and ex trinsic semiconductors under thermal equili

brium.


n-Type semiconductor. (a) Schematic band diagram.

(b) Density of states. (c) Fermi distribution function (d) Carrier concentration. Note

that np = ni

2.

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ENE 311