ENE 311 Lecture 5
Feb 05, 2016
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
Number of electrons per unit volume
• 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)
Number of electrons per unit volume
• 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
Number of electrons per unit volume
• From equation (5),
Number of electrons per unit volume
• 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
Number of electrons per unit volume
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”.
Number of electrons per unit volume
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
Intrinsic carrier concentration
• At room temperature, kT = 0.0259 eV and (E – EF) >> kT, so Fermi function -can be reduced to Maxwell Boltzman
n distribution.
Intrinsic carrier concentration
Intrinsic carrier concentration
Intrinsic carrier concentration
• 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.
Intrinsic carrier concentration
• Similarly, we can obtain the hole density p in the valence band as
(7)
= effective density of states in the valence band
Intrinsic carrier concentration
(a) Schematic band diagram. (b) Density of states. (c)Fermi distribution function.(d) Carrier concentration
Intrinsic 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
Intrinsic carrier concentration
• From (8);
Intrinsic carrier concentration
• The Fermi level of an intrinsic semiconducto r can be found by equating (6) = (7) as
Intrinsic carrier concentration
Intrinsic carrier concentration
Ex. Calculate effective density of states NC andNV for GaAs at room temperature if GaAs ha s and .
Intrinsic carrier concentration
Soln
We clearly see that the only difference betwee n NC and NV is the values of effective electro n and hole mass.
Intrinsic carrier concentration
Soln
Intrinsic carrier concentration
Ex. From previous example, calculate intrinsic carrier density ni for GaAs at room temperat
ure where energy gap of GaAs is 1.4 eV.
Intrinsic carrier concentration
Soln
Intrinsic carrier concentration
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.
Donors and Acceptors
The figure shows schematic bond pictur - -es for n type and p type.
Donors and Acceptors
• - 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.
Donors and Acceptors
• - 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.
Donors and Acceptors
• The ionization energy or binding energy, pro ducing a free charge carrier in semiconduct
or, can be approximately expressed by
Donors and Acceptors
Donors and Acceptors
Ex. Calculate approximate binding energy for donors in Ge, given that r = 16 and = 0.12
m0.
Donors and Acceptors
Soln
Donors and Acceptors
(a ) donor ions and (b ) acceptor ions.
Donors and Acceptors
• - 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
Donors and Acceptors
• - For a p type, the argument is similar. Theref ore, NA
- - or free hole density or ionized accep tor atom density is written as
Donors and Acceptors
We can obtain the Fermi level dependence on temperature for three cases:
• very low temperature
• intermediate temperature
• very high temperature.
Donors and Acceptors
1 . Very low temperature
Donors and Acceptors
Donors and Acceptors
2. Intermediate temperature
Donors and Acceptors
Donors and Acceptors
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 .
Donors and Acceptors
• From (6), we have
Donors and Acceptors
• - Similarly to p type, we have
• This is valid for both intrinsic and ex trinsic semiconductors under thermal equili
brium.
Donors and Acceptors
n-Type semiconductor. (a) Schematic band diagram.
(b) Density of states. (c) Fermi distribution function (d) Carrier concentration. Note
that np = ni
2.