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Chapter 13 Semiconductor Statistics
(半導體統計)
1. Energy Bands; Fermi Levels; Electrons and Holes
(能帶, 費米位, 電子和電洞)
A semiconductor is a system with electron
orbitals grouped into two energy bands
separated by an energy gap
In a pure semiconductor (純半導體) at T = 0 K
all valence band (價帶) otbitals are occupied
and all conduction band (傳導帶) orbitals are
empty, and is an insulator (絕緣體)
For T > 0 K, finite electrical conductivity follows
from presence of conduction electrons (傳導電子)
in conduction band and from unoccupied orbitals
(hole 電洞) in valence band
Denote conduction band edge (傳導帶端) by ec and valence band edge (價帶端) by ev
the difference is the energy gap (能隙)
eg = ec - ev
(13-1)(5/14)
(ex) Silicon semiconductor Si (矽半導體)
fcc diamond structure (鑽石結構), a = 0.543 nm
Si at (0, 0, 0) and (a/4, a/4, a/4)
covalent bond (共價鍵): 3s3p3 valence bands (價帶)
first Brillouin zone (FBZ) of fcc in k-space with
primitive axis vectors: a* = (2p/a)(-x + y + z),
b* = (2p/a)(x – y + z), c* = (2p/a)(x + y – z)
Band structure (能帶結構) e(k) (k-dependent)
indirect energy gap (間接能隙)
eg = ec(X) – ev(G) = 1.14 eV (at 300 K)
valence band edge ev at G point (k=0, 0, 0)
conduction band edge ec at X point
(k=2p/a, 0, 0)
Si ingot
(13-2)
Si wafer
If a pure semiconductor (純半導體) is electrical neutral, concentration of conduction
electrons (傳導電子濃度) ne is equal to the concentration of holes (電洞濃度) nh
ne = nh
For impurity doping (加雜質), let nd+ the concentration of positively charges donors
and na- the concentration of negatively charged acceptors with net ionized donor
concentration (淨游離施子濃度) Dn  nd+ - naelectrical neutrality condition is then (see next section)
ne – nh = Dn = nd+ - naElectron concentration is calculated from the Fermi-Dirac distribution function
fe(e) = 1/[exp(e – m)/t +1]
where chemical potential m is called Fermi level (費米位) in solid state physics
concentration of conduction electrons in conduction band (傳導帶)
ne = Ne/V
= [SCB fe(e)]/V
(e ≥ ec)
concentration of holes in valence band (價帶)
nh = Nh/V
= [SVB (1 – fe(e))]/V
= [SVB fh(e)]/V
(e ≤ ev)
with the probability that an orbital e is unoccupied (“occupied by a hole”)
fh(e)  1 – fe(e) = 1/[exp(m – e)/t + 1]
(13-3)(5/19)
(Classical regime)(古典區域)
For a nondegenerate semiconductor (非簡併半導體) with electron concentrations ne
in the classical regime fe << 1 if exp[(e – m)/t] >> 1 for all e ≥ ec, or
exp[-(ec – m)/t] << 1
then fe reduced to classical Boltzmann distribution function
fe(e) ~ exp[-(e – m)/t] (<< 1)
Hole concentrations nh in the classical
regime fh << 1 if exp[-(m – e)/t] >> 1
for all e ≤ ev, or
exp[-(m – ev)/t] << 1
then fh reduced to classical distribution
function
fh(e) ~ exp[-(m – e)/t] (<< 1)
Fermi level m lies inside the energy gap
and is separated from both band edges
(ec – m) > t
(m - ev) > t
(13-4)
conduction electron concentration ne in classical regime
ne = SCB exp[-(e – m)/t]/V
= exp[-(ec – m)/t].{SCB exp[-(e – ec)/t]}/V (e  e – ec: relative to edge ec)
= exp[-(ec – m)/t].nc
Near the conduction band edge ec, electrons behaves like a free electron with
effective mass (有效質量) me*
e(k) – ec = ħ2k2/2me* (see Si and GaAs band structure e(k))
For free classical particle with mass m and spin s = ½
n = l.nQ.Zint
(internal partition function Zint = 2s + 1 = 2 for spin ½)
= exp(m/t).nQ*
see Chap 6
nQ* = 2.nQ
= 2.(mt/2pħ2)3/2
Quantum concentration nc for free conduction electron with effective mass me*
nc = 2(me*t/2pħ2)3/2
conduction electron concentration in conduction band
ne = nc.exp[-(ec – m)/t]
(m  m – ec = -(ec – m): Fermi level relative to edge ec)
in classical regime ne << nc since exp[-(ec – m)/t] << 1
(13-5)
Similarly, hole concentration nh in classical regime
nh = SVB exp[-(m - e)/t]/V
= exp[-(m - ev)/t].{SVB exp[-(ev – e)/t]}/V (e  ev – e: relative to edge ev)
= exp[-(m - ev)/t].nv
Near the valence band edge ev, holes behaves
like a free particle with effective mass mh*
ev - e(k) = ħ2k2/2mh*
Quantum concentration nv for free hole
nv = 2(mh*t/2pħ2)3/2
hole concentration in valence band
nh = nv.exp[-(m - ev)/t] (m  ev - m)
in classical regime nh << nv
(ex) Si: me* = 1.06 m, mh* = 0.58 m (average)
nc = 2.7 x 1019 1/cm3, nv = 1.1 x 1019 1/cm3
(ex) GaAs (砷化鎵): crystal and band structure e(k)
Ga at (0,0,0) and As at (a/4,a/4,a/4)
me* = 0.07 m, mh* = 0.71 m (average) (Table 13.1)
direct energy gap (直接能隙) eg = 1.43 eV at G point
(13-6)
GaAs ingot
(Law of mass action)(質量作用定律)
In the classical regime, product nenh is independent of Fermi level m
nenh = ncnv.exp[-(ec – ev)/t]
= ncnv.exp(-eg/t)
where energy gap
eg  ec - ev
In a pure semiconductor (純半導體) ne = nh = ni
intrinsic carrier concentration (本性載體濃度)
ni = (ncnv)1/2.exp(-eg/2t)
For any concentration in the classical regime, mass action law of semiconductors
nenh = ni2
(Intrinsic Fermi level)(本性費米位)
For an intrinsic semiconductor (本性半導體) ne = ni
nc.exp[-(ec – m)/t] = (ncnv)1/2.exp(-eg/2t)
insert eg = ec - ev
exp(m/t) = (nc/nv)1/2.exp[(ec + ev)/2t]
Fermi level for intrinsic semiconductor
m(T) = (1/2)(ec + ev) + (kT/2).log(nc/nv)
(at T = 0 K, in middle of gap)
= (1/2)(ec + ev) + (3kT/4).log(mh*/me*) (position temperature dependent)
(13-7)
2. n-type and p-type Semiconductors (n型和p型半導體)
(Doners and acceptors)(施子和受子)
Semiconductors used in devices usually have impurities intentionally added
Donor (D) impurities in Si: increase
conduction electron concentration
e.g. P(3s23p3), As(4s24p3)
Acceptor (A) impurities in Si: increase hole
concentration
e.g. B(2s22p), Al(3s23p), Ga(4s24p)
n-type semiconductor: more conduction
electrons than holes (ne > nh)
p-type semiconductor: more holes
than conduction electrons (nh > ne)
Approximation of fully ionized impurities
D  D+ + (-e) (nd = nd+)
A + (-e)  A- (na = na-)
(13-8)
The electrical neutrality condition (電中性條件)
Dn = ne – nh = nd+ - nause mass action law ni2 = nenh = ne(ne – Dn)
quadratic equation for ne
ne2 – Dn.ne – ni2 = 0
positive root gives conduction electrons concentration
ne = (1/2){[(Dn)2 + 4ni2]1/2 + Dn}
with nh = ne – Dn, hole concentration
nh = (1/2){[(Dn)2 + 4ni2]1/2 - Dn}
extrinsic semiconductor (外性半導體): doping concentration is larger than intrinsic
concentration lDnl >> ni
[(Dn)2 + 4ni2]1/2 ~ lDnl + 2ni2/lDnl (extrinsic limit 外性極限)
in a n-type semiconductor Dn > 0 (ne > nh), in the extrinsic limit
ne ~ Dn + ni2/Dn ~ Dn
nh ~ ni2/Dn << ni
in a p-type semiconductor Dn < 0 (ne < nh)
ne ~ ni2/lDnl << ni
nh ~ lDnl + ni2/lDnl ~ lDnl
(13-9)
(Fermi level in extrinsic semiconductor)(外性半導體之費米位)
Fermi level for extrinsic, nondegenerate semiconductor is obtained from
ne = nc.exp[-(ec – m)/t] or nh = nv.exp[-(m – ev)/t]
m = ec + t.log(ne/nc)
= ev – t.log(nh/nv)
(ex) Fermi level in Si as a function
of temperature T for various doping
concentration (Dn = 1012-1018 1/cm3)
Dn >> ni ~ 5 x 109 1/cm3
quantum concentration
nc = 2.7 x 1019 1/cm3
nv = 1.1 x 1019 1/cm3
(note) small decrease of energy gap
with temperature
eg(T) = ec - ev = 1.14 eV at 300 K
Fermi level for extrinsic semiconductor approaches band edge with decreasing
temperature T or with increasing doping concentration Dn
(13-10)
(Degenerate semiconductors)(簡併半導體)
When doping concentration Dn is increased and approaches the quantum
concentration, classical distribution function is no longer valid, need to use
Fermi-Dirac distribution function
Number of electrons in a Fermi gas (see Chap 7)
N = de.D(e)f(e)
= (V/2p2)(2m/ħ2)3/2.de.e1/2.f(e)
For conduction electrons in conduction band of semiconductor, replace
1. N by neV
2. electron mass m by effective mass m*
3. energy e by e - ec
obtain conduction electron concentration
ne = (1/2p2)(2me*/ħ2)3/2 .ecde.(e – ec)1/2/[1 + exp(e - m)/t)]
let x  (e – ec)/t, h  (m – ec)/t and use the quantum concentration nc = 2(me*t/2pħ2)3/2
ne/nc = (2/p1/2)0 dx.x1/2/[1 + exp(x – h)]
= I(h) (Fermi-Dirac integral)
or ne = nc.I[(m – ec)/t] for degenerate semiconductor
(13-11)(5/21)
Exact relation r = ne/nc = I(h) plotted
with h = (m – ec)/t near or above
conduction band edge ec
1. for ne << nc, m < ec (h < 0)
Fermi level m inside energy gap
since (ec – m)/t = -h >> 1, exp(x – h) >> 1
ne/nc ~ (2/p1/2).eh.dx.x1/2.exp(-x)
= (2/p1/2).eh.G(3/2)
= exp[-(ec - m)/t]
nondegenerate semiconductor
(非簡併半導體) with h = log r
2. for ne > nc, m > ec (h > 0)
Fermi level m in conduction band
degenerate semiconductor (簡併半導體)
in metallic range
use Joyce-Dixon approximation
h ~ log r + (1/81/2)r – (3/16 - 31/2/9)r2
with modified mass action law
nenh = ni2.exp[-ne/(81/2nc) + …]
(13-12)
(Impurity levels)(雜質能位)
The lowest orbital of an electron bound to
a donor corresponding to an impurity level
with ionization energy (游離能) Ded
ed = ec - Ded
(ex) P donor in Si, Ded = 45 meV
The impurity level of an acceptor with
ionization energy Dea
ea = ev + Ded
(ex) B acceptor in Si, Dea = 57 meV
(Occupation of donor and acceptor levels)(施子與受子能位佔據率)
Grand partition function for a donor level
Z = exp[(0.m – 0)/t] + exp{[1.m - ed(↑)]/t} + exp{[1.m - ed(↓)]/t}
= 1 + 2.exp[(m – ed)/t]
probability of ionized donor (donor orbital empty (electron in conduction band):
N = 0, e = 0)
f(D+) = 1/Z
probability of neutral donor (donor orbital occupied by one electron: N = 1, e = ed)
f(D) = exp[(m – ed)/t]/Z = 1 – f(D+)
(13-13)(5/24)
Grand partition function for an acceptor level
Z = exp{[0.m - 0(↑)]/t} + exp{[0.m - 0(↓)]/t} + exp[(1.m – ea)/t]
= 2 + exp[(m – ea)/t]
probability of ionized acceptor (acceptor orbital occupied by one electron (hole in
valence band): N = 1, e = ea, spin up or down)
f(A-) = exp[(m – ea)/t]/Z
= 1/{2.exp[(ea – m)/t] + 1}
probability of neutral acceptor (acceptor orbital empty: N = 0, e = 0)
f(A) = 2/Z = 1 – f(A-)
Difference of concentration D+ and ADn = nd+ - na= nd.f(D+) + na.f(A-)
= nd/{1 + 2.exp[(m – ed)/t]} + na/{1 + 2.exp[(ea – m)/t]}
rewrite neutrality condition
n-  ne + na= nh + nd+
 n+
(13-14)
(ex) a n-type Si at 300 K containing
both donors (nd) and acceptors (na)
Logarithmic plot of n+(m) and n-(m) as
functions of position of Fermi level m
actual Fermi level m ~ ed where total
positive charges equal total negative
charges n+(m) = n-(m)
for n-type semiconductot with acceptor
na << nd << nc
ne ~ Dn
~ nd+
= nd/{1 + 2.exp[(m – ed)/t]}
= nd/{1 + 2.(ne/nc).exp[(ec – ed)/t]}
= nd/[1 + 2.(ne/ne*)]
with ne*  nc.exp(-Ded/t) is the electron
concentration when m = ed
positive solution
ne = (ne*/4){[1 + (8nd/ne*)]1/2 – 1}
(13-15)
3. p-n Junctions (p-n接面)
A p-n junction is made from single crystal
semiconductor modified in two separated
regions
(a) doping distribution
n-region (left): donor impurity atoms nd
major carriers: electrons
p-region (right): acceptor impurity atoms na
major carriers: holes
two sides are in thermal equilibrium
(T = constant) and in diffusive equilibrium
(Fermi level m = constant everywhere)
(b) build-in electrostatic potential (內建靜電位)
due to space charge near junction Vbi = Vn + Vp
(c) energy bands shifted by eVbi = ecp – ecn
(d) space charge dipole (空間電荷電雙極)
inner electric field E required to generate Vbi
(13-16)
(n-doped)
(p-doped)
Assume doping concentrations in the extrinsic but nondegenerate range
(n-side) ni << nd << nc
(p-side) ni << na << nv
and assume fully ionized doping
ne ~ nd; nh ~ na
conduction band edge energies (ec) (傳導帶端能量)
(n-side) ecn = m – t.log(nd/nc)
(m = ec + t.log(ne/nc) = constant everywhere)
(p-side) ecp = m – t.log(nep/nc)
(two sides in diffusive equilibrium)
= m – t.log(ni2/nanc) (p-side: ni2 = nepnh ~ nepna)
energy band shift (能帶移)
eVbi = ecp – ecn
= t.log(nand/ni2)
(use ni = (ncnv)1/2.exp(-eg/2t))
= eg – t.log(ncnv/ndna)
(ex) for nd ~ 0.01 nc, na ~ 0.01 nv
eVbi ~ eg – 9.2kT = 0.91 eV for Si (eg = 1.14 eV) at 300 K (kT = 25 meV)
The electrostatic potential f(x) must satisfy the Poisson equation
d2f(x)/dx2 = -r(x)/e (SI)
where r(x) is the space charge density (空間電荷密度) and e the permittivity of
semiconductor
(13-17)
In the vicinity of junction (x = 0), charge carriers no longer neutralize the impurities
space charge density must be positive on n-side and negative on p-side
(n-side: x < 0)
Fermi level m = ec(x)+ t.log[ne(x)/nc] = constant at diffusion equilibrium
let potential origin f(-∞) = 0 (f(x) < 0)
then position variation of conduction band edge
ec(x) = ec(-∞) – ef(x)
(> ec(-∞))
position variation of conduction electron concentration
ne(x) = nc.exp[-(ec(x) – m)/t]
= nd.exp[ef(x)/t] < nd
(donor concentration nd = nc.exp[-(ec(-∞) – m)/t])
Poisson equation
d2f/dx2 = -r(x)/e
(donor space charge density r(x) = e[nd – ne(x)])
= -(e/e)[nd – ne(x)]
= -(e.nd/e)[1 – exp(ef/t)]
multiply by 2(df/dx)
2(df/dx)(d2f/dx2) = (d/dx)[(df/dx)2]
= -(2e.nd/e).(d/dx)[f – (t/e).exp(ef/t)]
integrate with condition f(-∞) = 0
(df/dx)2 = -(2e.nd/e)[f + t/e – (t/e).exp(ef/t)] (df/dx on n-side)
(13-18)
at the interface x = 0, assume -f(0) = Vn >> t/e
x-component electric field at interface x = 0 (介面電場)
E = -df/dx
= [(2e.nd/e)(Vn – t/e)]1/2
(for n-side, x  -0)
1/2
= [(2e.na/e)(Vp – t/e)]
(for p-side, x  +0)
1/2
= {[2e.na.nd/e(na+nd)](Vbi – 2t/e)}
(Vbi = Vn + Vp)
p-doped
on n-side the electric field is the same as if all
electrons had been depleted from the junction
to a distance
wn = eE/end
no depletion for lxl > wn
on p-side
wp = eE/ena
total depletion width (總削弱寬度)
w = wn + w p
= (eE/end) + (eE/ena)
= 2(Vbi – 2t/e)/E
(ex) for na = nd = 1015 1/cm3, e = 10 e0,
Vbi – 2t/e = 1 V, E = 4.25 x102 V/m, w = 470 nm
(13-19)
n-doped
(Reversed-biased abrupt p-n junction)(反向偏壓斷裂p-n接面)
reversed biased (反向偏壓): voltage V applied
with p-side at negative voltage
drives conduction electrons from p-side to n-side
dives holes from n-side to p-side (left to right)
form abrupt junction: very little current flows
I~0
energy bands shifted by e(lVl +Vbi)
not in diffusive equilibrium (m not constant )
quasi-equilibrium state (似平衡態) with two
quasi-Fermi levels (似費米位)
mp – mn = elVl
electric field at interface (x = 0)
E = {[2e.nand/e(na + nd)][(lVl + Vbi - 2t/e]}1/2
junction thickness
w = 2(lVl + Vbi - 2t/e)/E
(13-20)
n-doped
p-doped
(I-V characteristics of a p-n junction diode)(p-n接面二極體I-V特徵)
forward-biased (前向偏壓) Vf: voltage V applied
with p-side at positive voltage, drives conduction
electrons from n-side to p-side, dives holes
from p-side to n-side
I-V curve for a p-n junction diode
(note) Si diode (see figure) and (Ga,Al)As diode are good
wide-range temperature sensors, use forward voltage Vf(T)
at constant current I (10 mA)
temperature range: 1.4 K – 500 K
(13-21)
4. Nonequilibrium Semiconductors (非平衡半導體)
(Quasi-Fermi levels)(似費米位)
Nonequilibrium concentration arise when a
forward-biased (前向偏壓) p-n junction injects
electrons into p-side and holes into n-side
Excess carriers eventually recombine with each other with recombination time
(再結合時間) 10-3-10-9 s > 10-12 s (time required for conduction electrons or holes
to reach thermal equilibrium at 300 K)
Orbital occupancy distributions of electrons and holes are very close to equilibrium
Fermi-Dirac distributions in each bands separately
but total number of holes is not in equilibrium with total number of electrons
two different quasi-Fermi levels (似費米位) mc and mv for two bands in the steady
state (穩定狀態) or quasi-equilibrium condition (似平衡條件)
fc(e,t) = 1/[1 + exp[(e – mc)/t]
(conduction band)
and
fv(e,t) = 1/[1 + exp[(e – mv)/t]
(valence band)
(13-22)
(Current flow: drift and diffusion)(電流:漂移和擴散) (see Chap 14)
Any conduction electron flow in a semiconductor must be caused by the positiondependent conduction band quasi-Fermi level mc(r)
in the linear region with weak gradient, total electrical current density (電子電流密度)
Je  mc
where Je = (-e)(electron flux density) [C/(m2.s)]
current density is proportional to the conduction electron concentration ne
Je = menemc
with proportionality constant me the electron mobility (電子移動率) [m3/(V.s)]
for ni << ne << nc (extrinsic but nondegenerate, ne = nc.exp[-(ec – mc)/t])
mc = ec + t.log(ne/nc)
Je = meneec + metne
gradient in the conduction band edge arises from a gradient in the electrostatic
potential and thus from electric field
ec = -ef = eE
introduce an electron diffusivity (電子擴散率) from Einstein relation (see Chap 14)
De = met/e
[m2/s]
then
Je = emeneE + eDene
(13-23)
For hole with hole current density (電洞電流密度)
Jh = (+e)(hole flux density) [C/(m2.s)]
current density is proportional to the hole concentration nh
Jh = mhnhmv
with proportionality constant mh the hole mobility (電洞移動率)
for ni << nh << nv (extrinsic but nondegenerate, nh = nv.exp[-(mh – ev)/t])
mh = ev - t.log(nh/nv)
Jh = mhnhev - mhtnh
gradient in the valence band edge arises from a gradient in the electrostatic
potential and thus from electric field
ev = -ef = eE
introduce a hole diffusivity (電洞擴散率) from Einstein relation (see Chap 14)
Dh = mht/e
then
Jh = +emhnhE – eDhnh
first term: by electric field E (same direction as electrons)
second terms: diffusion by concentration gradient
(opposite direction q = +e)
(13-24)
(example) Light-emitting diode (LED) (發光二極體)(1962)
When a diode is forward biased, electrons are able to recombine with holes within
the device, releasing the energy in the forms of photons (electroluminescence)
Need semiconductors with direct energy gap (直接能隙), ec and ev at same k
(note) indirect gap semiconductor creates both phonons (聲子) and photons
(ex) semiconductor materials
red LED (610-760 nm):
(Al,Ga)As, Ga(As,P). (Al,Ga)(In,P)
green LED (500-570 nm):
(In,Ga)N/GaN, GaP, (Al,Ga)P
blue LED (450-500 nm):
ZnSe, (In,Ga)N
RGB white LED
(13-25)
(example) Injection laser diode (注射雷射二極體)(1962)
When by electron injection the occupancy fe(ec) of the lowest conduction band
orbital becomes higher than the occupancy fe(ev) of the highest valence band orbital,
the population of inverted
laser light with radiation energy ec – ev = eg can then be amplified by stimulated
emission (LASER = Light Amplification by Stimulated Emission of Radiation)
condition for population inversion (數目反置) fc(ec) > fv(ev)
with quasi-Fermi distribution
mc – mv > ec – ev = eg
Need semiconductors with direct energy gap
(In,Ga)N
blue-violet laser (405 nm) (blue-ray disc)
(Al,Ga)As
green laser (532 nm) (laser pointer)
(Al,Ga)(In,P) red laser (635, 650, 670 nm)
(ex) micrograph of a laser diode taken from a
CD-ROM drive, chip displaying various thin-film
deposited layers on substrate:
n-type layer, p-type layer, metal layer and Au wire
(13-26)(6/2)
(ex) Doule-heterostructure (雙異質結構) injection laser (1970)
p-type AlAs/GaAs/n-type AlAs
population inversion for laser action:
apply a forward-biased voltage V
eV = mn – mp
> egl (active layer energy gap)
1. electrons flow from right into active
layer and form a degenerate electron
gas, potential barrier by the wide
energy gap egh on the p-side form a
quantum well (量子井) that prevents
electrons escaping to the left
2. holes flow from left into active layer,
but cannot escapes to right
Diode wafer provides electromagnetic cavity (電磁共振腔)
for laser with l = 838.3 nm (infrared)
(13-27)
Herbert Kroemer, Z. I. Alferov
2000 Nobel physics prize
“for developing semiconductor heterostructures
used in high-speed- and optical-electronics”
(ex) A Febry-Perot resonant cavity edge-emitting laser
(n-(Al,Ga)As/active region(p-GaAs)/p-(Al,Ga)As heterostructure on n-GaAs substrate)
(ex) bandedge diagram of a NpP
N-(Al,Ga)As/p-GaAs/P-(Al,Ga)As
heterostructure
(a) equilibrium bandstructure (Ef ≡ m)
(b) forward biased
(quasi-Fermi levels Efc ≡ mc, Efv ≡ mv)
(13-28)
(example) Solar cell (太陽電池)
A solar cell is a device that converts energy of sunlight directly
into electricity by the photovoltaic effect (光伏特效應)
(ex) polycrystalline Si (多晶矽) photovoltaic cells
laminated to backing materials in a module
(ex) basic structure of Si based
solar cell and its working
mechanism (current through
electron and hole diffusion)
(13-29)
(Report and Exercise 13) (due day: 6/2-6/9)
1. Intrinsic conductivity and minimum conductivity (本性電導率及最小電導率).
(modified TP Problem 13-2)
The electric conductivity (電導率) is
s = J/E = (Je + Jh)/E = e(neme + nhmh)
where me and mh are electron and hole mobilities (電子和電洞移動率).
(a) Find the net ionized impurity concentration Dn = nd+ - na- for which conductivity is
minimum smin.
(b) By what factor is it lower than the conductivity of an intrinsic semiconductor si?
(c) Give numerical values of ratio smin/si at 300 K for Si with me = 1350 cm2/Vs and
mh = 480 cm2/Vs.
(d) Calculate intrinsic conductivity si for Si at 300 K (use data in Table 13.1).
2. Resistivity and impurity concentration (電阻率和雜質濃度). (TP Problem 13-3)
For a doped Ge crystal with resistivity r  1/s = 20 W-cm, take me = 3900 cm2/Vs and
mh = 1900 cm2/Vs
(a) What is the net impurity concentration Dn if the crystal is n-type?
(b) What is lDnl if the crystal is p-type?
(13-30)
3. Mass action law for high electron concentration (高電子濃度之質量作用定律).
(TP Problem 13-4)
when ne is no longer small compared with nc, use the Joyce-Dixon approximation to
derive the modified mass action law
nenh = ni2.exp[-ne/(81/2nc) + …]
4. Injection laser (注射雷射). (TP Problem 13-9)
Use the Joyce-Dixon approximation to calculate at T = 300 K the electron-hole pair
concentration in GaAs that satisfies the inversion condition mc – mv > eg for laser
operation, assuming no ionized impurities.
(13-31)