sistivity

3.1 Calculate the donor binding energy for GaAs (e, = 13.2, m,
the approximate donor
PROBLEA.
0.067 m).
Calculate valuesfor-the Fermi function f(C) at 300 Kand plot vs. energy in eV
aein Fig.3-14. Choose EF= leV and make the calculated pointscloser to
gether near the Fermi level to obtain asmooth curve. Notice that
f(E) varies
ouite rapidly withina few kT of Ep. Show that the probability that a state AE
above Er is occupied is the same as the probability that the state AE below E
isempty.
3 An unknown semiconductor has E = 1.1eV and N, =N,. It is doped with
10! cm - donors, where the donor level is 0.2 eV below E. Given that E, is 0.25
eV below E, calculate n; and the concentration of electrons and holes in the
semiconductor at 300 K.

34 At room temperature, an unknown indirect band gap, intrinsic, cubic semicon

ductor has the following band structure: There are 6X minima along the (100)
directions If m(T) = 0.065mo, mX) = 0.30 m, (for each of the X minima)
the I mini
and m = 0.47mo, at what temperature is the number of electrons in
the
maand the X minimaequal if the T to X energy separation is 0.35 eV, and
bandgap is 1.7 eV (m = free electron mass)?
Chapter 3
conduction band decreases with
mass of electrons in a
3.5 Since the
effective
according to Eq.
(3-3), comment on the elec.
the band
creasing curvature of GaAs compared with
the indirect Xor L
mass in the I valley of
tron effective reflected in the
difference
3-10.) Howis this effective mass
valleys. (See Fig. Appendix III? From Fig.3-10.
mobilityfor GaAs and
GaP shown in
electron conductivity ofGaAs if
fl-valley elec-
expect to happen to the vallev
what would you suddenly promoted to the L
field were
trons drifting in an electric vs. 1000/
of Si from Eq. (3-23)and the plot of n,
gap
3.6 Calculate the band cannot be measured directly from a
semilogarith
The slope
(Fig.3-17). Hint:
points on the plot and take the natural los
from two
micplot; read the values
the solution.
arithm as needed for number ot
doped with 10° cm boron atoms anda certain
3.7 (a) ASisample is What is tha
shallow donors. The Fermi level
is 0.36 eV above E, at 300 K.
donor concentration N,?
acceptor atoms and a certain
number of
(b) ASi sample contains 10'° cm In 0.26 ev
shallow donors. The In acceptor
level is 0.16 eV above E,, and E, is
atoms are un-ionized (i.e., neutral)?
above E, at 300 K. How many (cm) In
Eqs. (3-15) and (3-19). If no = 10'6 cm
3.8 Show that Eq. (3-25) results from E,in Si at 300 K?
where is the Fermi level relative to
intrinsic level E, to the center of the
band
3.9 Derive an expression relating the
gap E/2. Calculate the displacement
of E, from E/2 for Si at 300K, assum
and holes are 1.lm, and 0.56m,
ing the effective mass values for electrons
respectively.
material; it is to be operated at 400 K.
3.10 A semiconductor device requires n-type application?
Would Si doped with 10!5 atoms/cm of arsenic be useful
in this
Could Ge doped with 10 cm antimony be used?
3.11 Anew semiconductor has N, = 10' cm, N, = 5X
10 cm andE =2e. It
electron, hole, and in
it is doped with 10 donors (fully ionized), calculate the diagràm.
trinsic carrier concentrations at 627°C. Sketch the simplified band
showing the position of Er.
occurs
3.12 (a) Show that the minimum conductivity of a semiconductor sample
when n = n V , . Hint: begin with Eg. (3-43) and apply Eq. (3-24).
(b) What is the expression for the minimum conductivity Omin' conductivity.
(c) Calculate omin for Si at 300K and compare with the intrinsic with
doped
3.13 (a) ASi bar 0.1 mlong and 100 um in cross-sectional area is applied.

10" cm phosphorus. Find the current at 300 K with 10 V

Repeat for a Si barl pm long. an
(b) How long does it take an average electron to drift l um in pure Si at

electric field of 100 V/cm? Repeat for 10Vicm. con-

electron
3.14. (a) ASisample is doped with 10" boron atoms/cm'.What is the
centration ng at 300 K? What is the resistivity? r e q u i r e m e n t s

(b) AGesample is doped with 3 X10!" Sb atonms/cm". Using the 300 K.

of space charge neutrality, calculate the electron concentration no at
 Energy
Bands and Charge Carriers in Semiconductors
113

Ror a Si conductor of length 5 um, dopedn-type at 10 cmcalculate the cur-

rent density for an applicdi voltage of 2.5V acrOss its length. How about for a volt-
age of200V? The clectron and hole nobilities are IS00 cm/Vs and 500cm'/Vs
resctivel, in the ohmic region for electric fields below 10 V/em. For higher
pclds electrons and holes have a saturation velocity of 10' cm/s
In along semiconductor bar (E: -2 eV). conduction band electrons come in
fromthe leftin the positiveer direction with a kinetic energy of 3eV. They move
fromlocation Ato BtoClo D. Between Aand B, the electric field is zero: be-
Iween locations B
andiC,
( there is a linearly varying voltage increase of 4
V; be-
twcen C and D, the field is again zero. Assuming no scattering, sketch a
simplified band diagram describingthe motion of these electrons. Assuming
that these electrons can be describedIas plane waves, with afree-electron mass.
ite down the wave function of the electrons at D. Leave your result in terms
afan arbitrary normalization constant.
Assume that a conduction electron in Si (e, = 1350cmN-s) has a thermal
oreyofKT, related to its mean thermal velocity by Erh =(miy,)/2. This
elestron is placed in an clectric fieldof 100 V/cm. Show that the drift velocity
of the electron in this case is smallcompared with its thermal velocity. Repeat
for afield of 10'V/cm. using the same value of . Comment on the actual mo
bilitv effects at this higher value of the field.
U8 U'se Eq. (3-45) to calculate and plot the mobility vs temperature u(T) from 10 K
to SO0 K for Sidoped with N, = 10*,10, and 10 donors cm.Consider the
mobility to he determined by impurity and phonon (lattice) scattering. Impu
riy scattering limited mobility can be described by
Hy =3.29 x 10!5
N; (m;/m,) in(! + ¿) -
where

:=3 x 10'e,T(m*/m,)(N;)l
Assume that the ionized impurity concentration N; is equal to N, at all tem
peraturcs.
Ihe conductivityeffective mass n÷ for Siis 0.26 mo. Acoustic phonon (lattice)
Cateting limitedmobility can be described by
Hac=1.18 x10 c,(m:/m) 'TE,)
where the stiffness (c) is given by
,=1.9 x 10 dyne cm for Si
dtheit conduction band acoustic deformation potential (EA) is

I-9.5 ¢V for Si
Rework Prob. 3.18 COnsiderng carrier freeze-out onto donors at low T. That
S. COnsIder

I+ exptE/kT)
 Chopter 3 Consider the donor ionization
114
as the ionized
impurity
concentration.
energy
meV for Si.
(E,)to be 45 p-type
semiconductor bar 500
um wide
are made on a and
3.20 Hall
measurements
Bare displaced 2 um with
Hall contacts A and respect to,
20 umthick. The of 3 mA.
mA. The voltage
each other in the
direction of current
flow
S wWb/cm' )
between A
field of 10kG (1kG = 10 pointing out of
IB with a magnetic field direction is reversed the
sample is 3.2 mV. When the magnetic
plane of the the hole concentration and
voltage changes to -2.8 mV. What is the hole mobility?
samplesuch as that shown in Fig. 3-25, it is difficult to
shown
3.21 In soldering wires to a
precisely. If Bis displaced slightly down the
align the Hall probes Aand B
erroneous Hall voltage results. Show that the true
length of the bar from A, an
obtained fromn two measurements of VAR. With the mag
Hall voltage Vy can be in the -z-direction.
neticfield first in the +z-direction and then
atoms/cm. What would voa.
3.22 A sample of Si is doped with 10 phosphorus
pect to measure for its resistivity? what Hall voltage would you expect in a
sample 100 um thick if I, = 1mA and B, = 1kG = 10 Wb/cm29