Physics 3
1) What does Schrodinger equation describe? What interpretation and properties have the
wave functions being solutions to this equation?
The general form of the Schrodinger equation looks as follows:

    i

t



where   T  V ,

T - kinetic energy operator

V - potential energy operator

T 
 2 2
2m

V - depends on circumstances
So we may write the equation in such form:
V  
  2 

 i
2
2m x
t
The Schrodinger equation is equivalent to the Newton’s Law for the classical mechanics. It
describes the state of a material particle, its position and momentum but in an indeterministic
way. The solution of the Schrodinger equation looks as follows:
  Fei kx wt   Re - the wave function
 (r,t) takes complex waves , so it can not have any physical interpretation. That is why we
multiply this function by its complex conjugate and we obtain the density of probability:
 * r , t  r , t   
2
 p r , t  - density of probability of finding the particle of positron r and
time t.
That is why

wholespace
Pr , t d 3  1
- three dimensional case
The above formula is called the normalization condition. The wave function  must be
continuous everywhere (also one to one and out)  we can not have two waves of  at one
point in the space (of the same time); neither can we hace a discontinuity!
The Schrodinger’s equation plays the role of the Newton’s Laws and conservation of energy
in classical mechanics – i.e., it predicts the future behaviour of a dynamic system. It is a wave
equation in terms of the wave function, which predicts analitically and precisely the
probability of events outcome. The detailed outcome is not strictly determined, but given a
large number of events, the Schrodinger’s equation will predict the distribution of results. It
describes the state of a material particle. Its position and momentum but in an indeterministic
way.
According to quantum mechanics, behaviour of a particle can be predicted only with a given
probability. Solution to the Schrodinger equation do not give a position of a particle or its
momentum but gives a function of a propagating wave which determines probability of
finding a particle in a given place and time:
 ri t  ri t     pri t 
2
Were p ri t 
time dependent S equation
2 2

   r ri t   ih
2m
t
2 
2
2
2


, r r , t  - outer force field being the
x 2 y 2 z 2
source of a patern time – independent: -
2 2
  r   r r  r   r 
2m
 r  - positron function
solution:
 r, t    r exp  iwt 
Since  r, t  takes complex rules it canot have physical interpretation.
These we use  has to finite
2) Define and interpret physically the following quantum-mechanical phenomena: tunneling
and resonant tunneling.
Tunneling – quantum mechanical effect by which a particle can penetrate a barrier into a
region of space that would be forbidden by ordinary classical mechanics, tunneling is a direct
result of the wavelike properties of particles, the wave associated with a particle „decays”
through a barrier. The amplitude decreases but the amplitude of the wave on the other side is
large enough that there is a finite probability of finding the particle there.
3) Explain the system of quantum numbers in the hydrogen atom. Describe electron
configuration of the neutral atom of germanium (32 Ge)
germanium (32) Ge
electron configuration: [Ar] 3d10 4s2 4p2
shell structure: 2.8.18.4
4) Show, compare and explain the electron configuration of atoms of silicon 14Si and carbon
6
C.
Silicon (14) Si
electron configuration: [Ne] 3s23p2
shell structure: 2.8.4
Carbon (6) C
electron configuration: [Ne] 2s22p2
shell structure: 2.4
5) Explain the electronic configuration of
31
Ga and
33
As atoms in their ground state. What
would say about their properties as dopants in the silicon crystal?
gallium (31) Ga
electron configuration: 3d10 4s2 4p1
shell structure: 2,8,18 (3)
n-type for Si
Arsenic (33) As
electron configuration: [Ar] 3d10 4s2 4p3
shell structure: 2,8,18 (5)
n-type for S
6) Explain the origins of the differences in electrical properties of metals and insulators.
INSULATORS:
At absolute temperatures all energy bonds (except the gap) are fully occupied by electrons
which means that electrons can not move when the electron field is applied neither can holes,
because there are no holes in fact. Of course, everything depends on the electron field and the
temperature. One can easily imagine a situation when the applied voltage is so huge that no
matter how big the energy is the current will flow or such a big temperature. That is why in
fact there is no sharp demarcation line between insulators and semiconductors. We may
assume that an insulator is a material that has energy gap greater than 2eV.
METALS:
If the valence bond at absolute temperature is not completely occupied by electrons, which
means that electrons can easily move within the conical lattice and we observe the current
flow, the material is called metal. Metals are the best conductors.
7) Why does aluminum conduct electrical current at absolute zero temperature and silicon
does not?
Is the conductivity of the extrinsic silicon increasing with temperature or decreasing ? (in
room temperature range)
Generally one can express the conductivity by means of the following equation
  qu  pp  where:
u,p – are numbers of electrons and holes
u
, up - are are mobilities of electrons and holes
All variables placed in the equation are vulnerable to temperature changes. In median (room)
temperature range number (concentration) of majority carriers constant. Thus the condictivity
is changing due to mobility dependence on temperature. Increase of temperature causes
vibrations of cristal lattice and so we observe more frequent collisions between carriers
and lattice. As for room temperature range is concerned we may sum up the behaviour of
conductivity in two points:
-
increase of temperature is not enough to increase the number of free connices
-
increase of temperature causes the increase of vibration frequency and so increases
the probability that the connices have collisions with the lattice.
As a result we observe decrease of conductivity within the range of room temperatures
8) What does the Fermi-Dirac function describe? Difine the Fermi level.
(8)
What is the difference between the depletion state and the strong inversion state of the
semiconductor surface region? (of the NOS system)
Depletion state : (the whole answer refers to the ‘p-type’ seminars
* ps < pB  NA
ps – holes in the surface region
pB - holes in both region
* ns < pB < NA
f - Fermi potential
s
- surface potential
* O <  s < 2 f
* V FB < V G < V T
V FB - flat band voltage
V G - voltage applied to the junction
V T - threshold voltage
Electric field that is applied causes reduction of majority conices concentration and increase
of minority carriers concentration. However, the gate voltage is not high enough to cause the
minority carrier concentration to be greater than the equilibrium majority carrier
concentration. That is why the semicondictor surface is potentially depleted in free carriers.
Strong inversion state:
*
ps < < pB
*
ns > pB  NA
*
 s > 2 f
*
VG> V T
-
the applied voltage is high enough to make the minority carriers dominate the
electric charge density at the semiconductor surface. In a strong inversion state
even a small increase of the surface potential s above 2 f results in a huge
increase of minority carrier concentration.
(9) - What interpretation and properties has the wave function representing an electron
according to quantum mechanics ?
9) Why are the light-emitting diodes fabricated of gallium arsenide and not of silicon?
why the LED fabricated of galium arsenide and not silioc?
LED s are made of direct – gap semiconductors, e.g. GaAsP, AlGaAs for which a
radiactive recombination process is different from the ore for the indirect gap semiconductors.
radiactive recombination process on electric signal into an electromagnetic radiation.
In a direct gap semiconductors the momentum of an electron having the minimum energy in a
conduction bond equals to the momentum of an electron having the max energy in the relanee
bond. In indirect gap semiconductors, electrons momentum are the same. In this case the third
particle called phenon is necessary to exchange the momentum between the electrons.
Probability of occurrence of radiactive process where 3 particles are needed is much less than
the probabilityn occurrence of a nonradiactive process ( 2 particles). in gap semiconductors
are less efficient of radiactive recpmbination than in direct gap semiconductors.
Radiactive recombination process – when an electron makes a transition from the conduction
bond into relenee bond and part of its energy is emitted in the form of photon.
10) which direct-gap semiconductor can generate light of shorter wave: made of
GaAs ( E g  1.47eV ) whether made of GaN ( E g  3.44eV ) ? Why?
(10) Which direct – gap semiconductor can generate light of shorter wave:
a)
GaAs  Eg  1,47eV
b)
GaU  Eg  3,44eV
Changing eV into y (should you forget)
LeV = 1,602.10-19 y
Ad a) Eg = 1,47eV = 1,47 . 1,602.10-19 y = 2,35 . 10-19y
E=uV
V=
Eg =
hc
hc
 g 
g
Eg
g 
19,824  10 26
 8,43  10 7  843m
19
2,35  10
Ad b) g 
c

this is „ni“
h= 6,63  10 34 ys
c  2,99  108 m / s
19,824  10 26 ym
19,824  10 26

 3,598  10 7  359,8m
19
19
3,44  1,602  10 y
5,51  10
As we can see the semiconductor made of GaN can produce a shorter wave !
11) The crystalline SiO 2 glass known as the quartz glass is transparent for the ultraviolet light
while the amorphous SiO2 known as silica glass is not. What can it result from?
12) What are the donors and acceptors in the semiconductor crystal? Give examples for both
kinds of dopants in silion.
Donors (n-type) – atoms having one valence electron more than the atoms of semiconductor.
Acceptors (p-type) atoms having one valence electron less than the atoms of semiconductor
The atoms of dopants which give the free electron after introducing to the semiconductor are
called donors.
The atoms of dopants which result in appearance of the holes are called acceptors.
Si – 4 valence electrons.
Introducing B to Si  B –3 valence electrons  there is a lack of one electron to bond an
atom of B with 4 atoms of Si  the hole appears  Si becomes a p-type semiconductor.
n-type: incorporating P(phosphorous) As (arsenic), Sb(antimony) to the crystal in place of Si
atom, then this atom (e.g As) will have 4 covalent bonds and one unbonded electron, because
the free electron atom have one extra electron to ‘donate’ it is called a donor atom/
p-type: a trivalent atom (baron) is substituted into the crystal lattice. The result is that an
electron is missing from one of the four possible covalent bonds.
13) Which electronic device can operate faster: made of silicon or made of gallium
arsenide?Why?
Galium arsenide devices can operate faster due to the carrier mobility which is approximately
five times the mobility for silicon. A higher mobility corresponds to a higher velocity of
carriers for a fixed electric field. This property makes possibility of faster switching what
makes the device operate faster.
14) Which silicon device can operate faster: based on operation of electrons whether of holes?
Why?
15) Characterize the thermal generation-recombination process in silicon and GaAs. Compare
the life-times of electrons in these semiconductors.
Ud 
qrc
Fe
me
rc
- time between collissions (relaxation time)
me - electron effective mass
Fe - electron field intensity
The effective mass of Si equal to 0,97 mb : much bigger than the effective mass of GaAs
which is equal 0,067 mb {GaAs is better for high frequency use }
16) How does the conductivity of extrinsic silicon change with temperature increase inn the
room temperature range?
S  q n n  p p 
n,p – number of electrons and holes
All factors of the equation are temperature dependent
In the median temperature range the concentration of majority carriers is constant,
temperature behaviour of the conductivity will be determined by temperature dependence of
the carrier mobility.
   0T

3
2
An increase of the temperature results in the fact that the probability of the electron interacts
with the lattice and its defects increase to what in turn results in shortening of the mean free
time between collisions. As a result a decrease of mobility is observed thus the conductivity
decreases too.
17) Explain behavior of conductivity of the extrinsic semiconductor with temperature.
The change of conductivity with the change of temperature in low and high temperature
ranges results from behavior of electrons and holes concentration. At low temperatures when

ni is less than n0 a temperature growth results in an exponential increase of a number of free
electrons. If the temperature is higher than Ti the semiconductor becomes practically intrisinc
(n=p=ni )… Due to exponential dependence of concentrations on temperature in ranges of
low and high temperature the conductivity changes in a similar way
--) What are the causes of differences in optical properties of solids (absorption egde,
spectrum and intensity of electromagnetic radiation) ?

hc
Eg
Absorption edge – the maximum wave length of the electromagnetic radiation than can be
absorbed by a given semiconductor.
Exemplary absorption spectrum:
Electromagnetic radiation is generated when electron makes a transition from the conduction
to the valence bond (radioactive recombination process) and part of its energy is diluted in the
form of photon.
CAUSES OF DIFFERENCE:
As we can see  depends on two constants hc and Eg. Only Eg can be changed so different
materials have different energy levels and gaps (different energy structures).
Absorption spectrum depends on two things absorption coefficient and wavelength, these
once more depend on the energy structure of a solid.
As we can remember, electromagnetic radiation generated in
direct and indirect-gap
semiconductors is different and it also depends on the type of semiconductor energy structure
{here remember about GaAs and Si}.
18) What are the donor dopoints and acceptor dopints? Give examples
Donor impunities – 5 valence electrons in outermost orbit, it has one electron more than is
needed for the covalent bonding. We talk about donor dopants when one or the
semiconductors atom is exchanged with an atom having one valence electron more than atoms
of semicondictor.
15P
1s22s2p63s2p3
35As
4s2p3
Acceptor dopants – 3 valence electrons
19) How to increase the rate of generation – recombination process in a given semiconductor
and a given temperature?
Generation – recombination process
System under non equilibrium has a tendency to return back to equilibrium to these rates are
not balanced when we perturb (disturb the order) the semiconductor causing the concentration
of one type of carrier to exceed its equilibrium value. Example: When extra minority carriers
are injected into one end of a semiconductor bar, electron and hole densities fall by to obtain
equivalent values by causing the rate of recombination to exceed the rate of generation.
20) What voltage is created on the contact of two solids? Is it possible to use this voltage as a
source of energy? Why?
Because, Si is effective mass is much bigger, electronic mobility for GaAs is 4-5 times greater
than for Si.
The effective mass of Si is equal to 0,97 mb is much bigger than the effective mass of GaAs
which is 0,067 mb < GaAs is better for high frequency
At time interval depends manly on temperature and mainly on concentration, mobility
depends also on them and the material through the effective mass
Si effective mass is much bigger, electron mobility is 4-5 times greater than for Si
21) Why does the Schottky diode operate very fast?
The current in the Schottky diode is carried by majority carriers (electrons), hence the storage
capacitance associated with minority carriers current does not exist. The result is a very short
time of switching from the forward conducting mode (forward bias) to the reverse blocking
state (reverse bias) less than caps (usually 4 orders of magnitude smaller than for the pn
junction). Therefore, Schottky diode cab be when small capacitance and short switching time
is required, eg.: for high frequency signals ( up to 50 GHz). As time interval depends mainly
on temperature and impunity concentration, mobility depends also on them and on the
material through the effective mass.
22)
Explain the physical mechanism of the depletion capacitance. How does it depend on
the bias voltage and small frequency signal?
The capacitance G (junction capacitance or depletion layer capacitance) represents a process
of the charge and discharge of the depletion layer edge by majority carriers. Since the
depletion layer width xd charges with the voltage, the capacitance G is also the bias voltage
dependent.
G is given by:
G
dm  ds

q s Nd

A s A
dV
dV
xd v 
2Vc  V 
and:
Charge of the bias voltage
Am   As  gNdxd where:
1
 2
2
xd   s V0  V 
 qNd

Increase of gate voltage (in this region) decreases the capacitance and makes wider the
depletion layer. But as we can see from the graph, it is the case for HF (high frequency) the
same as we had on our laboratories. When we consider the LF (low frequency) case we can
see that this capacitance increases on some point.
23) Define the threhold voltage of the metal-oxide semiconductor system.
The threshold voltage of an MOS transistor is defined as a potential that must be applied at
the gate in order to achieve the ‘strong inversion’ condition:  s  2 F where  s is the
surface potential and  F is the Fermi potential.
Vt  Vg  s  2F   MS 
where Qox is the
Qox Qox
1

 2 f  VFB 
Qox Qox
cox
gate
4qN A E  2 E
charge. QSUB is the subtract depletion charge. cox is the gate
capacitance.
VFB is the flat-band voltage ( voltage for which the surface potential is equal to 0 ) and VFB is
the Fermi potential
The threshold voltage VT equals the sum of the flat-band voltage twice the bulk potential and
the voltage across the oxide due to the depletion layer charge:
VT  VFB  2 F 
2 S qdNa2 F  VSB 
Cax
The gate charge corresponding to the on set of the strong inversion state is called the
threshold voltage. A formula for VT can be obtained if the semiconductor charge density
s at the onset of strong inversion  s  2 is expressed with the use of the depletion
approximation by  qN A xd s  2  .
4 s qN A F  2
1
VT  VG ns  p B   V g  s  2 F   VFB 
Ci
 2 F
24) What is the difference between the depletion state and strong inversion state of the
semiconductor surface region?
As a mere positive voltage then the flat-band
voltage is applied, a negative charge builds up
in the semiconductor. The positive charge on the gate pushes the mobile holes into the
substrate. Therefore, the semiconductor is depleted of mobile carriers at the interface and a
negative charge, due to ionized acceptor ions, left in the space charge region. Inversion occurs
at voltages beyond the threshold voltage. In inversion there exists a negative charged
inversion layer at the oxide semiconductor interface carriers which are attracted to the
interface by the positive gate voltage. (As are further increases the gate voltage, the depletion
layer width barely increases further since the charge in the inversion layer increases
exponentially with the surface potential.