Download Lab instruction Lab Semiconductor physics Course Modern Physics

Lab instruction
Lab
Semiconductor physics
Course
Modern Physics
Room
Lab code
5342
Content
HF
ASSIGNMENTS:
Measure the Hall effect and the electrical conductivity as a function of
temperature for the semiconductor Ge.
LITERATURE:
Tipler, Llewellyn, Modern Physics, kap 10.6 – 10.7.
B.G. Streetman, Solid State Electronic Devices, (Prentice-Hall 1980)
chap. 3.3.3 and 3.4.2. – available at the lab.
Instructors:
Daniel Hedlund ([email protected])
Tian Bo ([email protected])
Feb 2016
1
Preparatory Questions
In order to be able to attend the semiconductor physics laboratory, you must answer the
preparatory questions. The answers to the preparatory questions should be sent to
Studentportalen at least Monday 15 February. Each group submits a single answer to
Preparatory Questions.
1) How can the electrical conductivity be increased in semiconductors (except
by changing the impurity level)? Compare to a metal.
2) Will a semiconductor conduct at 0 K?
3) Find the concentration of holes in the valence band at thermal equilibrium.
4) Show how the energy gap can be estimated from the diagram ln(σ) vs 1/T.
5) Discuss how the doping level can be estimated from the diagram ln(n) vs
1/T.
2
1. Introduction
Semiconductors are a group of materials, which have conductivities between those of insulators
and metals. The resistivity of a semiconductor has strong temperature dependence. You can
change the electrical conductivity of a semiconductor by introducing a controlled concentration
of impurities into the material, and this is called doping. If, for example, a silicon crystal is
doped with phosphorus, four of the five valence electrons of phosphorus will form covalent
bonds with surrounding silicon atoms. The remaining valence electron will be weakly bound to
the phosphorus atom and if the electron is excited by adding some energy, it becomes free and
will be donated to the conduction band. The silicon crystal then becomes n-doped. The crystal
may also be p-doped and it is done with atoms that will give free holes. The most common
semiconductor materials are silicon (Si), germanium (Ge) and (gallium arsenide) GaAs.
In this experiment, all groups will examine n-doped Ge. Hall Effect measurements will be
used to determine the doping concentration. The bandgap of the examined semiconductor, the
Hall coefficient, and the electron mobility will also be determined. You will receive data also
from previous experiments on n-doped InSb.
2. Theory
If a conductor is subjected to a current I and a magnetic field B is applied to be perpendicular
to the current, a potential difference occurs across the conductor in a direction which is mutually
perpendicular to I and B . This phenomenon is called the Hall Effect.
How does it occur?
Consider a specimen as in Fig.1. If a current I flows across the sample in the x -direction, charge
carriers q move along the x -direction. Here, q represents e for electrons and e for holes,
where e is the elementary charge. If I is in the  x̂ -direction, e moves along the  x̂ direction and e moves along the  x̂ -direction. One should be careful about  and 
directions for the different cases. In Fig. 1, in order to simplify, only the case of n-doped
materials is shown. If a magnetic field B , which is perpendicular to I and the surface of the
sample, is applied to the sample, the magnetic force (the Lorentz force ) FB causes a deflection
of the electrons in the - y -direction, Fig.1b. Hence, the electrons accumulate on one side of the
sample and an excess of positive ions are present on the opposite side.
3
(a)
(b)
(c)
Figure1. Hall effect for an n-doped material. (a) Current flow through the specimen without magnetic
field. (b) A magnetic field B is applied, the electrons deflect and accumulate on one side of the
specimen. (c) The accumulated electrons give rise to an electric field E y which causes a balance between
the electric force and the magnetic force along the y -direction. At the newly reached balance, the
electrons flow along the x -direction without deflecting.
4
After a short while, an electric force Fe as a result of an electric field E y that is caused by the
accumulation of the charges at the edges will be equal to FB . Thus, the charges will continue to
move in the x -direction, Fig.1c, with the drift velocity v .
Figure 2. The Hall Effect in a semiconductor plate.
With the notation of Fig.2, we have the balance of the forces:
FB + Fe = 0
qv ´ B + qE y = 0
the relation between the magnitudes:
qvB  qE y
gives:
E y = vB
(1)
Note also that v and E y change direction when q changes sign. The following relation
between the current, the current density j and the drift velocity is achieved:
I  abj
(2)
j  nc qv
(3)
5
U H  Eyb
(4)
Here nc is charge carrier concentration (number/unit volume), U H is the Hall voltage. U H is
negative for electrons (i.e., E y is pointing to  ŷ ), and positive for holes (i.e., E y is pointing to
the  ŷ -direction).
Inserting eq. (2), (3) and (4) in (1) gives
I
U
B= H
abnc q
b
(6)
i.e.
nc 
IB
U H qa
(7)
and the Hall coefficient is defined as:
RH =
Ey
jB
(8)
Inserting eq. (1) and (3) to (8) or inserting eq. (2) and (4) into (8) leads to the relation
RH 
UHa
1

IB nc q
(9)
It is seen that the sign of the Hall coefficient is given by the sign of the charge carriers, or
equivalently the sign of the Hall voltage.
The conditions for the validity of the expressions above are that one type of charge carrier is
dominating and that the drift velocity is constant. If both electrons ( n ) and holes ( p ) are present
as significant charge carriers (which is often the case in semiconductors) their different
mobilities (  n and  p ) must be accounted for in the expression of RH .
RH =
1 p - k 2n
e (kn + p)2
(10)
where k  n /  p , n is electron concentration and p is hole concentration. For strongly n doped or p -doped materials Eq. (10) is reduced to Eq. (9).
6
The sample voltage U R (see Fig.2) over the plate can be expressed as:
U R  RI  I 
c
Ic

’
ab  ab
(11)
where the conductivity  is

I c
.
U R ab
(12)
For semiconductors the conductivity may generally be expressed as
  nen  pe p
(13)
In the case of dominating n -doping, we find that the mobility may be calculated by
substituting Eq.(6) and (12) in Eq.(13) as
n 

ne

UH c
URB b
(14)
and analogously for  p in the case of dominating p -doping.
The temperature dependence of the mobility may be expressed as
n, p  CT 
(15)
where  is called the temperature exponent of the mobility ( C is a constant). The value of α
depends on the scattering mechanisms which influence the electron and hole mobility. When
lattice phonon scattering is dominating (i.e. at high temperature), the temperature dependence is
  3 / 2 . When impurity scattering is dominating (i.e. at low temperature),   3/ 2 .
7
3.1. Experiment: n-doped Ge
Phywe
Power Supply
IB
Multimeter
IP
UH
12V
AC


DC
T
D
HC
Multimeter
IP
IB
Carrier Board
IB
IP
UH
UR
Multimeter
UR
T
D
HC
Figure 3. Experimental Setup of n-doped Ge measurements.
1.
Measure the Hall voltage, U H , as a function of sample current I .
a)
Set the display in “temperature” (TP) mode on the Carrier Board. Record the
current temperature.
b)
Set the display in “current” (IP) mode. ( I P denotes I on the equipment)
c)
d)
e)
Fix the sample current as I =0 A.
Apply a magnetic field of 250 mT. Use the calibration table in the Appendix.
Check the U H value. If it is different from zero, set it to zero using the “ U H
compensate” button on the carrier board.
8
f)
Measure U H as a function of I between -30 mA and 30 mA in steps of about 5
mA
g)
Present the results in a table.
h)
Using the data you obtained to determine the value of Hall coefficient RH of the
sample material at room temperature. (Hint: try to do linear fitting of U H - I graph to get the
value RH . This method gives the least error.)
2.
Measure U H as a function of B
a)
b)
c)
Set the magnetic field to zero.
Apply 30 mA sample current.
Check the U H value. If it is different than zero, set it to zero using the U H
d)
compensate button on the carrier board.
Change the magnetic field between 0 and 300 mT in steps according to the
calibration table. Measure U H .
e)
f)
g)
Return the magnetic field to zero.
Change the polarity of the coil-current.
Check the U H value when the magnetic field is zero. If U H is different than zero,
set it to zero using the U H compensate button on the carrier board.
h)
Change the magnetic field between 0 and -300 mT in steps according to the
calibration table. Measure U H .
i)
j)
Present the results in a table.
Using the data you obtained to determine the value of Hall coefficient RH of the
sample material at room temperature.
k)
Compare the result with those obtained in the first experiment and discuss.
3.
Measure U H and U R as a function of temperature.
a)
b)
c)
Change back the polarity of the coil-current.
Apply 30 mA sample current.
Check the U H value. If it is different than zero, set it to zero using the U H
d)
e)
compensate button on the carrier board.
Set the magnetic field to 300 mT.
Start the measurements by activating the heater and measure up to at maximum of
170°C. Measure U H and U R for every 5 degrees. (In the beginning of the cooling
f)
g)
process, the temperature will rise very fast so it is difficult to record the data. One
way to solve this problem is to use a video camera to video-tape the multimeters
and the temperature display during the heating process.)
Present the measured values in a table and plots.
Plot a graph to show the Hall coefficient\s dependence on the temperature.
9
h)
Plot ln( ) as a function of 1/T, ln(nc ) as a function of 1/T and ln(  ) as a
function of ln(T ) . Determine the values of bandgap, doping concentration and
temperature exponent of the mobility from the above graphs. Compare the bandgap and
temperature exponent of the mobility values with theoretical values (Physics handbook
or Hofmann).
10
Appendix
Sample dimensions:
a = 1 mm
b = 10 mm
c = 20 mm
Calibration Table for n-doped Ge-setup
B(mT)
300
270
250
230
200
170
150
130
100
70
50
30
0
-30
-50
-70
-100
-130
-150
-170
-200
-230
-250
-270
-300
I(mA)
1131
1020
947
872
768
655
582
499
403
290
208
135
0
-79
-162
-233
-346
-460
-528
-603
-713
-825
-903
-985
-1114
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3.2. FYI - Experimental Setup of Te-doped InSb
HP
UH
UR
UT
I
 
Philips
IB

M
P
LN 2
p
M
S
M
P
B
G
G
IB
I
UH
UR
UT
s
B
Figure 4. Experimental Setup of Te-doped InSb measurement.
12
On the Hall plate, thin wires are soldered, allowing U H and U R to be measured when a current
I passes through the plate, see Fig. 5. To protect the Hall plate and the wires while cooling, the
plate is mounted inside a sample holder of brass. Be very careful with the sample holder, do not
drop, hit or handle it carelessly! The thin wires are connected to cables exiting on the top of the
sample holder. These cables have different colors and Fig. 4 shows how to measure on the Hall
plate. The sample holder should not be opened because it has to be nitrogen tight. On the Hall
plate, a thermocouple is mounted (see Fig. 5) which has its reference point outside the sample
holder, placed in ice water. A calibration table is given below, to convert voltage in mV to
degrees Celsius.
white
blue
green
red
yellow
black
thermocouple
Figure 5. The Hall plate of Te-doped InSb and connections for U H , U R and I .
a) Check the connections according to Fig.4 and Fig.5.
b) Place the reference soldering from the thermocouple in ice water and check the
temperature.
c) Cool down by slowly lowering the sample holder into the Dewar filled with liquid
nitrogen, while at the same time noting the thermo voltage, (thermocouple). Wait until the
lowest temperature possible is reached. Prepare a table of U R , and U H , measured at
various U T ranges from the lowest temperature to room temperature in steps of 0.1 mV.
(In the beginning, the temperature will rise very fast so it is difficult to record the data.
One way to solve this problem is to use a video camera to videotape the multimeters and
the temperature display during the heating process.)
13
d) When it reachs the lowest temperature ( U T ~5.1 mV), apply a sample current of ≈ 20
mA, which should be kept as constant as possible during the experiment. If it changes,
note the new values continuously.
e) Switch on the current supply for the magnet and adjust the current to set a magnetic field
of 1000 Gauss(0.1T) at the position where the sample is placed. The current supplied
should be around 2A.
f) Take out the sample holder from the Dewar. Place it between the poles of the
electromagnets. The Hall plate should be perpendicular to the magnetic field and
centered between the electromagnets. The sample holder has marks indicating the
orientation of the Hall plate.
g) Take the first measuring point of U H , U R , I and U T at the possible lowest temperature.
Then, continue to take data points at every 0.1 mV up to room temperature.
h) After reaching the room temperature, turn off the magnetic field and measure U H and U R .
Ideally, U H should be zero if there is no magnetic field. But since the solder points for
U H might not be exactly perpendicular to the current path, U H is often non-zero. Correct
the
data,
that
you
formula: U H (T )  U H ,meas (T ) 
obtained
during
U H ( B  0)
U R (T )
U R ( B  0)
.
the
measurement,
with
the
i) Present the measured values in a table.
j) Calculate the values nc ,  , and  .
k) Plot a graph to show the Hall coefficient\s dependence on the temperature.
l) Plot ln( ) as a function of 1/T, ln(nc ) as a function of 1/T and ln(  ) as a function of
ln(T ) . Determine the values of bandgap, doping concentration and temperature exponent
of the mobility from the above graphs. Compare the bandgap and temperature exponent of
the mobility values with theoretical values (Physics handbook or Hofmann).
m) Discuss the physical reasons why silicon and for example not InSb, has been the
dominating semiconductor material for electronic components.
14
Appendix
Sample dimensions: a = 1 mm
b = 10 mm
Calibration table to convert voltage in mV to degrees Celsius:
c = 20 mm
15
4. Report Writing
Each group submits a single report to the lab instructor latest 25 March 2016. Send it to
“Studentportalen”. The report should be written in English.
The report should include:
1.
2.
3.
4.
5.
6.
The purpose of the experiment (eg.what physical quantities we aim measure).
A brief description of the theory.
The experimental methods.
The results and discussions.
The conclusions.
Appendix( Include the tables of measured values and intermediate calculated values here)
16