14 Sep 07 Semi.1

SEMICONDUCTOR BEHAVIOR AND THE HALL EFFECT The object of this experiment is to study various properties of n- and p-doped germanium crystals. The temperature dependence of the electrical conductivity, and the Hall Effect in germanium, will be demonstrated. The sign of the charge carriers responsible for conduction, the charge carrier concentration, and the charge carrier mobility will be determined for the two types of crystals. Theory: A rigorous treatment of the electrical behavior of semiconductors involves the application of quantum mechanics through the Schrodinger equation and of statistical physics through the Fermi-Dirac distribution, and this is best left to a class in solid-state physics. In this discussion the free-electron and nearly-free-electron models will be used. Where necessary, results from solid-state physics will be presented without derivation and the reader is directed to the list of references at the end of this experiment for further information. The free-electron model Applying the Schrodinger equation to the relatively simple situation of a free electron confined within a solid yields quantization of the electron's momentum and hence energy. That is, the electron can only occupy certain energy levels. The electrons in a sample of material at very low temperature will fill the energy levels from the lowest on up so that the system is in its ground (lowest total energy) state. Any given energy level can contain at most two electrons, one with spin up and one with spin down (Pauli exclusion principle). The maximum energy an electron can have in the ground state is called the Fermi energy, EF. At zero temperature (0 K) all the electron energy states below the Fermi energy are occupied, and all those above are vacant. At finite temperature, the probability that an energy state is occupied is a function of energy and temperature, the Fermi-Dirac distribution. The effect of higher temperature is that electrons near the Fermi energy are able to 'jump' into unoccupied higher-energy states. Thus, the situation of filled and vacant states at T = 0 becomes one of filled, partially filled, and vacant states at T > 0. In discussing the electronic properties of materials the core electrons (those in completely filled orbits) play little or no role. It is the valence electrons, those from partially filled shells whose orbits encompass more than one atom, that most affect the properties of a material. The limits of the free-electron model can be shown by considering the situation when a voltage is applied across a sample in which the valence electrons are assumed to be free. If the sample has dimension l in the direction of the applied voltage and cross section A, then the current I (rate of charge flow through the sample) and current density j = I/A are

14 Sep 07 Semi.2

I = noevA (1) j = noev (2) where no is the number density of valence electrons, e is the electronic charge, and v is the electron velocity. The applied voltage produces an electric field E. Since an electron at the Fermi energy can be considered free because of the unlimited empty energy states above it, the only force acting on the electron is that due to the applied electric field. F = –eE (3) It should be immediately obvious that this model is incorrect. A constant force acting on the electron means constant acceleration, increasing electron velocity, and increasing current through the sample the longer the field is applied. This is contrary to experience whereby an applied voltage produces a constant current. Writing Ohm's Law in terms of conductivity, σ, current density, and electric field: σ = j/E = noev/E (4) The result of applying equation (3) is: σ = e2not/m (5) where t is the time that the voltage has been applied and m is the electron mass. This states that conductivity depends directly on the density of valence electrons, a reasonable statement. However, it also states that the conductivity increases with time!, a statement that does not make sense. The fault in this derivation lies in the assumption of free electrons. The valence electrons do in fact interact with the atoms of the material. This can be accounted for by assuming that the valence electrons move through the material, generally unaffected by the atoms, until a major interaction such as an elastic collision occurs. The equation derived using the free-electron model can be suitably modified by replacing t, the time for which the electric field has been applied, with τ, the mean time between interactions of the valence electron with the atoms of the material. σ = e2noτ/m (6) Although not shown explicitly, σ is temperature dependent. An electron interacts with an atom only when it gets within a certain distance of it. At higher temperatures, the atom vibrates

14 Sep 07 Semi.3

around its equilibrium position as a result of its thermal energy. Thus it represents a fuzzy volume in space that is larger than the actual size of the atom and so at higher temperature an electron is more likely to interact with any particular atom. As a result the mean time between interactions, τ, becomes shorter and σ decreases with increasing temperature. [In the above expression it has been implicitly assumed that a single value for τ can be assigned to a given material at a specific temperature. In fact, τ depends on the energy of the electrons (charge carriers).] The substitution of the 'mean free time' in the conductivity equation is equivalent to assigning an average, steady, drift velocity, vD, to the valence electrons. Equation (4) becomes: σ = noevD/E (7) Defining the mobility, µ, as the drift velocity per unit electric field: σ = noeµ (8) The nearly-free-electron model Although the modified free-electron model can account for the temperature dependence of the conductivity of conductors, it cannot account for the temperature dependence or the low values of the conductivities of semiconductors. A more realistic treatment of the effects of the atoms on the valence electrons is to substitute a position-dependent potential function into the Schrodinger equation. Because of the regular spacing of atoms in a solid, the atoms of the material (considered without their valence electrons) represent uniformly spaced, deep, attractive potential wells for the valence electrons. At certain values of electron momentum the electron wavefunction and thus the electron probability function have a zero at the location of each atom. When this occurs there are discontinuities in the electron energy distribution. These discontinuities are called energy gaps and there are no allowed states with energies in these regions. The size of the energy gaps depends on the details of the potential function V(x). The regions of allowed energy states are called energy bands. Because of this, this model is also known as the band theory of solids. Each band contains one energy level for each atom in the sample of material. Since each energy level can accommodate two electrons, at low temperature (T = 0) atoms with even numbers of valence electrons will have bands that are either completely filled or empty and atoms with odd numbers of valence electrons will have the upper band half-filled. In this simple band model the three factors that are important in determining the electronic properties of a material are:

1) the temperature, 2) the size of the energy gap,

14 Sep 07 Semi.4

3) the relative locations of the Fermi energy and the energy gap. In this experiment the behavior of germanium, a semiconductor, which has a valence of 4, will be studied. Consider a sample of pure germanium (intrinsic semiconductor). At low temperature the valence electrons are all in the two lowest bands, no electrons being in the third band. The highest occupied band is called the valence band and the lowest vacant band is called the conduction band. The population of the various energy levels in the valence and conduction bands can be shown graphically.

EGAP

Valence Band

Conduction Band

EF

2 0 0 T > 0

2 T = 0

Figure 1

Electron energy increases vertically and the number of electrons in each energy level is shown horizontally. The solid line shows the Fermi-Dirac (F-D) distribution. At low temperature the F-D distribution is a step function between 0 and 2, centred at the Fermi energy. The valence band is completely filled and the conduction band is completely empty. Thus at low temperature the material behaves like an insulator. At higher temperature, the curvature of the F-D distribution shows the finite probability of some electrons from the higher levels of the valence band acquiring enough energy through thermal excitation to cross the energy gap and reach the allowed levels in the conduction band. Electrons that reach the conduction band behave like free electrons because of the unoccupied levels above them. Thus a semiconductor such as germanium at room temperature will conduct to a certain extent. Three factors of note are: 1. The number of electrons in the conduction band is relatively small compared to a pure

conductor such as a metal. 2. The number of electrons in the conduction band is highly temperature dependent due to the

temperature dependence of the F-D distribution. At higher temperature the F-D distribution broadens along the energy axis (higher energy levels become populated, lower energy levels

14 Sep 07 Semi.5

become vacated) and narrows along the probability axis (a wider spread of levels will be partially occupied). This increased number of conduction electrons with increased temperature 'overrides' the decrease of mean free time that occurs with increased temperature. As a result, the conductivity of a semiconductor increases with increased temperature contrary to the behavior of a conductor.

3. When an electron jumps into the conduction band it leaves a vacant state or 'hole' at the top of the valence band. The hole behaves like a positive charge and thus when an electric field is applied it can contribute to the conduction by interchanging energy states with an electron in the valence band.

To reiterate the quantum nature of the conduction process: conduction in semiconductors at room temperature may take place via two distinct and independent quantum-mechanical modes of electron motion, which can most simply be described as classical conduction by 'conduction' electrons with charge –e and effective mass mE*, and by holes with charge +e and effective mass mH*. This classical picture of holes and conduction electrons is possible only because the major quantum features of the electronic motions in the solid can be buried in the 'effective mass' parameter. To obtain a quantitative description of the conductivity of a semiconductor, equation (8) is used: σ = neµ (8) where n is the number density, e is the charge, and µ is the mobility of the charge carrier. In a pure (intrinsic) semiconductor the number of electrons in the conduction band equals the number of holes in the valence band: nE = nH (9) The total conductivity is σ = nE eµE + nH eµH (10) σ = nE e(µE + µH ) (11) (addition because electrons and holes have opposite charge and move in opposite directions). nE is determined by integrating the F-D distribution over the occupied states in the conduction band. The result is n UT e

EkT

E

GAP

=−

23

2 2 (12) where U is a constant that depends on the density of electron states in the conduction band and k is the Boltzmann constant. Substituting for nE in equation (11) yields

14 Sep 07 Semi.6

σ µ µ= +

−2

32 2eU T e

EkT( )E H

GAP

(13) The effect of the T 3/2 factor is extremely small compared with that of the exponential factor. Therefore the expression for σ is often written as: σ =

−( )constant

GAP

eE

kT2 (14) A plot of ln(σ) versus 1/T will be linear with slope –EGAP/2k. If σ is measured as a function of temperature, the energy gap of the material can be determined. From Ohm’s Law in terms of conductivity, current density, and electric field (equation (4)), for a sample of fixed size the conductivity is inversely proportional to the voltage drop across the sample if the current through the sample is kept constant:

σ = j/E = I

AEI

AV V= ∝

l 1

where l and A are the sample length and cross-sectional area respectively. Therefore, for constant current through the sample, a plot of ln(1/V) versus 1/T will be linear with slope –EGAP/2k. To this point the discussion has concerned pure (intrinsic) semiconductors, for example a crystal of pure germanium. Of greater interest and importance is the behavior of extrinsic or doped semiconductors. A doped semiconducting device is made from a semiconductor such as germanium that contains a small amount of some impurity. Consider a germanium sample to which a small number of phosphorous atoms have been added. Phosphorous has a valence of 5 compared to germanium with a valence of 4. Four of the phosphorous valence electrons occupy levels in the germanium valence band. The energy level of the extra phosphorous valence electron is just below the conduction band of germanium; thus very little additional energy is needed for this valence electron to jump into the conduction band. This additional energy is provided by thermal excitation at room temperature, so adding phosphorous to the germanium sample results in the addition of electrons to the conduction band without the formation of holes in the valence band. Since the phosphorous provides an additional electron it is called a donor impurity. Since electrons are negatively charged and there are now more electrons than holes, the sample is said to be an n-type extrinsic semiconductor. The electrons are called the majority charge carriers and the holes are the minority charge carriers. Now consider a germanium sample to which a small number of indium atoms have been added. Indium has a valence of 3 and the energy level of the 'missing' valence electron is just slightly higher than the valence band in germanium. Thus at room temperature valence band electrons from germanium atoms will jump into the vacant energy level of the indium atoms. This

14 Sep 07 Semi.7

results in the creation of holes in the valence band of germanium without the addition of electrons to the conduction band. Since the indium accepts an electron from the valence band it is called an acceptor impurity. The sample is said to be a p-type semiconductor, holes are the majority charge carriers, and electrons are the minority charge carriers.

In the case of impurity (extrinsic) semiconductors, the temperature behavior of conductivity in the extrinsic range is due to the temperature dependence of charge carrier mobility (i.e. there is a decrease in conductivity with increased temperature).

In the range of temperatures from 100 to about 300 K, the mobilities in germanium vary as follows:

Electron Mobility = (4.9 × 107) T –1.66 (15)

Hole Mobility = (1.05 × 109) T –2.33 (16)

Note that because of the direct dependence of conductivity on mobility, equations (15) and (16) also describe the functional dependence of conductivity on temperature for doped germanium.

The Hall Effect

The Hall effect refers to a phenomenon observed in conductors and semiconductors whereby a transverse voltage is generated in a current-carrying sample that is placed in a magnetic field perpendicular to the direction of current flow.

The sign of the majority charge carrier, the charge carrier concentration and the charge carrier mobility can be determined from the Hall effect.

A basic understanding of the Hall effect can be obtained by applying the modified free-electron model and considering the forces acting on a conduction electron due to an applied voltage, Vx, in the x direction and a magnetic field, Bz, in the z direction.

Figure 2 The applied voltage produces an electron drift velocity, vx, in the –x direction. The magnetic field causes a force in the –y direction. The conduction electrons are thus deflected toward the

x

y

z

t

w

l

vx

VH

Bz I

Vx + –

-

+

–

14 Sep 07 Semi.8

bottom of the sample as they flow through the sample. Negative charge will thus accumulate on the bottom side of the sample. This accumulation of charge creates a potential difference (voltage) across the sample and thus causes an electric field in the –y direction. Accumulation of charge will cease once the force on the conduction electrons due to this electric field is equal and opposite to the force due to the magnetic field. The transverse voltage due to the accumulation of charge is called the Hall voltage, VH. Now consider (on your own) the result if positive charge carriers are assumed. The polarity of the Hall voltage indicates the sign of the majority charge carrier of the sample. Let the sample have dimensions l (in x direction), w (in y direction) and t (in z direction). The Hall field Ey has magnitude VH/w. At equilibrium, eEy = evxBz (17)

From equation (1), the electron drift velocity can be written as vx =I

neA

Substituting this expression and Ey = VH/w into the force equilibrium equation (17), yields:

eVw

eI

neAB V

neB Iw

AR

B It

Hz H

zH

z= ⇒ =⎛⎝⎜

⎞⎠⎟ =

⎛⎝⎜

⎞⎠

1 (18)

where the cross-sectional sample area, A, has been expressed as wt (width × thickness). The factor RH is called the Hall coefficient, and in this simple model has the value RH = –1/ne (19) where n is the conduction electron density and e is the electronic charge. This expression for RH gives reasonable results when applied to pure conductors. In the case of semiconductors, however, a more rigorous analysis taking into account band structure and carrier scattering mechanisms yields the following expressions for the Hall coefficients of n- and p-type germanium: Rn = –0.93/nEe for n-type (20) Rp = 1.4/nHe for p-type (21)

14 Sep 07 Semi.9

Apparatus Two samples are provided: one of p-type germanium and one of n-type germanium. The impurity levels in each sample are such that at room temperature they behave as extrinsic semiconductors (conduction due to charge carriers produced by impurity atoms, conductivity depends on impurity concentration and decreases with increasing temperature as per equations (15) and (16)). At temperatures greater than about 340 K (67 °C) the samples behave as intrinsic semiconductors (conduction due to thermally-generated electron-hole pairs, conductivity increases with increasing temperature as per equation (14)). Each crystal (with dimensions 20 mm × 10 mm × 1 mm) is mounted on a supporting plate containing the necessary connections to the crystal as well as some additional components. The applied voltage, Vx, is connected across terminals 2.1 and 2.3. Between terminals 2.2 and 2.3 is a current stabilizer/limiter. The voltage drop across the crystal is measured between terminals 2.1 and 2.2. Terminals 3 allow measurement of the Hall voltage. Adjusting control 5 allows compensation of interfering voltages that may be superimposed on the Hall voltage. i.e. When the magnetic field is 0, control 5 is adjusted to give 0 Hall voltage. Terminals 4 allow connection to the heating grid. A copper-constantan thermocouple connected to the crystal allows temperature measurement by measuring the thermoelectric voltage developed across terminals 7. Refer to the included graph and equations for the conversion between thermocouple voltage, VT, and absolute temperature. The magnetic field for the Hall effect measurements is provided by a pair of coils and a solid iron core. The calibration curve of magnetic field Bz as a function of coil current Im is included.

14 Sep 07 Semi.10

The following circuit connects the various components:

Figure 3 Procedure and Experiment Check that the circuits are connected as shown in the circuit diagram (Figure 3) and ask the instructor to confirm that the equipment is ready for use. Turn on the meters to the appropriate settings, check that the heater circuit rheostat knob is fully CCW and turn on the power supply. 1. TEMPERATURE DEPENDENCE OF CONDUCTIVITY OF N-GE SAMPLE Check that the n-Ge sample is connected and that it is raised out of the magnet. Adjust the 560 Ω potentiometer fully clockwise until the current through the sample reaches

its constant control value of about 25 mA. Turn on the heating coil current by turning the 5 Ω rheostat fully clockwise. As soon as the

thermocouple voltage reaches 5.0 mV back off the rheostat (turn it slightly counter-

14 Sep 07 Semi.11

clockwise) until the thermocouple voltage stops increasing. Do not allow the voltage reading to exceed 5.4 mV.

Measure the voltage, Vx, along the sample as a function of thermocouple voltage, VT. Take

readings of Vx at 0.2 mV intervals of VT for values of VT from 5.0 to 0.0 mV. Because of the relative insensitivity of the thermocouple voltmeter, the following procedure

will be used to record the Vx values: Take a Vx reading when the thermocouple voltage has just changed to the desired value, and again when the thermocouple voltage has just changed from the desired value. For example take the reading of Vx when thermocouple voltage has just changed from 5.1 to 5.0 mV and again when thermocouple voltage has just changed from 5.0 to 4.9 mV. The value of Vx corresponding to a thermocouple voltage of 5.0 mV is then the average of these two readings.

2. ROOM TEMPERATURE CONDUCTIVITY OF N-GE SAMPLE Set the control current at 4.0 mA by adjusting the 560 Ω potentiometer. Record the value of applied voltage, Vx. as a function of applied control current in 4 mA

intervals from 4 mA to 24 mA.

After completing the measurements, reduce the control current to 0 via the 560 Ω potentiometer and disconnect the lead in the 15V terminal of the power supply.

Ask the instructor to change the sample. 3. ROOM TEMPERATURE CONDUCTIVITY AND HALL EFFECT OF P-GE SAMPLE Plug in the magnet coil circuit and after demagnetizing the electromagnet by working down

from 4.0 A in 0.5 A intervals, set the magnet current at 2.3 A (giving a magnetic flux density of 3.04 kG or 0.304 T).

Set the control current at 4.0 mA by adjusting the 560 Ω potentiometer. With the sample raised out of the field, adjust the compensation control so that the Hall

voltage meter reads 0. Record the value of applied voltage, Vx. CAREFULLY move the sample down into the centre of the magnetic field and measure the Hall voltage. DO NOT FORCE THE SAMPLE BETWEEN THE POLE FACES. (The pole face gap is only slightly larger than the sample thickness.)

Using the above procedure measure the applied voltage, Vx, (when sample is out of field) and

Hall voltage, VH, (when sample is in field) as a function of applied control current in 4 mA intervals from 4 mA to 24 mA. At each setting of control current the sample must be raised from the field, the applied voltage read, the compensation control adjusted to read 0 for Hall voltage, the sample lowered into the field, and the Hall voltage read.

Examine the connections of the voltmeter used to measure the Hall voltage and determine the

polarity of the Hall voltage (i.e. is the top of the sample positive or negative relative to the bottom of the sample?). Knowing the direction of the current (left to right when facing sample), the direction of the magnetic field (‘out’ of the sample), and the polarity of the Hall

14 Sep 07 Semi.12

voltage, determine the polarity of the majority charge carriers (refer to the discussion concerning the effect on the charge carriers’ motion of there being an applied magnetic field).

4. DEPENDENCE OF HALL VOLTAGE MAGNITUDE ON MAGNETIC FIELD - P-GE SAMPLE Reduce the magnetic field to 0 by reducing the magnet current. Adjust the 560 Ω

potentiometer until the control current is at its maximum value (about 25 mA). With the sample raised out of the magnetic field, adjust the compensation control to read 0 for Hall voltage. Lower the sample into the magnetic field, and for magnet current values from 0.5 to 2.5 A in 0.5 A intervals, record the Hall voltage.

5. Perform procedure 1, TEMPERATURE DEPENDENCE OF CONDUCTIVITY, for the p-Ge sample. Analysis: To avoid unit problems, use SI units (meters, kilograms, seconds, Volts, Amps, Tesla, ...) throughout the calculations. The thermocouple calibration curve and/or calibration equations can be used to determine sample temperature in Kelvin at each data point. The results are:

Thermo-couple Voltage, VT

(mV)

Sample Temp, T (K)

1/T (10–3/K) ln(T)

5.0 406 2.46 6.006 4.8 403 2.48 5.999 4.6 399 2.51 5.989 4.4 395 2.53 5.979 4.2 390 2.56 5.966 4.0 387 2.58 5.958 3.8 382 2.62 5.945 3.6 378 2.65 5.935 3.4 373 2.68 5.922 3.2 369 2.71 5.911 3.0 365 2.74 5.900 2.8 361 2.77 5.889 2.6 356 2.81 5.875 2.4 352 2.84 5.864 2.2 348 2.87 5.852 2.0 343 2.92 5.838

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1.8 338 2.96 5.823 1.6 334 2.99 5.811 1.4 330 3.03 5.799 1.2 325 3.08 5.784 1.0 320 3.12 5.768 0.8 315 3.17 5.753 0.6 310 3.23 5.737 0.4 305 3.28 5.720 0.2 301 3.32 5.707 0.0 296 3.38 5.690

TEMPERATURE DEPENDENCE OF CONDUCTIVITY ♦ Plot ln(1/Vx) vs. 1/T. From the high temperature (intrinsic) portion of the curve determine a

value for the energy gap of germanium (see equation (14) and recall that σ is directly proportional to 1/Vx). (Boltzmann’s constant is k = 8.617 × 10–5 eV/K.) The accepted value for EGAP for germanium is 0.67 eV.

♦ Plot ln(1/Vx) vs. ln(T) for the first 8 or so low temperature (extrinsic) points. Compare the

slope of this linear, low temperature portion of the curve with the value predicted from equation (15) or equation (16) depending on which sample's data are being plotted. Recall that 1/Vx is proportional to conductivity which in turn is directly dependent on charge carrier mobility.

ROOM TEMPERATURE CONDUCTIVITY AND HALL EFFECT ♦ Plot the applied voltage vs. control current characteristic (Vx vs. Ic) for each sample and from

the slope and the sample dimensions determine the conductivity. Recall Ohm's Law, V = IR, and σ = 1/ρ = l/RA, so the slope of Vx vs. Ic is R = l/σA. Thus the conductivity is given by

σ =l

( )slope A

♦ Plot Hall voltage as a function of control current for constant magnetic field for the p-Ge

sample. Noting from equation (18) that the slope of this graph is R B

tH z , determine the Hall

coefficient. ♦ From the p-Ge sample conductivity and the Hall coefficient determined above, determine the

charge carrier concentration and the charge mobility (equations (21) and (8)). The hole mobility for pure germanium is 0.182 m2/V·s.

DEPENDENCE OF HALL VOLTAGE MAGNITUDE ON MAGNETIC FIELD ♦ Using the magnet calibration curve and the Hall voltage versus magnet current data obtained

for the p-Ge sample, plot magnetic field versus Hall voltage. What is the significance of the shape of this plot?

14 Sep 07 Semi.14

ANALYSIS SPREADSHEET ♦ An Excel spreadsheet is available for use in tabulating the data, performing the required

calculations, and producing the required graphs. ♦ The spreadsheet is obtained from the lab manual web page:

http://physics.usask.ca/~bzulkosk/modphyslab/phys381.htm : ♦ Enter your data in the Data sheet. ♦ The data is automatically transferred to the Sample 1 and Sample 2 sheets, which are

formatted for printing. These sheets also contain the results of most of the calculations that are required.

♦ The required graphs are automatically produced and can be viewed by selecting a sheet with a tab labelled Chart x.x.

References: Experimental Physics, R.A. Dunlap, 1988, Oxford University Press, QC 33.D86 Introduction to Semiconductor Physics, Adler, Smith, & Longini, 1964, John Wiley & Sons, QC 612.S4A23 CRC Handbook of Chemistry and Physics, 56th Ed., CRC Press Principles of Electronic Instrumentation, Diefenderfer, 1972, W.B. Saunders Company Laboratory Physics, Meiners, Eppenstein, & Moore, 1969, John Wiley & Sons Experiments in Modern Physics, Melissinos, 1966, Academic Press, QC 33.M52 Semiconductors, Smith, 1959, Cambridge University Press, QC 612.S4S65