UMEÅ UNIVERSITY Department of Physics 2004-04-06 Agnieszka Iwasiewicz Leif Hassmyr Ludvig Edman SOLID STATE PHYSICS HALL EFFECT
1. THE TASK To measure the electrical conductivity and the Hall voltage for InSb at different temperatures and from these data determine: band gap temperature dependence for the charge carriers' mobility. temperature dependence for the charge carriers' concentration. 2. ADDITIONAL LITERATURE 1. N.W. Ashcroft and N.D. Mermin, Solid State Physics (mainly chapter 28) 2. J. R. Hook and J. E. Hall, Solid State Physics 3. H. M. Rosenberg, The Solid State 3. AIM OF THE LAB The lab is providing you with a possibility to examine the Hall effect in a semiconductor. You will be asked to use the knowledge gained during the lectures in order to solve a practical task. On the way, you will work through some exercises to recall the important parts of the theory of conductivity mechanisms in a semiconductor. You will deepen the understanding of the phenomena taking place in a semiconductor at different temperatures. You will also practice the experimental skills by planning your measurements, using a thermocouple, a switch and a voltmeter, as well as you will learn how to work safely with liquid nitrogen. 4. THEORY 4.1. Electrical conductivity in a pure semiconductor For a semiconductor to conduct electrical current, the charge carriers have to be present. The charge carriers are electrons in the conduction band as well as holes in the valence band. When electron is being excited from the valence band to the conduction band, two charge carriers appear, since an electron-hole pair is created. The conductivity is determined both by the concentration of conduction electrons (n c ) and holes in the valence band (p v ), and by the charge carriers' mobility (μ c for electrons, μ v for holes). Exercise 1 Write down the expression for electrical conductivity of an intrinsic semiconductor, keeping in mind the two types of charge carriers: σ = The number of electrons thermally excited to the valence band depends on the temperature. The fraction of the total number of electrons excited across the gap at temperature T Eg 2kT B is roughly of the order e. 2
Exercise 2 What does the magnitude of the mobility depend on at different temperatures? An ideal, perfectly pure semiconductor (with no impurities) is called an intrinsic semiconductor. In such a semiconductor every excited electron leaves behind a hole in a valence band, and besides these there are no other charge carriers. Therefore, the charge carrier densities n ( T) = p T = n T. The charge carrier at each temperature for electrons and holes are equal: c h( ) i( ) density ni ( T ) can be determined using a formula: 32 Eg 34 2kT B 1 2kT B ni( T) = 2 ( mcmv) e, (1) 4 π h where m c and m v are the effective masses (products of the principal values of the effective mass tensor) for electrons and holes, respectively. A complete derivation of this formula can be found e.g. in the book by Ashcroft and Mermin. Exercise 3 E e kt B The Boltzmann factor (the term of type) appearing in the formula for the charge carriers concentration shows, that the energy of one thermally excited charge carrier equals half of the energy gap between the conduction and the valence band. Explain why: The temperature dependence of conductivity is completely determined by the temperature dependence of mobility and charge carriers density. For pure semiconductors at relatively high temperatures, the temperature dependence of mobility is proportional to T -3/2. This fact, together with ( ) Eg 32 2kT B ni T T e results in the temperature dependence of the conductivity: Eg 2kT B σ e. (2) The energy gap is also temperature dependent. There are two sources of this behavior: Thermal expansion of the lattice results in the expansion of the periodic potential experienced by electrons The effect of lattice vibrations varies with temperature, depending on the phonon distribution at each temperature. A typical temperature dependence of the energy gap is quadratic at very low temperatures and linear at higher temperatures. For the sake of our experiment it is correct to assume the dependence of the form: E g (T) = (1-αT) E g0. 3
Summarizing, one can write the formula for the conductivity in a form: σ = Ae E g0 α 2k B e E g0 2k B T (3) where A is a constant. Taking a logarithm of the above formula leads to: lnσ = ln Ae E g0 α E g0 2k B + ln 2k B T e = C E g 0 2k B T, (4) C being a constant. Exercise 4 What property of a semiconductor can be determined from the above formula? In which way? 4.2. Electrical conductivity in a doped semiconductor Usually, semiconductors are not free from impurities. One can also introduce the impurities (dopants) in order to influence the conductivity of a semiconductor. There can be two types of the impurities present: donor atoms, capable of donating an additional electron to the conduction band ( N d ionized atoms per unit volume) and/or acceptor atoms, able to absorb an electron from the valence band, and therefore creating a hole ( N atoms which accepted an electron per unit volume). a Both types can influence the conductivity and it is mostly visible in the low temperature range. The behavior caused by the presence of impurities is called the extrinsic behavior. The additional conduction electrons density, originating from the ionized donor atoms, will be denoted as n d. The additional holes density will be denoted by p a. The charge conservation and neutrality of a semiconductor as a whole demands: nc + nd + Na = pv + pa + Nd (5) where all the negatively charged contributions were collected on the left hand side of the equation, and positive-charged items on the right hand side. The above equation may be rewritten to get the expression for the (negative) charge carrier density: nc + nd = p. v + pa + Nd Na (6) 123 123 n p Let us assume that the semiconductor is doped only with donor atoms. Then the charge carrier density formula gets a bit simplified. The character of the temperature dependence of the number of carriers in a unit volume is shown in figure 1 below: 4
Figure 1. The logarithm of a charge carrier density for a doped semiconductor, plotted versus the reciprocal temperature. Exercise 5 What is the physical origin of the temperature dependence of charge carrier density shown in figure 1? Consider the three distinct regions separately. Hint: A sketch of the energy levels in a semiconductor can be helpful. Exercise 6 Sketch the conductivity dependence on temperature. Chose the axes scales in a convenient way. 4.3. The Hall effect The Hall effect arises when a current I passes through a conductor exposed to a magnetic field. If the current density j is not parallel to the direction of the magnetic field B, an electric field E (Hall field) will arise in the direction of j B. The Hall field is maximal when the current is directed perpendicularly to the magnetic field. In this case, we may choose a convenient coordinate system, such that the magnetic field is pointing in the positive z-axis direction, and the current density is along the x-axis as shown on a figure 2. 5
Figure 2. The Hall effect geometry. The relation between the current density, electric and magnetic field vectors is now reduced to the equation binding their components: Ey = RH jxbz (7) where R is the Hall coefficient, a constant which value depends on the sample material. H In semiconductors there are two types of charge carriers, and for the relatively small magnetic fields the following equation holds: 2 2 1 pμv nμc RH = (8) 2 e pμ + nμ ( ) The Hall effect is a direct consequence of the fact that the electric current consists of a number of moving charged particles, which experience the Lorentz force in a magnetic field. The sign of the Hall coefficient determines the type of the dominant charge carriers. Exercise 7 Check the table values of the mobility for electrons and holes in InSb (indium antimonide): μc =... μ =... v What can be said about these values? Can we assume something on the basis of this comparison? v c Exercise 8 Keeping in mind the outcome of the above exercise, write down the simplified equations for the conductivity and the Hall coefficient in case of InSb: σ = R H = 6
Exercise 9 How can we determine the mobility and the concentration of charge carriers? μ c = n = 5. EXPERIMENTAL SETUP The semiconductor sample to be examined is an InSb-plate with connection wires soldered onto it in accordance with figure 3. 7 Figure 3. The InSb-plate with connection wires. This plate is mounted vertically in a copper cylinder which can be placed between the poles of an electromagnet. Please do not dismount the copper cylinder. An identical InSb plate, not connected to the measurement setup is available in the lab as a model of your sample. The plate's vertical plane is parallel to the marking Ι-ΙΙ on the holder. The construction is such that the plate is in an area where the magnetic field is homogenous when the holder is at its lowest position. The temperature of the sample will be measured with a Chromel-Alumel thermocouple. One of the junctions of the thermocouple is in a close contact with the sample. The other should be placed in an ice bath, since the ice/water equilibrium point is used as a reference point for the thermocouple s voltage. The thermocouple voltage can be recalculated to the temperature by use of a calibration table, provided in the folder next to your experimental setup. The information about your sample dimensions and a wiring diagram can be also found in this folder. The sample is connected to a constant-current generator and a switch. The switch allows for the use of just one digital voltmeter (DVM) in order to measure the voltages from many sources, one after another. The voltages you can measure are (according to the labels on the switch box): HALL SP. the Hall voltage, measured between the contacts C and D on the sample (see figure 3) THERMOSP. the thermocouple voltage OHMSK SP. the Ohmic voltage along the current direction, measured between the contacts A and B on the sample (see figure 3)
STRÖM the voltage measured over a 10Ω resistor, use for the determination of the current The same box is additionally equipped with a switch for changing the current direction (below the voltage switching knob). The magnetic field is provided by an electromagnet. The electromagnet is supported by a stabilized DC-voltage aggregate. Set the current through the magnet coils to be 0.75 A. The current from the power supply should be controlled by the additional ampere meter for higher accuracy. The current must be maintained at a constant level throughout the experiment. The magnitude of the magnetic field can be determined with a Gauss meter. Exercise 10 Why must the current be maintained at a constant level? Liquid nitrogen (LN 2 ) will be used for cooling the sample. You will find a special insulating container for LN 2. It has a form of interconnected tubes, one thin and one thick. The container is attached to a foundation that can be slowly raised or lowered. The container is to be pre-cooled with a bit of LN 2, and filled with liquid nitrogen just a moment before the use. The thin tube is designed to fit between the poles of the magnet. When the container is raised, the copper cylinder is surrounded and filled by the liquid nitrogen and the sample is cooled. To warm up the sample, lower the container and the temperature will slowly increase. 6. EXPERIMENTAL TASKS Before you start the experiment, read through the tasks and the notes given below. Prepare a detailed plan of work and consult the measurement procedure with the lab supervisor. 1. Get familiar with the measurement equipment and prepare your workspace (e.g. fill the thermos with ice and let it rest for a moment to get a stable thermocouple s reference). 2. Determine the conductivity of the sample at different temperatures. What do you need to measure? Plot 1 lnσ = lnσ T (logarithm of conductivity versus the reciprocal temperature). 3. Determine the energy gap (band gap) E g. At which temperature can it be determined? 1 4. Plot n= n T 5. Plot 1 μc = μ c T 8
6. Plot ln μ ln μ ( ln ) Note 1: c = c T and determine the temperature dependence of mobility at higher temperatures. Compare your result with the theory. The conductivity of the semiconductor will unfortunately be affected by the magnetic field (magnetoresistance). In order to be able to compensate for this effect, two experiments must be made, both from 77 K to room temperature. In the first run the ohmic voltage is measured without any magnetic field (check that the Hall voltage is zero during this measurement). In the second run, the Hall voltage is determined for a constant magnetic field. Try to perform the measurements at the same temperature points (every 10 K) for both measurement series. Note 2: The sample is very small and it is possible that the Hall voltage contacts C and D are not exactly opposite to each other. Therefore, measuring the Hall voltage over the sample, one must consider that there exists an unwanted ohmic voltage component, a resistive voltage drop along the current direction. Exercise 11 Find a way how to eliminate the unwanted ohmic component from the Hall voltage measurement between the contacts C and D. Illustrate the problem with a right figure. GOOD LUCK! 9