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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 nen 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 11 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