6. THE TEMPERATURE DEPENDANCE OF RESISTANCE 1. Objective Determining temperature coefficient of metal and activation energy of self-conductance of semiconductor sample. 2. Equipment needed Metal and semiconductor samples in electrical owen (experiment stand), experiment control unit (see appendix). NOTE: student must have a personal storage media (floppy disk, memory stick) and an A4 format sheet of paper. 3. Theory A single atom is a quantum mechanical complex where electrons are orbiting at selected energetic levels only. No electron can be located outside these orbits i.e. other energetic levels are forbidden. According to exclusion principle only one electron can occupy selected quantum state. Unexcited (in main or default state) atom has electrons distributed along different energetic levels so that the summar energy of the atom is minimal. When a crystal is formed, distance between separate atoms r reduces up to a balanced value (see figure 6.1) and so forces between atoms increase. Now the crystal constitutes a quantum mechanical system. Due to mutual action allowed energetic levels of separate atoms shift somewhat forming zones of energy levels. This is best described by Pauli exclusion principle. Figure 6. 1 Position of zones on energy axis is similar to position on energy levels if separate atoms -19 forming the crystal. Width of an energy zone is about 1 ev (1eV = 1,6 10 J), distance between energy levels in a zone is about ~10-22 ev. There are so-called forbidden zones between valid energy zones. Highest valid (allowed) energy zone filled with electrons at temperature 0 K (absolute zero) is called a valence band. Conduction band is allocated energetically higher than a valence band and is separated from the latter by a band gap (see figure 6.2). 1
conduction band band gap valence band Figure 6.2 Since electrical properties of a solid are mainly determined by valence electrons, we limit the theoretical part to dealing with them only. Width of the energetic gap W between valence and conduction bands is called activation energy. In essence this is energy needed to convert a valence band electron with highest allowed energy to a free electron (electron of a conduction band). It must be noted that during a process of crystal forming electrons belong to their atoms i.e. they are localised in space with dimensions of about one atom. Only electrons of conduction band are delocalised and can move freely in the whole crystal. Solids are classified by their zone structure as follows: conductors, semiconductors and dielectric materials. In conductors valence band and conduction band partially overlap: ( W = 0 ). A relatively small amount of energy is needed to turn valence band electrons into conduction electrons. Such energy could be obtained from energy of thermal agitation of atoms (about ~ 0,03 ev at room temperature). In case of semiconductors W ~ 1eV. Around absolute zero temperature valence band is filled, no electrons are present in conduction band and semiconductor acts as a dielectric material. At higher temperatures (T > 0 K) electrons can get energy from thermal agitation to jump into conduction band. In dielectric materials W > 3 ev. Valence band is filled and there are no electrons in conduction band (at normal conditions). Conduction property on materials is expressed by conductivity, depending on concentration of free charge carriers and their mobility: σ = e n b, where e is value of charge, n concentration of charge carriers, b mobility. Mobility of charge carriers is defined as speed acquired by them in unitary electric field. In metals valence electrons are conduction electrons and their concentration is nearly independent of ambient temperature changes. That means that change of conductivity depends on change of charge carriers mobility. Reason for this lies in the fact that amplitude of oscillations of atoms (and interactions between free electrons and lattice ions increase) rises together with temperature rise meaning that charge carriers mobility decreases. Summing up: conductivity of metals decreases together with ambient temperature rise. In quite wide temperature region this dependency can be considered linear: 2
R = R 0 (1 + α t), (1) where R 0 is resistance at 0 0 C, t temperature 0 C, α temperature coefficient of resistance 1 1 (pure metals have α = K ). 273 Function (1) is represented in ( R) R 0. t, axis with a straight line with ascent α R0 and free term Let us take a closer look at conduction mechanisms in semiconductors. After acquiring additional energy W, a valence band electron crosses over to conduction band leaving vacancy ( hole ) behind. Hole can be filled by another electron creating a vacancy in another place of the lattice. Process of forming and filling holes in crystal lattice can be considered as movement of positive charge + e. If external electric field is applied to semiconductor, movement of electrons with known direction i.e. electric currrent can be observed in addition to chaotic movement of carriers caused by thermal agitation. Both electrons in conduction band and holes in valence band can have equal responsibility in forming current flow. Semiconductors (i.e. Ge, Si) where free electrons and holes are formed by electrons drifting from valence band to conduction band, are called intrinsic semiconductors. Concentration of free electrons and holes in intrinsic semiconductor can be calculated using Boltzmann distribution law: W 2kT p n = N e =, (2) where p is concentration of holes, n concentration of electrons, N concentration of 23 J W atoms, T absolute temperature, k Boltzmann constant ( 1,38 10 ) and energy K 2 needed to release one charge carrier ( W would be spent forming 2 carriers: electron and hole). It can be noticed from formula (2) that ambient temperature rise causes exponential rise of concentration of charge carriers. That is the reason why decreasing at the same time mobility of charge carriers has negligible effect. Conductivity of a semiconductor can be calculated as follows: W W 2kT 2kT e n ( b+ + b ) = N e ( b+ + b ) e = e σ = σ, where b + and b are mobilities for a hole and an electron, e elementary charge value, σ conductivity at temperature, when all electrons from the valence band can be considered 1 having passed over to conduction band. Since R ~, one can write the temperature σ dependence of resistance as follows: where R is a coefficient of proportionality. Taking a logarithm of formula (3) gives us: W 2kT R R e =, (3) 3
W ln R = ln R +. (4) 2kT Formula (4) displays a linear dependence between 1 ln R and. It follows that plotting T calculated dependence having with a slope W 2k ln R values on y-axis and. W can easily be found knowing graph slope. 1 on x-axis gives a linear graph T Additional data about the experiment stand and data processing guide can be found in appendix. 4. Experimental procedure 1. Ask instructor to give an exact task. 2. Read instructions for using the experimental stand. 3. Print your results and hand to instructor for checking. 4. Save your results electronically as an MS Excel file (see appendix for instructions). 5. Using a spreadsheet program (i.e. MS Excel) plot results on a graph using values of temperature in degrees of Centigrade on x-axis. Values of functions characterising temperature dependence of metal and semiconductor samples R m = f ( t) and R p = f ( t) must be on 2 different y-axis with different resolutions (see appendix for instructions). 6. Use supplied program Lineaarne regressioon for following data processing (see appendix for instructions). 7. Using graph R m = f ( t) characterising temperature dependence R m of metal sample find the value of resistance s temperature dependency coefficient α (see appendix for instructions). 1 8. Using program Lineaarne regressioon Plot graph ln R p = f showing semiconductors T resistance temperature dependence, find graph slope and calculate activation energy W. Note: T = t + 273, 15! 9. Calculate expanded combined measurement uncertainties U c ( α ) and U c ( W ) for temperature dependence coefficient α and activation energy W, using uncertainties for other values found from plotted graphs. 5. Questions and tasks 1. How are energy zones in solid materials formed? 2. What is the difference between conductors, semiconductors and dielectric materials according to zone theory? 3. Which particles are free charge carriers in metals? And in semiconductors? How are they formed? 4. What are resistance, resistivity and conductivity? 5. Define the units of resistance, resistivity and conductivity. 6. Why resistance of metals increases and resistance of semiconductors decreases on ambient temperature rise? 4
7. Define resistance temperature dependence coefficient and activation energy. 8. Why should resistance be measured both on increase and decrease of ambient temperature? 6. Literature 1. Halliday, D., Resnick, R., Walker, J. Fundamentals of Physics. 6th ed. New York, John Wiley & Sons, Inc., 2001, 29-2, 30-1, 32-3. 5
APPENDIX Experiment stand for work Temperature dependence of resistance is described in present appendix. Also instructions for using the stand and processing data with program Lineaarne regressioon (Linear regression) will be given. Note that before starting the experiment student should ask instructor exact task (numbers of metal and semiconductor samples, temperature region and step. Student must have a personal storage media (memory stick) for storing results of his work. 1. The experiment stand The stand consists of a thermostat, sample resistors for investigation, temperature sensor, power supply, analog to digital converter and computer. Simplified diagram of the stand is displayed on figure 1. Constant current generators electronic change-over switch analog-todigital converter computer Program Takistus 1 2 3 heating coil 1 metallic resistance 2 temperature sensor 3 semiconductor resistance thick-walled copper cylinder fan for rapidly cooling Figure 1. Simplified diagram of experiment stand The thermostat consists of a massive copper cylinder with sample resistors and temperature sensor as shown on figure 1. Resistors are fed with stabilised (constant) current so voltage drop on resistor is proportional to its resistance. Analog to digital converter unit converts this value to digital form for a computer. Exact sample for your work is selected by an electronic switch from computer program. Computer is also used for automatic registration of measurement data, saving it and further 6
processing. Thermostat is controlled manually in both directions of temperature change. The device also has a built-in safety switch deactivating heater if temperature in the device raises higher than about 70 C. Figure 2 displays thermostat with its main parts. Figure 3 displays thermostat's control panel with description of control organs. Figure 2. Thermostat Figure 3. Thermostat control panel All electrical connections are pre-made, no need to connect anything. Student must be familiar to Windows operating system. 2. Using the experiment stand 1. Switch on the thermostat (power switch on figure 3). 2. Switch on computer. Use credentials: user name tudeng and password tudeng for logging in. These will be needed for printing results. 3. Start program Takistus double-clicking displayed here icon on computers desktop: 7
Main window of program Takistus openes: This window lets you to set up your experiment (select samples for measuring etc.) 4. Fill in the fields with your data (first name, surname, group code): Starting and ending time of the experiment are recorded automatically. 5. Select given by instructor numbers of metal and semiconductor samples from drop-down menus. Example here displays m 2 and p 4. 6. Choose measurement mode. You have 3 choices: a) each new measurement is taken after given time interval s b) each new measurement is taken after temperature has changed by given step value C, c) manual control manual. Chose option given by instructor. If measurement control by temperature C is selected, you must also indicate direction of temperature change. Selecting starts experiment with temperature decreasing and + in increasing mode. Selection box temperature direction will be available only 8
if mode C is selected. Enter temperature step into field located left from mode symbol C like it is shown in example. If mode with control by time interval s was chosen, enter required time step into field left from mode symbol s. Example on previous page displays a time step of 2s chosen. Now you have entered all required data and experiment can be started. Let's assume that we have a case of increasing temperature. If the thermostat is cold, there can not be any other options anyway since it can not be cooled below ambient temperature. 7. Set heater power knob to position 8 if not instructed other way by instructor and turn heater power on (rightmost switch on figure 3). Red control light will be illuminated and temperature of investigated sample will start increasing. Immediately click on button Start in programs main window. ONLY NOW you started recording of measurement data. Computer starts a subroutine that records data read from experiment units analog to digital converter and already converted to resistance value into corresponding rows of data table together with measurement number and temperature. When temperature reaches to value given by instructor (this value is usually lower than 80 C) switch heater off and start ventilator (see figures 2 and 3 for corresponding switches). If thermostat's temperature reaches to about 80 100 C, automatics cuts power to heater but ventilator must always be started manually! If in temperature controlled mode C you must now reverse temperature change direction now (from the drop-down menu left to the START button). Recording of measurement data continues and will not be stopped until you click the Stop button. Click it when temperature has decreased to a value given by instructor. 8. Switch off ventilator. One case of successful experiment data is displayed. In this example data was collected only during increase of temperature. 9
If something went wrong, click the Reset button and start the experiment again. 9. Now the data you collected must be saved. Click File tab. A standard Save As dialog box opens. Save your data to drive C's folder Üliõpilased (Students) and subfolder with your group s code (for example YAFB-11). Give your file name First name Surname. If subfolder of your group does not exit, create it now. Save your file also to your personal storage device to be on the safe side. Print your data by selecting File Print. Close program window. 10
3. Processing results 3.1. Preparing data for linear regression program Start program Excel from computers desktop. Then select File Open. From a dialog window that opens find a field Files of type and select All files (*.*). The field looks in displays folders and files. Find your data file and open it. A window Text Import Wizard opens now. Select Delimited, Start import at row: 1, File origin: Windows (ANSI). Press Next button, select Delimiters Tab and press Finish. Excel worksheet with your data is displayed. If some columns contain symbols like #######, simply drag this column wider with mouse. File type you opened is.ttt. Remember to save it as an Excel worksheet via File Save As and selecting Microsoft Excel Workbook (*.xls) as file type. Next we have to prepare data on worksheet for linear regression. Linear dependence of experimental data is assumed for conducting the linear regression, parameters of dependence will be found with method of least squares (sum of squares of deviations of data from straight trendline is required to be minimal). It is known that (see main part of instructions, formulas (1) and (3)) that the temperature dependence of metal samples resistance can be calculated as follows: (In last formula kt R = R 0 (1 + α t) same for semiconductor:: R = 2 R e. R is not measured at 0 C.) Presented formulas can be written in another form: And R = α R t + 0 R 0 W 1 ln R = + ln R 2k T As we can see, first case expresses a linear dependence between R, t and second formula similar dependence between ln R and T 1. It means that for approximation experimental data with a straight trendline following pairs of data will be needed: t (temperature C) and R (metal samples resistance), 1 (inverse value of temperature) and ln R (natural logarithm from semiconducting T samples resistance). First pairs you already have, others must be calculated. You can let Excel calculate values automatically. Enter a formula =1/(B12+273.15) into worksheets field E12 and =LN(D12) into field F12. Select both fields and position cursor to lower right corner of selection: cursor changes shape to +. Press mouse left button, keep it pressed and drag down until there are rows with data. This copies entered formulas to all needed fields and automatically changed formulas' pointers. Release button fields will be filled with calculated results. Now your data is prepared for linear regression. Do not close the worksheet yet!. W (1a) (3a) 11
3.2. Performing linear regression and displaying results graphically A program in form of Excel worksheet, called Lineaarne regressioon.xls quickly and easily finds parameters of wanted regression line together with parameters uncertainties. Make following selections for running the program (opening a new Excel worksheet): File Open Desktop Lineaarne regressioon.xls Open. (If you can not find file Lineaarne regressioon.xls from desktop, ask the instructor). Worksheet Lineaarne regressioon.xls consists of two sheets: Andmed (Data) and Graafik (Graph). First sheet has columns for entering values of x and y, fields for results regression graph calculations together with uncertainties and (in hidden form) all needed formulas. Note that you have actually two different Excel worksheets open: old one with your experiment data and new with formulas for Lineaarne regressioon. You can activate one or another using keyboard combination Ctrl+Tab. In order to find a regression graph for your data copy all x values from worksheet containing your data to clipboard (Ctrl+c). Activate the regression worksheet and select column with x values. Press mouse right button and select from displayed menu Paste Special Paste Values. Now you have entered the x values to worksheet Andmed (data). Repeat mentioned procedures with y values. A maximum of x-y pairs can be entered. Enter into corresponding field number of your data pairs and press Enter. If data was correct, results will be displayed in corresponding fields. On worksheet Graafik (graph) points corresponding to the pairs x-y will be displayed in x-y axis, approximated regression line and its parameters (copied from worksheet Andmed) will also be displayed. Correct starting points and/or intervals for X and Y axis if needed. Now perform last check of worksheet Graafik with File Print Preview. Print the graph, if everything is correct. Such processing must be carried out both for metal and semiconductor sample data sets. Now you should have two graphs with all relevant data. Acquired experimental information lets you calculate coefficient of temperature dependence α for resistance of metal sample, activation energy of semiconductor sample W and their uncertainties. Calculate α and W, using formulas (1a) and (3a). Present W in electronvolts (ev). Derive formulas for uncertainties independently. Plot a graph with temperature C on common x-axis and two y-axis (one with resistance of metal and the other with resistance of semiconductor. You can let Excel do your worksheet. First activate 3 columns (without headers): temperature, metal and semiconductor. Then: Insert Chart Chart Wizard opens. Select Chart type: XY (Scatter) and Chart subtype: upper window (sets of data as point diagrams). Next, select Series in: Columns. Next Title, enter in window Chart title: title of your graph, Value (X) axis: name of x-axis and unit, Value (Y) axis: name of y-axis and unit. Do not close window Chart Wizard. 12
Gridlines, select Value (X) axis Major gridlines and Value (Y) axis Major gridlines. There should be a coordinate grid on your graph now. Next, select Place chart As new sheet. Finish Activate one of graphs on the diagram by left-clicking on it. Press right button of mouse and select from drop-down menu Format Data Series (using left mouse button). In dialog-window that opened select Axis then Secondary Axis and finally OK. This adds another y-axis in suitable form. Activate point diagrams and draw approximation trendlines with command Add Trendline found from menu popping up from under mouse's right button. First approximation should use linear and second one exponential function. Print resulting graphs. 13