University of Toronto Department of Electrical and Computer Engineering ECE 330F SEMICONDUCTOR PHYSICS Eng. Annex 305 Experiment # 1 RESISTIVITY AND BAND GAP OF GERMANIUM TA: Iraklis Nikolalakos OBJECTIVE 1. Determine the resistivity and carrier concentration of a germanium sample at room temperature with the four-contact method. 2. Determine the bandgap of germanium through observation of the conductivity temperature dependence. EQUIPMENT 1 n-type sample of Ge in holder (Type A) 1 p-type sample of Ge in holder (Type B) 1 Breadboard 2 Digital Multimeters (Fluke 8000A) 1 Oscilloscope (HP 1200A) 1 dc Power Supply 1 Hot plate 1 thermometer -20 o C to +150 o C 1 Retort stand, 250 ml. beaker with glycerol (m.w. 92.09), clamps and dry ice PREPARATION (see Appendix for background information) 1. Express the resistivity, ρ, and carrier concentration, n, of an n-type semiconductor (See Fig. 1) in terms of the parameters I 21, V 34, x, W, t, and µ n (the electron mobility). 2. Express the carrier concentration of an intrinsic semiconductor (n = p = n i ) in terms of the band gap and absolute temperature T. Prove that if µ n and µ p vary as T -3/2, (due to lattice scattering at high temperature) the conductivity varies with temperature according to ln(σ) = A - (E g /2k B T) where E g is the energy gap, k B is Boltzmann's constant, T is the absolute temperature, and A is a constant. 3. Sketch a layout of the experimental equipment and sample probe connections for carrying out the experiment in Part 1. PART 1: FOUR-CONTACT RESISTIVITY METHOD The germanium sample has six ohmic contacts as shown in Fig. 1. The contact pairs 1,2 and 3,4 enable the resistivity of the sample to be measured reliably in a manner essentially identical to the -1-
"four point probe" method widely used in semiconductor device characterization. Contact points 5 and 6 are not used. Resistivity measurements are frequently inaccurate when only two probes are used to measure both the current and voltage. The non-zero current at the contacts is rectifying and the contact is non-ohmic. The four-contact method eliminates this problem by applying a known current through contacts 1 and 2 while measuring the potential difference between contacts 3 and 4. If the voltmeter is perfect (takes no current) it would not matter if contacts 3 and 4 were rectifying, and therefore the measurement of V 34 together with a knowledge of the mobility of the carriers and the dimensions x, W, and t, enables the carrier concentration to be calculated. 6 5 I 23 2 1 W t x 3 V 4 Fig. 1 Germanium sample with dimensions t = 0.11 ± 0.005 cm, W = 0.11 ± 0.005 cm, and x = 0.6 ± 0.1 cm. Contacts 5 and 6 are not used. Experiment: For one of the n-type or p-type samples: 1. Measure the voltage V 34 using a high-impedance differential voltmeter while the currents between contacts 1 and 2 is varied from 1.0 ma to +1.0 ma in steps of 0.2 ma. Find the mean value of V 34 /I. It is recommended that a 4.7 kω resistor be placed in series with the contact pair 1 and 2 to facilitate current control. 2. Assume for the n-sample that µ n = 3900 ± 10% cm 2 /V. s and for the p-sample that µ p = 1900 ± 10% cm 2 /V. s. Calculate the room temperature resistivity and carrier concentration. Estimate the uncertainty (probable error) of your values. PART II: BAND GAP The temperature dependence of conductivity of the sample is used to determine the bandgap of germanium. The sample is assumed to behave intrinsically (n = p = n i ). Experiment: 1. Use the method of Part 1, but with a constant current of I 12 = 0.5mA, and study the variation of conductivity with temperature over the range +140 o C to -5 o C at approximately 5 o or 10 o C intervals, depending on how quickly the conductivity varies. Warning: do not exceed 150 o C higher temperature can destroy the device. -2-
2. Plot a graph of ln (σ) versus 1/T. Over what temperature range does the sample behave as an intrinsic semiconductor? Examine the slope of ln (σ) in this range and explain the underlying physics. Determine the band gap, E g, from your graph in 1. OPTIONAL PROBLEM The measurement of the resistivity of a semiconductor slice (probe 1 to 2) is often made using four identically spaced point contact probes the four-point probe method as illustrated in Figure 2. I V I 1 2 3 4 t s s s Fig. 2 A thin semiconductor sample with four point contacts. The slice is assumed to extend infinitely in the plane. Assuming that t << s and that the slice is infinite in area, prove that the resistivity is given by: πtv ρ = I ln( 2) Hint: The current flow lines are the same as the electric field lines produced by two line charges (see Attwood, S.S., "Electric and Magnetic Fields", 3rd Ed., page 85, J. Wiley, 1949). An analogue for Gauss s law for electric field lines can be used for current density. The solution for the case t >> s (i.e. a thick sample compared to the probe spacing) is discussed in reference 3. REFERENCES 1. Lin, H.C., "Integrated Electronics", pp. 105-108, Holden-Day, 1967. 2. Smits, F.M., "Measurement of Sheet Resistivities with the Four-Point Probe", Bell System Technical Journal, 37, 711, 1958. 3. Allison, J., "Electronic Engineering Materials and Devices", McGraw Hill, 1971, pp. 140-141. 4. Bar-Lev, A., "Semiconductors and Electronic Devices", Prentice Hall, 1979, pp. 24-25, and pp. 38-41. TEXTBOOK REFERENCES -3-
1. Streetman, B.G., "Solid State Electronic Devices", 4th Ed. 1995, Ch.3. 2. Neaman., D.A., Semiconductor Physics and Devicess, Irwin 1992. APPENDIX OVERVIEW OF SEMICONDUCTOR ELECTRICAL PROPERTIES Carrier Transport in Semiconductors: The conductivity, σ, of a semiconductor depends heavily on the density of free carriers in the crystal. The natural breaking of crystal bonds solely by the sample temperature k B T provides both unbound electrons (negative charges) and broken bonds (positively charged holes ), which freely move through the crystal under the forces of electric field. The density of free electrons is n and these carriers are known as conduction band electrons. The density of holes is p and these carriers are known as valence band holes and treated as positive charges. All other electrons are assumed frozen in place in the bonds or the atoms, and do not play a role in electrical conductivity. At zero temperature, all valence electrons become tightly trapped at the rigid bond sites and the semiconductor becomes an insulator. As temperature increases, the density of broken bonds rises exponentially, increasing the conductivity by orders of magnitude. The crystal conductivity is expressed by σ = 1/ρ = e (µ n n + µ p p) (1) The mobility of electrons, µ n, and holes, µ p, are a measure of how far the free carriers can be accelerated by an external electric field before colliding with the thermally vibrating crystal atoms (phonons) or being scattered by the Coulomb forces of dopant (impurity) ions. These collisions impede the mobility, causing resistive heating of the crystal. The mobility is weakly temperature dependent, scaling as T -3/2 for lattice (phonon) scattering at high temperature and T +3/2 for ionizedimpurity scattering a low temperature. While mobility has a small temperature effect on the conductivity, the carrier densities (#/cm 3 ), n for electrons and p for holes, can most significantly affect the semiconductor conductivity. The n and p values can change values by orders of magnitude through two sources: - the controlled addition of impurity dopants into the semiconductor, producing either n-type (n>p; excess electrons) or p-type (p>n; excess holes) semiconductors will lock the n and p values i.e. will be insensitive to temperature up to a maximum temperature. This is an extrinsic semiconductor. - when impurity levels are low (or dopants are compensated), the density of electron and hole carriers fall into equilibrium (n = p = n i ) and become controlled by the crystal thermal energy, k B T. The semiconductor is then said to be intrinsic. The intrinsic carrier density is highly dependent on temperature, following: n i = N v N c exp E gap 2k B T (2) The strong temperature dependence in the exponential term originates with the laws of Fermi quantum statistics that governs the probability that a bond is broken at temperature T. The outcome is a powerful effect on the conductivity of a semiconductor. The energy gap, E gap, in the equation is fixed for a particular crystal type. Its value arises for electrons with energy (or wavevector k) that satisfy the crystal Bragg-reflection condition, i.e. λ el /2 = a, the crystal lattice spacing. A gap in allowed energy states arises for such electrons (or holes), imposing constraints on the carrier motion and therefore the physical properties (electric, thermal, optical) of the semiconductor. Note that small changes in E gap from one material to another will have a dramatic effect on the sample s conductivity -4-
even when all samples are at the same temperature. The physics of the energy bandgap is described in Neamen s book and will be discussed later in the course. The remaining terms in Eq. 2, N c and N v, are the effective density of states for electrons in the conduction band and holes in the valence band, respectively. These values are also slightly temperature dependent, their physics originating with the density of allowed energy states for electrons in a three-dimensional quantum well. Theoretical values are provided by N c = 2 2πm n * k B T h 2 3 2 and N v = 2 2πm* p k B T h 2 where m n * and m p * are the effective mass of electrons in the conduction band and holes in the valence band, respectively. These mass values differ from the electron mass by several factors and result from additional forces on the electrons and holes near the Bragg condition of a particular crystal. Their values are tabulated for several semiconductor materials in standard Semiconductor Physics or Devices textbooks. The conductivity of an intrinsic semiconductor (n = p = n i ) therefore encompasses many physical processes that are highly temperature dependent. In an extrinsic semiconductor, the n and p values are fixed by the doping levels and do not show nearly as strong a temperature dependence the exception is at a high temperature when n i becomes equivalent to the bigger of n or p. The electron and hole density in an extrinsic semiconducotor also follow a balance such that np = n i 2. A study of temperature dependence of semiconductor resistance provides a means to determine the density of broken bonds (n i ), the bandgap energy, and the mobility properties of a semiconductor. 3 2 (3) -5-