# Fall 2004 Ali Shakouri

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1 University of California at Santa Cruz Jack Baskin School of Engineering Electrical Engineering Department EE-145L: Properties of Materials Laboratory Lab 5b: Temperature Dependence of Semiconductor Conductivity Fall 2004 Ali Shakouri 1.0 Learning Objectives After successfully completing this laboratory assignment, including the assigned reading, the lab questions, and any required reports, the students will be able to: 1. Determine the bandgap energy for a semiconductor from a linear regression of measured data in the intrinsic region. 2. Explain why semiconductor conductivity changes with temperature. 3. Distinguish between intrinsic and extrinsic regimes and identify the applicable temperature range from an examination of measured data. 4. Express the mathematical relationships between carrier concentration and temperature in the intrinsic, extrinsic and ionization regimes. 5. Express the functional relationship between mobility and temperature in the intrinsic. Extrinsic and ionization regimes. 2.0 Reference: 1.Chapters 5.3 of S.O Kasap. Electrical Engineering Materials and Devices. I 2 nd. Ed Chapter 4.3 of B.Streetman, Solid-State Electronic Devices, 4 th Edition, Theory 3.1 Conductivity of a Semiconductor The conductivity of a semiconductor is given by: σ = q (µ n n + µ P p) (1) where µ n and µ P refer to the mobilities of the electrons and holes, and n and p refer to the density of electrons and holes, respectively. Recall that in a doped semiconductor, majority carriers greatly outnumber minority carriers, so the Equation 1 can be reduced to a single term involving the majority carrier. 3.2 Effects of Temperature and Doping on Mobility of a Semiconductor Conductivity of a material is determined by two factors: the concentration of free carriers available to conduct current and their mobility (or freedom to move). In a semiconductor, both mobility and carrier concentration are temperature dependent. Thus it is important to view the conductivity as a function of temperature which is expressed by: σ = q [µ n (T) n(t) + µ P (T) p(t)] (2) There are two basic types of scattering mechanisms that influence the mobility of electrons and holes: lattice scattering and impurity scattering. We have already discussed lattice scattering in the context of metals; we know that lattice vibrations cause the mobility to decrease with increasing temperature. Lab 5: Semiconductor conductivity 1

2 However, the mobility of the carriers in a semiconductor is also influenced by the presence of charged impurities. Impurity scattering is caused by crystal defects such as ionized impurities. At lower temperatures, carriers move more slowly, so there is more time for them to interact with charged impurities. As a result, as the temperature decreases, impurity scattering increases and the mobility decreases. This is just the opposite of the effect of lattice scattering. The total mobility then is the sum of the lattice-scattering mobility and the impurityscattering mobility. Figure 1 shows how the total mobility has a temperature at which it is a maximum. The approximate temperature dependence of mobility due to lattice scattering is T -3/2, while the temperature dependence of mobility due to impurity scattering is T +3/2 (see Figure 1). In practice impurity scattering is typically only seen at very low temperatures. In the temperature range we will measure, only the influence of lattice scattering will be expected. µ (cm 2 /Vs) (log scale) T 3/2 T -3/2 Lattice scattering Impurity Scattering T(K) (log scale) Fig. 1 Approximate temperature dependence of mobility with both lattice and impurity scattering 3.3 Temperature Dependence of Carrier Concentration The carrier concentration in a semiconductor is also affected by temperature. The intrinsic carrier concentration is governed by: 2πkT 3 / 2 n i = 2 [ ] 3 / 4 E g (m 2 n m p ) exp (3) h 2kT where the exponential temperature dependence dominates n i (T). To determine the total carrier concentration, we must also consider space-charge neutrality: 2 n n(t) = N+ D (T) N - i ( T ) A + n( T ) and 2 n p(t) = N - i ( T ) A(T) - N+ D (T) + (4) p( T ) For a doped semiconductor, the temperature dependence of electron concentration can be seen in Figure2. At very low temperature (large 1/T), negligible intrinsic electron-holepairs (EHPs) exist (n i is very small), and the donor electrons are bound to the donor atoms. This is known as the ionization (or freeze-out) region. As the temperature is raised, increased ionization occurs and at about 100K all of the donor atoms are ionized, at which point the carrier concentration is determined by doping. The region where every Lab 5: Semiconductor conductivity 2

3 available dopant has been ionized is called the extrinsic (or saturation) region. In this region, an increase in temperature produces no increase in carrier concentration. Referring to Equation 4, this is the region where N + D (T) = N D, N - A(T) = N A, and n i (T) << N D N A. At high temperatures, the thermally generated intrinsic carriers outnumber the dopants (n i > N D N A ). In this intrinsic region, carrier concentration increases with temperature as shown in Equation 3 because n i has become the dominant term of Equation Intrinsic n o (cm -1 ) Extrinsic Ionization /T (K -1 ) Figure 2. Carrier Concentration vs. reciprocal temperature for silicon doped with donors/cm Temperature Dependence of Conductivity for a Semiconductor Remember that Equation (1) showed that conductivity depends on both carrier concentration and mobility, so there are a variety of possible temperature dependencies for conductivity. For instance, at fairly low temperature (less than 200 K), the dominant 3 / 2 scattering mechanism might be impurity scattering (µ α T ) while the carrier concentration is determined by extrinsic doping (n = N + D ), therefore, conductivity would be seen to increase with temperature (σ α T ). Other possibilities, depending on the material, doping, and temperature will show different temperature dependence of conductivity. One particularly interesting case occurs at high temperatures (above 400k or higher) when carrier concentration is intrinsic (Equation 4) and mobility is dominated 3 / 2 by lattice scattering ( µ lattice α T ). In such cases, the conductivity can easily be shown to vary with temperature as: 3/ 2 Lab 5: Semiconductor conductivity 3

4 σ exp E g 2kT (5) In this case, conductivity depends only on the semiconductor bandgap and the temperature. In this temperature range, measured conductivity data can be used to determine the semiconductor bandgap energy, E g. 4.0 Experimental Procedure In this experiment we measure the resistivity of an unknown sample as a function of temperature, from about 80 K up to 540 K, using a cryostat with a heating stage. A cryostat is an enclosure, which can be cooled to very low temperature. The system is very easy to use and should work perfectly if you follow a few simple steps. Below is an outline to how the apparatus works, what you should expect to measure, and how to do it. The results will show the saturation region (extrinsic) and the intrinsic region of conductivity. The temperature will not be low enough to observer the ionization region. 4.1 The cryostat The apparatus consists of a micro-miniature refrigerator (inside the small metal vacuum enclosure which contains a window through which you can see the semiconductor sample), a tank of high-pressure nitrogen for the refrigeration fluid, and a vacuum pump and manifold. The apparatus uses the Joule-Thompson expansion of a high-pressure gas to achieve enough cooling to liquefy the gas in a small reservoir underneath the sample mount. The refrigerator is made up of a laminated set of glass slides, each of which has micro-channels etched into it, providing a serpentine path for the gas to proceed through a heat exchanger. The sample is mounted on a cold mount at the end of the refrigerator with the thermal grease, which makes excellent thermal contact between the sample and the refrigerator. The sample can be cooled to 80 K from room temperature in about 10 minutes. The refrigerator also has a heater built in (thin wires) with which we can heat the sample or control the temperature stability. 4.2 Procedure Please use the instruction manual located next to each station: 1. Close all valves on manifold. (See drawing located near instrument). 2. Switch on backing pump. (It may be on already.) 3. Connect vacuum meter to power. 4. Connect temperature controller to power. 5. Open valve to begin pumping on cryostat). 6. Watch the reading on vacuum meter, the pressure will begin to fall. 7. When the pressure drops to about 20 millitorr, pass pure N 2 at 500 psi to purge the system for about 10 minutes. 8. While purging, make the electrical connections. Connect the leads from the two ends of the sample to the ohm meter. Lab 5: Semiconductor conductivity 4

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