Band Structure and Electrical Conductivity in Semiconductors Amrozia Shaheen, Wasif Zia, Asma Khalid and Muhammad Sabieh Anwar LUMS School of Science and Engineering Tuesday, September, 13, 2011 Semiconductors are one of the technologically most important class of materials. According to the band theory of solids, which is an outcome of quantum mechanics, semiconductors possess a band gap, i.e., there is a range of forbidden energy values for the electrons and holes. In this experiment, we will calculate the energy band gap in the intrinsic region and the temperature dependence of the majority carrier mobility in the extrinsic region. KEYWORDS Semiconductor · intrinsic conduction · extrinsic conduction · energy band gap · conduction band · valence band · conductivity · resistivity · mobility · unijunction transistor · temperature control · low temperature physics Approximate Performance Time 2 weeks. 1 Objectives In this experiment, we will, 1. understand how conductivity in semiconductors depends on carrier concen- tration and mobility, and how these depend on temperature, 2. distinguish between intrinsic and extrinsic temperature regimes and identify the applicable temperature range from an examination of measured data, 3. appreciate and utilize the advantages of the four-probe resistance measure- ment technique, 4. calculate the energy band gap for doped Si and pure Ge, 1 5. calculate the temperature dependent coefficient α of the majority carriers, 6. through experimental realizations, appreciate a physical understanding of the band gap structure of semiconductors. References [1] C. Kittel, \Introduction to Solid State Physics", John Wiley and Sons, (2005), pp. 216-226. [2] S. O. Kasap, \Principles of Electronic Materials and Devices", Boston, McGraw-Hill, (2006), pp. 378-405, 114-122. [3] B. S. Mitchell, \An Introduction to Materials Engineering and Science", New Jersey, John Wiley and Sons, Hoboken, (2004), pp. 550-557. [4] S. M. Sze, \Semiconductor Devices", John wiley and Sons, (2002), pp. 41. [5] J. Chan, \Four-Point Probe Manual", EECS 143 Microfabrication Technol- ogy, (1994), http://www.inst.eecs.berkeley.edu. [6] \Low Level Measurements Handbook", pp. 16-19=3, 50-51=4, http://www.keithley.com. [7] A. Sconza and G. Torzo, \An undergratuduate laboratory experiment for mea- suring the energy gap in semiconductors", Eur. J. Phys. 10 (1989). [8] \CN1500 Series Multi-Zone Ramp and Soak Controller", http://www.omega.com. [9] \Discrete PID controller", http://www.atmel.com. [10] http://allaboutcircuits.com. 2 Theoretical introduction 2.1 Semiconductors The available energies for electrons help us to differentiate between insulators, conductors and semiconductors. In free atoms, discrete energy levels are present, but in solid materials (such as insulators, semiconductors and conductors) the available energy states are so close to one another that they form bands. The band gap is an energy range where no electronic states are present. In insulators, the valence band is separated from the conduction band by a large gap while in good conductors such as metals the valence band overlaps the conduction band. Semiconductors have a small band gap between the valence and conduction bands that allows thermal excitation of electrons from the valence to conduction band. The overall picture is shown in Figure (1). 2 Conduction band Band gap Conduction Conduction Eg E band band Overlap f E E f region Eg f Valence band Electron Energy Valence Valence band band Insulator Semiconductor Metal Figure 1: Simplified diagram of the electronic band structure of insulators, semi- conductors and metals. The position of the Fermi level is when the sample is at absolute zero temperature (0 K). The Fermi level is an important consequence of band theory, the highest occupied quantum state of electrons at absolute zero temperature. The position of the Fermi level relative to the conduction band is an important parameter that contributes to determine the electrical properties of a particular material. The position of the Fermi level position is also indicated in Figure (1). For a semiconductor, the electrical resistivity lies between a conductor and an insulator, i.e., in the range of 103 Siemens/cm to 10−8 S/cm. An externally applied electrical field may change the semiconductor's resistivity. In conductors, current is carried by electrons, whereas in semiconductors, current is carried by the flow of electrons or positively charged holes. Q 1. Explain (or sketch) the temperature dependence of resistance for metals and semiconductors. Why does the resistance of a semiconductor decrease with increasing temperature? 2.2 Intrinsic and extrinsic semiconductors An intrinsic semiconductor is a pure semiconductor having no impurities. In an intrinsic semiconductor, the numbers of excited electrons and holes are equal, i.e., n = p as shown in Figure (2a). An extrinsic semiconductor on the other hand is the one in which doping has been introduced, thus changing the relative number and type of free charge carriers. An extrinsic semiconductor, in which conduction electrons are the majority carriers is called an n-type semiconductor and its band diagram is illustrated in Figure (2b) and the one in which the holes are the majority charge carriers is called a p-type semiconductor and is indicated in Figure (2c). In extrinsic semiconductors, the dopant concentration Nd is much larger than the thermally generated electron- hole pairs ni and is temperature independent at room temperature. Q 2. Why is doping introduced in semiconductors? How does it effect the 3 Conduction band E E Ec c c E f n Ef i E fp E E v v Ev Valence band (a) (b) (c) Figure 2: Energy band diagrams for (a) intrinsic, (b) n-type, and (c) p-type semi- conductors. Ef is the Fermi energy level, and the letters i, n, p indicate intrinsic, n and p-type materials. Ec and Ev are the edges of the conduction and valence bands. conductivity of a semiconductor? 2.3 The ubiquitous role of semiconductor devices Semiconductor devices are the foundation of the electronic industry. Most of these devices can be constructed from a set of building blocks. The first building block is the metal-semiconductor interface as shown in Figure (3a). This interface can be used as a rectifying contact, i.e., the device allows current in one direction as in ohmic contact. By using the rectifying contact as a gate, we can form a MESFET (metal-semiconductor field-effect transistor), an important microwave device. p-type n-type Metal Semiconductor semiconductor semiconductor (a) (b) Oxide Semiconductor Semiconductor Metal Semiconductor a b (c) (d) Figure 3: Basic device building blocks of (a) metal-semiconductor interface, (b) p-n junction, (c) heterojunction interface, (d) metal-oxide-semiconductor structure. The second building block is the p-n junction, a junction of p-type and n-type materials indicated in Figure (3b). The p-n junction is the key compound for numerous semiconductor devices. By combining two p-n junctions, we can form the p-n-p bipolar transistor, and combining three p-n junctions to form a p-n-p-n structure, a switching device called a thyristor can be formed. 4 The third important building block is the heterojunction interface depicted in Fig- ure (3c). It is formed between two dissimilar semiconductors, for example gallium arsenide (GaAs) and aluminium arsenide (AlAs) and is used in band gap engi- neering. Band gap engineering is a useful technique to design new semiconductor devices and materials. Heterojunctions and molecular beam epitaxy (MBE) are the most important techniques in which required band diagrams are devised by continuous band-gap variations. A new generation of devices, ranging from solid- state photomultipliers to resonant tunneling transistors and spin polarized electron sources, is the result of this technique. The fourth building block is the metal-oxide-semiconductor (MOS) structure. It is a combination of a metal-oxide and an oxide-semiconductor interface indicated as in Figure (3d). The MOS structure is used as a gate and the two semiconductor- metal oxide junctions are the source and drain; the result is the MOSFET (MOS field-effect transistor). The MOSFET is the most important component of modern integrated circuits, enabling the integration of millions of devices per chip. 2.4 Conduction in intrinsic semiconductors The process in which thermally or optically excited electrons contribute to the conduction is called intrinsic semiconduction. In the absence of photonic excita- tion, intrinsic semiconduction takes place at temperatures above 0 K as sufficient thermal agitation is required to transfer electrons from the valence band to the conduction band [3]. The total electrical conductivity is the sum of the conductivities of the valence and conduction band carriers, which are holes and electrons, respectively. It can be expressed as σ = neqeµe + nhqhµh; (1) where ne, qe, and µe are the electron's concentration, charge and mobility, and nh, qh, and µh are the hole's concentration, charge and mobility, respectively. The mobility is a quantity that directly relates the drift velocity υd of electrons to the applied electric field E across the material, i.e., υd = µE: (2) In the intrinsic region the number of electrons is equal to the number of holes, so Equation (1) implies that, σ = neqe(µe + µh): (3) The electron density (electrons/volume) in the conduction band is obtained by integrating g(E)f(E)dE (density of states×probability of occupancy of states) from the bottom to top of the conduction band, Z 1 ne = g(E)f(E)dE: (4) Ec 5 E E E E Conduction n = band e ni E Ec c Ef Ef E E v v Valence nv= ni band 0 0.5 1.0 g(E) n (E) and n (E) f(E) e h (a) (b) (c) (d) Figure 4: Band gap structure of an intrinsic semiconductor. (a) Schematic band diagram, (b) density of states g(E), (c) Fermi distribution function f(E), (d) carrier concentration ne(E) and nh(E).