Reading: The first two readings cover the questions to bands and quasi-electrons/holes. See also problem 4. General Questions: 1. What is the main difference between a metal and a semiconductor or insulator, in terms of band structure?. What do we mean when we talk about a hole in a semiconductor? What are its properties? 3. Why does a good solar cell design make a poor light emitter? 4. How can band-gap engineering improve the efficiency of a solar cell? 5. What is the difference between spontaneous and stimulated emission of light? 6. Why does putting a quantum well in the middle of a pn-junction improve its ability to emit light by stimulated emission? 7. Why does modulation doping improve the speed of a field effect transistor? 8. Describe qualitatively how to start with a -dimensional electron gas (DEG), for example at a GaAs/AlAs interface or Si/SiO interface, and pattern it to a 1D or 0D quantum structure. 9. Why does confining conduction to a narrow 1D channel lead to quantized conductance? 10. Why are some carbon nanotubes metallic and some semiconducting? Problems: 1. Effective Mass Show that the one dimensional effective mass is given by the inverse curvture of the band structure E(k), 1 m * 1 d E(k) h dk Start from the basic relationships: F dp dt h dk for a quantum system with momentum p ħk and show that this dt definition of the effective mass m* lets us also use the standard form from Phys 11, F dp m*a, where the acceleration a is the time rate of change of the group velocity dt v g 1 de(k). Now F is just the external force, and the internal forces from the variation h dk in the energy with position (or wavevector) are included in the effective mass. 1
. PN junction Fields In class, we drew S-shaped fields through a pn junction. In ρ this problem, you will derive that shape, which comes from the field due to ionized dopants. When a donor (an atom with one more electron than the host) loses its +qn d electron to the conduction band, a positive ion is left behind. When an acceptor loses a hole (gains an electron), qn a a negative ion is left behind. In the junction region, W a electrons from donors end up on acceptors, leaving Wd x positive and negative ions on the n- and p- sides of the junction, respectively. We will model this as a region of uniform charge and constant width. Assume that a p-n junction is characterized by two sharply defined charge regions of charge density ρ as shown, where N a and N d are the density of acceptors and donors, respectively, and the space charge depletion widths are W a and W d. In this central region, the free carrier densities are orders of magnitude smaller than in the bulk of the device since the electrons and holes recombine. a) For x<0, you can write the Poisson equation (MKS units) as V x qn a where ε ε dielectric constant (κε o). We are using MKS so the answers will be in Volts, and not statvolts or esu. Integrate this equation and a similar one for x>0 to show that the barrier voltage is given by: V b V 1 + V q ε N W ( a a + N d W d ). Note that from charge neutrality, W d N d W a N a, so that it may also be written as: V b qw a N a 1 + N a ε. b) Sketch the potential as a function of x through the junction. Justify your linear, quadratic, or whatever you find, dependence of the potential, as well as which side is positive or negative. Show that the width of depletion region i (j is the other region) is given by: N d W i εv b qn i 1 + N i N j c) Show that in the case where N a «N d, the width of the depletion region to a good approximation may be written as W tot εμv b, where σ conductivity and μ σ mobility. These are related through σ i n i eμ i where i denotes electrons or holes. d) The junction capacitance is defined by C j dq dq dv b dw dw dv b. In the practical case for which one side of the junction is very lightly doped while the other is heavily doped (N a «N d ), show the capacitance may be written: C j εσ μv b
e) Si-doped GaAs (n-type) with carrier density ~ 10 16 cm -3 has room temperature mobility ~ 8000 cm /V-sec, for σ ~ 10 mho/cm, ε 13.ε o, V b ~ 0.7 V. Find the width of the depletion region and the junction capacitance in the case N a «N d. f) Sketch the bands for a double heterostructure pn-junction, with a central region of undoped GaAs (band gap 1.44 ev) of width ¼ what you found in (e) with n-type AlGaAs (n ~ 10 16 cm -3 ) on one side (band gap 1.75 ev) and p-type AlGaAs (p ~ 10 15 cm -3 ) on the other. Explain your reasoning. 3. D Subbands in a MOSFET The contact plane between two semiconductors with different band gaps, or a semiconductor and an insulator, gives rise to an offset in the bands at the interface. When an electric field is applied, electrons can be trapped by a triangular well at the interface. The potential energy for an electron in the conduction band may be approximated as V (z > 0) eez and V (z < 0), where the interface is at x 0. The wavefunction must go to zero for x < 0, and for x > 0 may be separated as ψ (x, y, z) u(z)e ik x x e ik y y, where u(z) satisfies the differential equation h d u + V (z)u εu. m dz The exact solution to the triangle well is an Airy function, but the variational trial function u(z) Azexp(-az) gives a reasonable approximation to the ground state energy.* a) Using the trial wavefunction, show that the energy is ε h a m + 3eE a. b) Show the energy has a minimum when a 3eEm h c) Show that ε min 1.89 h m 1/ 3 3eE / 3. In the exact solution, the factor of 1.89 is replaced by 1.78. As the electric field increases, the extent of the wavefunction in the z direction decreases. The function u(z) defines a surface conduction channel on the low energy side of the junction. The various eigenvalues ε of u(z) define what are known as electric subbands. The eigenfunctions cannot carry current in the z direction, but do carry a surface channel current in the xy plane. d) Explain how changing the applied electric field can move the ground state through the Fermi level and turn the interface layer on or off for conduction. 1/ 3. 3
4. Carbon Nanostructures Read any one of the thousands of papers on nanotubes, graphene or buckyballs. Choose one figure from these papers (that includes data or the results of a theoretical calculation). Explain the figure what happened to generate the result (i.e., what was the technique, sample geometry, etc.), what the result is, and what the authors say it means. Include a copy of the figure and the citatioin of the reference from which you took it. HINT: go to UW libraries home page, and select Articles and Research Databases. The two I find most useful are Web of Science and Engineering Village. Enter a few terms into the topic line (e.g. nanotube and transport and blockade). In Engineering Village you can ask it to restrict the result to General or Review papers. 4
Addendum: HINTS and definitions you might not have seen before: The Poisson equation is a differential form of Gauss Law: In Phys 1, you learned E da q encl 1 ρ d 3 x ε o ε o. There is a math theorem that relates the integral of a field closed surface enclosed volume over a surface to the integral of its divergence inside the volume, so that E da E d 3 x. Since the electric field is the gradient of the electric potential, we closed surface enclosed volume can rewrite the divergence as E V. In this case, the potential varies only in the direction perpendicular to the junction, and the charge density equals the density of dopants time the charge on each dopant, so we end up with V x qn a ε The group velocity is a standard definition for waves that have a small range of frequency and wavevector. For a standard wave of the form cos(kx ωt), the phase velocity (speed at which a constant phase front moves) v phase ω k, and the group velocity (speed at which information moves) v g dω. Making the correlation E hω, for an electron wave this becomes dk v g 1 de h dk. * A variational wave function is one that captures the basic physics of the situation (in this case, u 0 at z 0 and decays exponentially away from the interface) but which has simpler math than the true solution. The optimal form of the variational wave function is found by minimizing the energy for a wave function of that form. The expecation value of the energy ε u *εudz u *udz standard integral that you may find useful:. To solve, substitute for εu from the differential equation. There is a x n e μx dx n!μ n 1 0 5