Review of Elementary Solid-State Physics

Transcription

1 APPNDIX ON Review of lementary Solid-State Physics A1.1 A QUANTUM MCHANICS PRIMR A1.1.1 Introduction In quantum mechanics, the properties and motion of particles are defined in terms of a wave (or state) function, ; its magnitude squared gives the probability density of finding a particle at some point in time in a volume element dv. Or, put another way, the density of particles at some point in space is proportional to. Note that dv = 1, for properly normalized state functions, since the probability of the particle being somewhere is unity. In our case we are interested in both electrons and photons as particles. For photons this description is roughly equivalent to standard electromagnetic theory where the wavefunction is analogous to a normalized electric field. Maxwell s equations give the description of photon fields. In this appendix we shall focus more specifically on the properties of electrons. In quantum mechanics, measurements are limited in accuracy by the uncertainty principle, x p x /2, where p x is the momentum in the x-direction. (In electromagnetic theory the equivalent statement is that x k x 1/2.) The expected (or mean) value of some observation is calculated by operating on the wavefunction with the operator, A, corresponding to the observable, a. The operation to obtain Diode Lasers and Photonic Integrated Circuits, Second dition. Larry A. Coldren, Scott W. Corzine, and Milan L. Mašanović John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc. 509

2 510 RVIW OF LMNTARY SOLID-STAT PHYSICS the mean value is analogous to a standard weighted average, a = A dv, (A1.1) where a is a possible observation of the operator A. In many cases the operator simply multiplies the observable variable, in others it is more complex, such as momentum, p, whereitis i. The motion of particles is governed by Schrödinger s equation, 2 2m 2 + V = i t, (A1.2) where m is the particle s (e.g., electron) mass, V is the potential energy operator (same as observable), and ( 2 /2m) 2 is the kinetic energy operator (= p 2 /2m). Together, these two form the overall energy operator, the so-called Hamiltonian. The state function can be expressed as the product of space-dependent and timedependent factors, (r, t) = ψ(r)w(t). If we substitute into q. (A1.2) and divide by ψw, we obtain a function on the left that only depends on r and a function on the right that only depends on t. Thus, to be valid for all r and t, each side must equal a constant, : 2 2m From this we immediately have 2 ψ ψ + V = i w w t =. (A1.3) w(t) = Ce i(/ )t, (A1.4) from which we can identify = ω, where ω is the radian frequency of oscillation. For the time-independent part, 2 2m 2 ψ + Vψ = ψ. (A1.5) The general solution for a uniform potential can be written as the sum of two counterpropagating plane waves, ψ(r) = Ae ikz + Be ikz, (A1.6) where k 2 = 2m ( V ), 2 (A1.7) is found by substituting back into q. (A1.5).

3 A1.1 A QUANTUM MCHANICS PRIMR 511 A1.1.2 Potential Wells and Bound lectrons lectrons are confined by some potential depression in most situations. The most fundamental example is the atom, where electrons are bound by the confining potential of the positively charged nucleus. For the simple case of the hydrogen atom, V (r) = q 2 /[4πε 0 r], and analytic solutions to Schrödinger s time-independent q. (A1.5) can be found. However, for atoms with higher atomic numbers and many electrons, only numerical solutions are possible. Nevertheless, the electron always experiences some sort of confining potential. When solids are formed from these atoms, the more weakly bound electrons near the exterior of the atom are significantly influenced by the attractive potential of neighboring atoms. In fact, in covalently bonded solids, the outer valence electrons are shared by many atoms, and they develop wavefunctions that extend throughout the crystal. In such cases, the details of the original atomic confining potential are lost. Thus, we shall not dwell on that problem unduly. Rather, we shall investigate the properties of an electron in a simple rectangular potential well to develop the concepts of confined wavefunctions and discrete energy levels common to atoms. It will later be shown that by coupling together a series of such wells, a periodic potential is formed that leads to electronic properties very similar to those in real crystals. Thus, we can learn much about the properties of electrons in solids by taking this course. As is well known, one of the key results is that electrons in solids can behave much like free electrons with plane wave solutions and a parabolic k relationship as illustrated by q. (A1.7). However, they appear to have an effective mass, m, that is different from the free electron mass. Also, this effective mass approximation is usually limited to relatively low kinetic energies. Finally, we shall consider quantum-confined structures that include heterostructures to form much larger potential wells than for a single atom. Nevertheless, the mathematics is very similar, and we will be able to apply much of what we develop in this section. First, consider the one-dimensional potential well of width l shown in Fig. A1.1. The simplest method of solution is to recognize that there are three separate regions of uniform potential, where the solution to Schrödinger s equation will have the form of q. (A1.6). Then, if we assume that the effective electron mass is the same in all regions, we can develop a complete wavefunction by requiring that the value and slope of the constituent solutions in each of the three regions match at the boundaries. That is, we would not expect any discontinuity in the probability density function. Looking for bound solutions, for which < V 0, we can rewrite the general solution q. (A1.6) in each region. In the central region II, { A cos kx (symmetric solutions) ψ II = A sin kx (antisymmetric solutions), (A1.8) where k 2 = 2m/ 2. In region III, ψ III = Be γ x, (A1.9)

4 512 RVIW OF LMNTARY SOLID-STAT PHYSICS V 0 l/2 l/2 x I II III FIGUR A1.1: One-dimensional potential well for electron confinement. where γ 2 = 2m(V 0 )/ 2. In region I, ψ I = Be γ x, but by symmetry, we only need to use the single boundary condition at x = l/2 between regions II and III. At x = l/2, we have that ψ II = ψ III and ψ II = ψ III. For the symmetric solutions, this gives A cos ( ) kl = Be γ l/2, (A1.10a) 2 and Ak sin ( ) kl = Bγ e γ l/2. (A1.10b) 2 Dividing q. (A1.10a) by q. (A1.10b), we obtain the characteristic equation, k tan Similarly, for the antisymmetric solutions, we obtain ( ) kl = γ. (A1.11) 2 ( kl k tan 2 π ) = γ, (A1.12) 2 where cot x = tan(x π/2) has been used to illuminate the similarity between the symmetric and antisymmetric characteristic equations. The electron energy,, appears on both sides of these characteristic equations via k and γ, implying that only discrete values of will satisfy the requirement that the wavefunction and its derivative be continuous across the boundaries. Because the tangent function is periodic, multiple solutions can be found for, leading to a discrete set of wavefunctions that satisfy the boundary conditions.

5 A1.1 A QUANTUM MCHANICS PRIMR 513 V 0 3 n = 3 2 n = 2 1 n = 1 l/2 l/2 x FIGUR A1.2: nergy levels and wavefunctions of one-dimensional potential well. Three bound solutions illustrated. Figure A1.2 shows the first few wavefunctions drawn schematically on their respective energy levels over the potential well for reference. These clearly represent bound solutions. There are solutions with > V 0 but they are not bound, and their wavefunctions extend to ±. An interesting property of all the solutions is that they must be orthogonal. That is, if we multiply one wavefunction by the complex conjugate of another and integrate over all space, the integral must be zero. If the wavefunctions are normalized so that the integral of the product of a wavefunction times its own complex conjugate is unity, then the wavefunctions would be orthonormal. To determine the bound solutions, we need to solve the characteristic equations. For infinitely large V 0, such that the wavefunction goes to zero at the boundaries (i.e., ψ(l/2) = 0), the characteristic equation for both symmetric and antisymmetric cases becomes simply kl 2 = nπ 2, n = 1, 2, 3,... (A1.13) where odd (even) quantum numbers correspond to symmetric (antisymmetric) states. The corresponding discrete energy levels in terms of the quantum numbers are where n = n 2 1, (A1.14) 1 = 2 k 2 1 2m = 2 π 2 2ml 2 = 3.76(m 0/m)(100 Å/l) 2 in mev.

6 514 RVIW OF LMNTARY SOLID-STAT PHYSICS When V 0 is reduced from infinity, the discrete energies can still be found using (A1.14), however, the quantum numbers in this case are no longer simple integers, but are real numbers that we will refer to as n QW. For example, if V 0 = 251,the infinite-barrier integer quantum numbers n = 1, 2, 3, 4, 5 become n QW = 0.886, 1.77, 2.65, 3.51, To calculate n QW for an arbitrary V 0, we need to solve the characteristic equations given in qs. (A1.11) and (A1.12). Using q. (A1.14), combined with the definitions for k and γ given below qs. (A1.8) and (A1.9), the characteristic equations can be conveniently normalized: [ π ] tan 2 n QW = 1 n QW [n 2 max n2 QW ]1/2 (symmetric) (A1.15) [ π ] tan 2 (n QW 1) = 1 [nmax 2 n n2 QW ]1/2 (antisymmetric) (A1.16) QW where n QW n 1 and n max V 0 1. (A1.17) These equations can be solved graphically by plotting both the left-hand side (LHS) and the right-hand side (RHS) as a function of n QW. Figure A1.3 illustrates this procedure for four different values of V 0. RHS LHS n 2 max = n = 2 n = 3 n = 4 n = 5 n = 6 V 0 n = n QW FIGUR A1.3: Graphical solution to qs. (A1.15) and (A1.16). The intersections between the LHS and RHS of the equations yield the possible values of n QW for a given n max (or equivalently V 0 ). The odd (even) quantum numbers displayed next to each tangent curve correspond to the LHS of the symmetric (antisymmetric) characteristic equation.

7 A1.1 A QUANTUM MCHANICS PRIMR 515 Note that only a finite set of quantum numbers exist for a given potential barrier, V 0. The normalized variable, n max when rounded up to the nearest integer, yields the largest number of bound states possible for a given V 0. For example, with V 0 = 31, from q. (A1.17), we find that n max = Thus, only two bound states are possible under these circumstances. This is perhaps demonstrated more clearly by plotting the possible n QW as a continuous function of n max. Figure A1.4 gives all possible solutions for n max 6 (which covers nearly all practical ranges of interest). Note that all quantum numbers approach their integer limit as n max increases toward infinity. In addition, the quantum numbers cease to satisfy the equations (indicated by the open circles) when a given quantum number approaches the integer value of the next lowest state. The lowest quantum number can be n QW n QW (n 1) n max n = 1 state n = 2 n = 3 n = 4 n = 5 n = n max FIGUR A1.4: Plot of quantum numbers as a function of the maximum allowed quantum number that is determined by the potential height, V 0. The quantum numbers are related to V 0 and through q. (A1.17). The lower plot gives a close-up view of the curves (which have been shifted vertically to fit on the same scale).

8 516 RVIW OF LMNTARY SOLID-STAT PHYSICS approximated to within ±1% using the following formula: n QW 2 π tan 1 [ n max ( n max + 1 )]. (A1.18) A1.2 LMNTS OF SOLID-STAT PHYSICS A1.2.1 lectrons in Crystals and nergy Bands lectrons in crystals experience a periodic potential originating from the regularly spaced wells at the lattice ions. Figure A1.5 gives a schematic picture along one dimension of such a lattice. As predicted in Chapter 1, when N A atoms are coupled in such a manner, each atomic energy level of the constituent atoms splits into a band of N A discrete levels. However, this splitting is only significant for the uppermost energy levels where the two atoms interact. There are several approaches that have been applied to solve this problem. The Kronig Penney model approximates the actual periodic potential of Fig. A1.5 by a square wave potential, then uses the single rectangular well solution above as a starting point. However, the result is a complex transcendental equation that must be solved numerically. A second approach, which provides better closed-form analytic solutions, is the coupled-mode approach of Feynman et al. [1]. For accuracy some fairly complex functions need to be evaluated, but by leaving them in general form, we can still get a good picture of the nature of the solutions. The first step is to go back to Schrödinger s equation and consider a possible general solution for a perturbed system, such as the atom that has been placed into a crystal. The isolated atom had a set of orthonormal wavefunction solutions just as we obtained for the rectangular potential well. When we perturb the original Potential well of isolated atom y k y j Net confining potential a Lattice ions x FIGUR A1.5: Schematic of net potential variation along a one-dimensional crystal lattice. Wavefunctions of nominally bound (ψ j ) and free (ψ k ) electron states are illustrated.

9 A1.2 LMNTS OF SOLID-STAT PHYSICS 517 potential, a new set of orthonormal wavefunctions will exist. But now it may be impossible to solve Schrödinger s equation. It is common to use a superposition of the original set of orthonormal functions to express the new solutions. We shall use this kind of normal mode expansion later when we discuss optical solutions. In the present case we let = j w j (t)ψ j (r), (A1.19) plug into q. (A1.2), multiply by ψk, and integrate. Then, we have j w j (t) ψ k H ψ j dv = i j dw j dt ψ k ψ j dv, (A1.20) where the Hamiltonian, H = [( 2 /2m) 2 + V ], in which V includes the perturbation. Since the original basis functions are orthonormal, the last integral is zero 1 unless j = k. Using the shorthand notation, H kj ψk H ψ j dv, (A1.21) we finally have i dw k dt = j H kj w j. (A1.22) This is the desired coupled-mode equation, which is independent of the spatial variables. It illustrates that the probability density will flow back and forth among the various original states as a function of time to form the new states. Note that with k = j in q. (A1.21) we have the equation to determine the expected value of energy for that original wavefunction the eigenvalue that we have been evaluating previously. Thus, the diagonal terms in the H kj matrix are these energy eigenvalues for the respective unperturbed states. The off-diagonal terms represent the coupling strength between the various states. They determine the magnitude of the energy splitting experienced by some original state. It is also important to realize that for most of what we are doing here we do not have to know the actual form of the wavefunctions or even the magnitudes of the matrix elements, H kj. xperimental measurements are usually used to determine the actual values. Our first example is that of coupling just two identical atoms together. For sufficiently weak coupling, we can approximate the effect on a particular state by using only the basis function for that state from each atom in the summation. (Clearly, for vanishingly small coupling, these give the exact solution.) Then, q. (A1.22)

10 518 RVIW OF LMNTARY SOLID-STAT PHYSICS Wavefunctions of isolated atoms a x k 1 x k x k+1 (x k a) (x k + a) Lattice ions x FIGUR A1.6: One-dimensional lattice of coupled atoms to derive energy bands using the coupled-mode approach. can be expanded into two coupled-mode equations: 1 i dw 1 dt i dw 2 dt = H 11 w 1 + H 12 w 2, = H 21 w 1 + H 22 w 2. (A1.23) Letting the energy of the state in question H 11 = H 22 = 0, the coupling energy H 12 = H 21 =, and then, assuming solutions w j (t) = C j exp( it/ ) and plugging into q. (A1.23), we obtain a characteristic equation from which we must have = 0 ±. (A1.24) Thus, the original energy level at 0 for the isolated atom has split into two levels spaced equally on either side by the magnitude of the off-diagonal matrix element,. This same process for N A atoms leads to N A levels spaced symmetrically about the original level. Now we are ready to illustrate how energy bands are formed when a large number of atoms are coupled together in a crystal. First we consider a simple onedimensional crystal. Figure A1.6 illustrates a row of atoms spaced by a distance a, similar to the situation of Fig. A1.5. Our first approximation will be to neglect the perturbation from all atoms except nearest neighbors. Then we can consider a general atom, the kth atom, which can represent every atom in this long chain. From q. (A1.22), taking H 11 = H kk = 1 and H 12 = H kk ± 1 =, i dw k dt = w k w k + w k+1. (A1.25) Again letting w j (t) = C j exp( it/ ) and plugging into q. (A1.25), we obtain a characteristic expression, C k = 1 C k + [C k 1 C k+1 ]. (A1.26) 1 Actually, in some important cases of interest, ψ k and ψ j may include basis functions of laterally displaced atoms to better approximate the perturbed solution. Thus, for some terms in the summation the integral is only small rather than identically zero.

11 A1.2 LMNTS OF SOLID-STAT PHYSICS 519 However, since the subscript k corresponds to the general lattice location, x k, and the neighbors are at x k ± a, we can rewrite q. (A1.26) letting C k C (x k ) and C k±1 C (x k ± a). Then, we have a difference equation in terms of the spatial variable x. This is solved by letting C (x k ) = K exp(ikx k ). Finally plugging in this assumed solution, or, e ikx k = 1 e ikx k + [e ik(x k a) + e ik(x k + a) ], = cos ka. (A1.27) quation (A1.27) indicates that in this infinite one-dimensional crystal a continuum of energy values between = 1 ± 2 is allowed. This is the familiar energy band that solid-state and semiconductor engineers are always referring to. (A later section of this appendix will remind us that for finite crystals, the discrete levels in any real situation really are very closely spaced.) This same development of bands happens for all of the higher-lying energy levels when atoms are bonded together to form crystals. Thus, the next higherlying band at energy 2 also splits into a band due to nearest-neighbor coupling energy. Therefore, it provides a new band with = cos ka, where in direct bandgap semiconductors, the sign of is reversed. Also, the overlap of wavefunctions is larger for the higher lying energy levels. Thus, according to q. (A1.21), the coupling energy is larger, and the bands become wider. Figure A1.7 illustrates these two bands. As indicated, one period of the plot is sometimes referred to as a Brillouin zone. Since the curves repeat themselves for larger k-values, we usually need concern ourselves only with the first Brillouin zone. In semiconductors all states of all bands up to the valence band are full, and in the next higher-lying band, called the conduction band, they are empty at T = 0K. We could imagine that Fig. A1.7 represents the conduction and valence bands of a direct bandgap semiconductor such as GaAs or InP. The potentials affecting electrons in such semiconductors are a little more complicated than described by this simple example, in which only nearest-neighbor interactions are considered. So, the k plots are not perfect sine waves. Also, in these materials the valence band actually divides into two bands called the light-hole and heavy-hole bands. These originate because of the asymmetric wavefunctions involved, and the difference in overlap that can occur for different relative orientations when q. (A1.21) is evaluated. For a real three-dimensional crystal with lattice constants a, b, and c, the same procedures can be carried out using a three-dimensional version of q. (A1.22) with coupling coefficients x, y, and z, and a three-dimensional envelope wavefunction, w(x, y, z, t) = e it/ e i(k x x+k y y+k z z), (A1.28)

12 520 RVIW OF LMNTARY SOLID-STAT PHYSICS Δ Δ 2p a p a 0 p a 2p a k x Brillouin zone FIGUR A1.7: nergy bands created by a one-dimensional chain of coupled atoms. Two bands resulting from two original discrete states are shown. to obtain = x cos k x a + 2 y cos k y b + 2 z cos k z c. (A1.29) The real situation is still more complex than the first-order calculation resulting in q. (A1.29). Figure A1.8 illustrates the actual band structure for both GaAs and InP along the 100 and 111 directions. A1.2.2 ffective Mass Near the top of the valence band and near the bottom of the conduction band it is sometimes possible to approximate the shape of these k extrema by parabolas. In these cases the concept of an effective mass is useful, and simple expressions for the density of states are possible. However, the concept of an effective mass has also been extended to limited regions within nonparabolic bands where the parabolic approximation is still valid. To determine an expression for the effective mass and show that the parabolic band is desired, we follow a semiclassical approach in which we calculate the acceleration of an electron in a solid under the force of an applied electric field. The force q on a particle may be classically expressed as the time rate of change of its momentum, p. Quantum mechanically p = k. Thus, the force is F = q = dk dt. (A1.30)

13 A1.2 LMNTS OF SOLID-STAT PHYSICS 521 g 6 ev ev hh lh so GaAs c g = 1.42 ev v Δ so = 0.33 ev 6 ev c 0.52 ev hh lh so InP g = 1.35 ev v Δ so = 0.11 ev L Λ Γ Δ X 12 L Λ Γ Δ X <111> <100> <111> <100> FIGUR A1.8: Band structure of GaAs and InP. The conduction band as well as the heavyhole, light-hole, and split-off valence bands are labeled by, c, hh, lh, and so, respectively. (Reprinted, by permission, from K. J. beling, Integrated Optoelectronics, Springer-Verlag, 1993.) The velocity of particles is defined by their group velocity, v g = dω/dk = (1/ ) d/dk, which shows the proportionality of velocity to the slope of the k characteristic. Since the acceleration, acc., is the time derivative of the velocity, we can write acc. = dv g dt = dv g dk dk dt = 1 d 2 dk dk 2 dt. (A1.31) Dividing q. (A1.30) by (A1.31), and defining an effective mass, m = F/acc., we obtain m = 2 d 2 /dk 2. (A1.32) Thus, for parabolic bands, as observed for uniform potentials (e.g., q. (A1.7)), the electron will move much like a free particle, but with an effective mass, m, related to the curvature of the band. For nonparabolic bands, m is not constant and the local slope and curvature of the k relationship must be used to obtain the velocity and acceleration of a particle with energy.

14 522 RVIW OF LMNTARY SOLID-STAT PHYSICS A1.2.3 Density of States Using a Free-lectron (ffective Mass) Theory We just learned above that an electron in a crystal can behave much like a free electron moving in a region of uniform potential if it is at a point on the k diagram that is parabolic. This is a remarkable result, since we know that the potential within a crystal is very nonuniform. Nevertheless, this revelation allows us to treat some very complex problems. For example, if we consider a crystal of finite dimensions, d x, d y, d z, we can more or less ignore the crystal lattice potential which is periodic on the scale of the lattice constant a, provided that d j a (Fig. A1.9). But, we must use a different effective mass as determined by the curvature of the k diagram. By considering electron energies near band extrema, where the k curve tends to be parabolic, we can now consider reusing some of the same physics that we developed in Section A1.1.2 for electrons in very simple potential wells that had uniform potential regions. That is, we can now find the states in finite pieces of crystal or pieces with potential wells created by double heterostructures as described in Chapter 1. The simplest case is when the potential barriers are large so that we can assume that the bound wavefunctions go to zero at the boundaries. Then, from qs. (A1.13) and (A1.14), we have = 2 k 2 2m = 2 2m [k x 2 + k y 2 + k z 2 ], (A1.33) where we have included all three dimensions for completeness and assumed that the effective mass is the same in all directions. Applying the boundary conditions for a large barrier, k j d j = n j π, = 2 π 2 2m [ (nx ) 2 + d x ( ny d y ) 2 ( ) ] 2 nz +. (A1.34) d z ffective potential well Actual potential d 2 Lattice ions a d 2 x FIGUR A1.9: Potential plot for crystal (or quantum-well) of thickness d. The dashed well is an approximation to the actual potential.

15 A1.2 LMNTS OF SOLID-STAT PHYSICS 523 From q. (A1.34) we note that we have an energy state for each (n x, n y, n z ) set of quantum numbers. To determine the size of the energy spacing between states, we can evaluate the coefficient, 2 π 2 /2m 0 = 376 mev-nm 2, where we have used the free electron mass rather than the effective mass. In GaAs the electron mass, m = 0.067m 0, so we should use 15(376) = 5640 mev-nm 2 for the coefficient in the conduction band. From this we can see that the energy separation between states is quite small. For example, for a cube with d j = 1 μm, the difference in energy between the first two states, (211) (111) mev. Since kt 26 mev at room temperature, we see that this energy difference is less than one thousandth of a kt. On the other hand, for a cube with dimensions d j 10 nm, this energy difference is 170 mev, or more than 6kT at room temperature. From the above, we conclude that for dimensions d j 1 μm, quantum effects are not going to be very noticeable at room temperature, and the k diagram can be treated as describing a continuum of states. We shall refer to this as the bulk regime. On the other hand for d j < 100 nm, the discreteness of the energy levels indicated in q. (A1.34) must be considered. We shall refer to this as the quantum-confined regime. ven though the states may be very closely spaced in the so-called bulk regime, we still need to be able to count them to determine the carrier density and the energy to which they would have to be filled for a given carrier density. In the smaller structures, we again need an effective method of counting states. The method commonly used is to define a density of states, ρ, which when integrated over some range gives the number of states in that range. The density of states can be expressed in terms of a number of variables (e.g.,, p, ork) in a number of different coordinate systems. If N s is the number of states up to some point, we can generally state that N s (u) = V u 0 ρ(u) du, (A1.35) where u is the desired variable and V is the volume. Once we have this definition, we can then state that ρ(u) du = 1 V dn s(u). (A1.36) It should be realized that ρ(u) du can be defined and used regardless of the size regime in which we find ourselves. For the smaller structures, we find that it contains discontinuities and even impulse functions, but it is still a good function. To determine ρ(u) du for the various cases of interest, we follow a standard procedure: (1) determine the number of states by calculating the volume in state or n-space, N s (n); (2) substitute for the desired variable, n = f (u), which gives N s (u); and (3) apply q. (A1.36) to get the desired ρ(u) du. A few examples are useful for future reference. For the first example, we consider bulk dimensions, a spherical coordinate system, and energy as the variable. Spherical coordinates imply that we are considering

16 524 RVIW OF LMNTARY SOLID-STAT PHYSICS a spherical state space. quation (A1.34) is written in rectangular coordinates, but as stated after it, each set of quantum numbers, or each volume element in n-space, represents a state that can be occupied by an electron. Figure A1.10 illustrates this n-space. The first step is to calculate the volume N s (n) N s (n) = 4 3 πn (A1.37) The first factor just gives the standard expression for volume. However, we must multiply by 2, since two states actually exist at each allowed energy because of spin degeneracy. And only positive quantum numbers are allowed, so we have the factor of 1 8. Now, for the second step, we use q. (A1.34) in spherical coordinates, (identical to q. (A1.14)), solve for n in terms of, and plug back into q. (A1.37): ( 2m d 2 ) 3/2. (A1.38) N s () = π 3 2 π 2 For the third step, we now apply q. (A1.36), and use V = d 3, without loss of generality ρ() d = 1 2π 2 [ 2m ] 3/2 1/2 d. (A1.39) 2 n z n y n x FIGUR A1.10: State space or n-space in spherical coordinates. ach block corresponds to a particular state and has unit dimensions.

17 A1.2 LMNTS OF SOLID-STAT PHYSICS 525 quation (A1.39) is our final result. It will be of much use in calculating carrier densities, gain, and other quantities associated with bulk active regions in lasers. For the second example, we consider bulk dimensions, rectangular coordinates, and momentum as the variable. Then, the volume in n-space is, N s (n) = 2n x n y n z. The momentum in each direction, j,is p j = k j = πn j d j = hn j 2d j. (A1.40) Solving for n x, n y, and n z, and plugging into N s (n), we get N s (p x, p y, p z ) = 2 ( ) 2 3 p x p y p z (d x d y d z ). (A1.41) h Applying q. (A1.36) to (A1.41) gives the desired density of states, ( ) 2 3 ρ(p x, p y, p z )dp x dp y dp z = 2 dp x dp y dp z. (A1.42) h The density of all states with a given momentum, ρ(p), can be obtained from this result by setting dp x dp y dp z = 4πp 2 dp/8. The factor of 8 is required because (A1.42) defines the density of standing wave states, ρ SW (p x, p y, p z ), which do not distinguish between positive and negative values of momentum. Hence, the density is limited to the first quadrant. We can also define a density of plane wave states, ρ PW (p x, p y, p z ), which can travel in any direction. Using periodic boundary conditions, we have k j = 2πn j /d j instead of k j = πn j /d j, but now we consider both positive and negative values of n j as unique states. In three dimensions then, ρ PW (p x, p y, p z ) = ρ SW (p x, p y, p z )/2 3, and is distributed over all quadrants of momentum space. So for ρ PW (p x, p y, p z ), we can set dp x dp y dp z = 4πp 2 dp. In either case, we obtain ρ PW (p) = ρ SW (p) = 8πp 2 /h 3, from which we also obtain ρ(k) = (k/π) 2 (since ρ(k) dk = ρ(p) dp). For the third example, we consider a quantum well (small dimension in one direction), cylindrical coordinates, and energy as the variable. We shall let d x be the small dimension. As always the energies are given by q. (A1.34), but we need to develop a density of states for the y z plane, which will be summed for each n x. Figure A1.11a gives a plot of the energy,, relative to the k x k z plane for a quantum-well region using q. (A1.34). The lowest-lying states for n x = 1 and 2 are labeled by the quantum numbers (n x, n y, n z ). Since d x is small, there are no states near k = 0. Figure A1.11b replots the energy versus k z (or k y )ina two-dimensional graph for clarity. In part (a) only positive k x and k z are shown. Now to determine the density of states for this quantum well, we start with the disk of Fig. A1.12 and determine N s (n). The volume of the unit height disk in the first quadrant, multiplied by 2 for spin, is N s (n yz ) = π 2 (n2 y + n2 z ). (A1.43)

18 526 RVIW OF LMNTARY SOLID-STAT PHYSICS c 111 c p/d x 2p/d x k x v k y, k z k z (a) (b) FIGUR A1.11: (a) Three-dimensional k plot showing discrete jumps in k x due to small d x. (b) Projection perpendicular to the k z axis. n x n z n y FIGUR A1.12: Two-dimensional state space that occurs for each n x in a quantum well. Using the y and z terms from q. (A1.34) for yz and (ny 2 + n2 z ), letting d 2 = (dy 2 + d z 2 ), and assuming that the effective masses are the same in y and z, N s ( yz ) = m d 2 π 2 yz. (A1.44) Again, we apply q. (A1.36) and recognize that this density of states is for n x = 1. Thus, generally ρ() = 1 d x n x m π 2 H ( n x ), (for QW) (A1.45)

19 RADING LIST 527 r() 3m d x ph 2 Bulk 2m d x ph 2 m d x ph 2 Quantum well FIGUR A1.13: Density of states for an infinite-barrier quantum well and bulk material. If the barrier is not infinite, the quantum-well energies decrease slightly. If desired, the density of state plateaus can be decreased by using an effective d x = dx (a different one for each state) so that the extrema continue to intersect the bulk characteristic. where n x and are related by q. (A1.14) with l = d x, and H ( nx ) is the Heaviside unit step function. Figure A1.13 compares the densities of states for the bulk and quantum-well active regions, qs. (A1.39) and (A1.45), respectively. As can be seen the bulk curve forms an envelope for the steps of the quantum-well case, which correspond to the energies where the quantum numbers are (n x,1,1). RFRNCS 1. Feynman RP, Leighton RB, Sands ML. The Feynman lectures on physics. New York: Addison-Wesley; RADING LIST Solymar L, Walsh D. Lectures on the electrical properties of materials. New York: Oxford University Press; Chapters 3 to 8. Kittel C. Introduction to solid state physics. New York: Wiley; Chapters 7 and 8. Kroemer H. Quantum mechanics. nglewood Cliffs, NJ: Prentice Hall; Chapter 7.