Chapter 14Application III: The k · p Method for Bandstructure 14.1 Introduction ... 14.2 Perturbation theory For k · p theory, we will need some elementary results of time-independent perturbation theory. We list the results here. They can be found in any textbook of quantum mechanics. There are two levels of the solution - the first is degenerate perturbation theory, and the next non-degenerate theory. For k · p theory, the non-degenerate case is important. Assume that we have solved the Schrodinger equation for a particular potential with Hamiltonian H (0) H (0) |ni = "(0) n |ni, (14.1) (0) and obtained the eigenfunctions |ni and eigenvalues "n . Now let us see how the solution set (eigenfunction, eigenvalue) changes for a potential that di↵ers from the one we have solved for by a small amount. Denote the new Hamiltonian by H = H (0) + W , where W is the perturbation. If the eigenvalues are non-degenerate, the first order energy correction is given by "(1) n ⇡ hn|W |ni, 85 (14.2) Chapter 14. Application III: The k · p Method for Bandstructure 86 and there is no correction (to first order, in the absence of non-diagonal matrix elements) in the eigenfunction. This is just the diagonal matrix element of the perturbing potential. The second order correction arises from from non-diagonal terms; the energy correction is given by "(2) n ⇡ X |hn|W |mi|2 (0) m6=n "n (0) "m , (14.3) where |mi are all other eigenfunctions. The correction to the eigenfunction is |pi = |ni + X hm|W |ni |mi. "n "m (14.4) m6=n Thus, the total perturbed energy is given by "n ⇡ "(0) n + "(1) n + (0) "(2) n = "n + hn|W |ni + X |hn|W |mi|2 (0) m6=n "n (0) "m , (14.5) and the perturbed eigenfunction is given by the the equation before last. Some more facts will have a direct impact on bandstructure calculation by k · p method. The total second-order perturbation (2) "n arises due to the interaction between di↵erent eigenvalues. Whether interaction between states occurs or not is determined by the matrix elements hn|W |mi; if it vanishes, there is no interaction. Whether the states vanish or not can typically be quickly inferred by invoking the symmetry properties of the eigenfunctions and the perturbing potential W . (0) Let us look at the e↵ect of interaction of a state with energy "n with all other eignes(0) (0) tates. Interacting states with energies "m higher than "n will lower the energy "n by contributing a negative term; i.e., they push the energy down. Similarly, states with (0) energies lower than "n will push it up. The magnitude of interaction scales inversely with the di↵erence in energies; so, the strongest interaction is with the nearest energy state. This is all the basic results that we need for k · p theory. The last homework that needs to be done is a familiarity with the consequences of symmetry, which we briefly cover now. 14.3 Symmetry A brief look at the symmetry properties of the eigenfunctions would greatly simplify solving the final problem, and greatly enhance our understanding of the evolution of bandstructure. First, we start by looking at the energy eigenvalues of the individual orbital (14. In other words. the conduction-band minimum state is no longer |Si-like. The s-orbital is spherical. and hence symmetric along all axes. the Bloch lattice-function uc (k.and p-orbitals of atomic systems. However.e. r px = z z y x x (14. Application III: The k · p Method for Bandstructure 87 atoms that constitute the semiconductor crystal.i.one positive. one is typically worried about the bandstructure of the conduction and valence bands only. the individual atoms have the outermost (valence) electrons in in s. r) = uc (0. r) possesses the same symmetry properties as a |Si state1 . For semiconductors. |Xi. |Zi. The crystal develops its own bandstructure with gaps and allowed bands. |Y i. it has mixed |Si and p-characteristics.orbital Figure 14. have the symmetry of p-orbitals.and p-type orbitals. the valence electrons hybridize into sp3 orbitals that lead to tetrahedral bonding. it has spherical symmetry.orbital y x pz .8) z y s .6) x = 3 sin ✓ cos r y p py = = 3 sin ✓ sin r z p pz = = 3 cos ✓. 1 If the semiconductor has indirect bandgap. For direct-gap semiconductors.orbital (14. All semiconductors have tetrahedral bonds that have sp3 hybridization.1: s. Once we put the atoms in a crystal. for states near the conduction-band minimum (k = 0). The symmetry (or geometric) properties of these orbitals are made most clear by looking at their angular parts - s=1 p (14. In general.. and the other negative. the px orbital has two lobes .9) z y x px . It turns out that the states near the band-edges behave very much like the the |Si and the three p-type states that they had when they were individual atoms.Chapter 14. on the other hand. The spherical s-state and the p-type lobes are depicted in Figure 2. Let us denote the states by |Si. The states at the valence band maxima for all bands.7) py . the p-orbitals are antisymmetric or odd along the direction they are oriented . . integrated over all unit cell is zero. uy . since all of them are linear combinations of p-orbitals. with i = x. Figure 3 denotes these properties. So. however. let us say that we have the following Bloch lattice-functions that possess the symmetry of the s. even the conduction band minima states have some amount of p-like nature mixed into the s-like state. and it is immediately clear that hus |p|ui i = hus |pi |ui i ⌘ P. pz -type states . y.11) . Application III: The k · p Method for Bandstructure conduction band Direct E gap |S> Indirect gap u|S>+v|P> More indirect -> more |P> k bandgap valence bands 88 Linear combination of p-type states of the form a|X> + b|Y> + c|Z> HH LH SO Figure 14. we make the direct connection that uc is the same as us . (14. Next.e. the p-states are odd along one axis and even along two others. The momentum operator is written out as p = i~(x@/@x+y@/@y +z@/@z). we see that the Bloch lattice-functions retain much of the symmetries that the atomic orbitals possess. we look at the momentum-matrix element. Since we do not know the linear combinations of ux . &uz that form the valence bands yet. &uz . huc |p|uv i between the conduction and valence bands.us . So. whereas the Bloch lattice-functions of the valence bands usv are linear combinations of ux .Chapter 14. For indirect-gap semiconductors on the other hand. For direct-gap semiconductors. uy . This is easily seen by looking at the orbitals in Figure 2. the valence band states may be written as linear combinations of p-like orbitals. z. (14.10) i. we can immediately say that the matrix element between the conduction band state and any valence band state is huc |uv i = 0. uy .. &uz . let us look at the momentum-matrix elements hus |p|ui i. To put it in more mathematical form. the conduction band state at k = 0 is s-like. Without even knowing the exact nature of the Bloch lattice-functions. Then. Note that it does not matter which valence band we are talking about. py . The valence band states are linear combinations of p-like orbitals.2: The typical bandstructure of semiconductors. the product. it vanishes.and px . the s-states are even. ux . we are in the (enviable!) position of understanding k · p theory with ease. With a detailed look at perturbation theory and symmetry properties. (14. r) is the Bloch lattice function. we can see that the momentum operator along any axis makes the odd-function even. 6 1 uLH. m0 2m0 | {z } W where u(k.15) (14. Phys. r).4 k · p theory Substituting the Bloch wavefunction into Schrodinger equation.16) (14. 2 1 p (ux + iuy 2uz )." = p (ux + iuy + uz ).13) (14.14) (14. 3 1 uSO. from Figure 2.18) and note that hus |p|ui i = 0. but with two extra terms [H 0 + ~ ~2 k 2 k·p+ ]u(k. We also note that hus |pi |uj i = 0. it can be shown2 that the valence band states may be written as the following extremely simple linear combinations uHH.12) To go into a little bit of detail. since it is the derivative of that function. Again. (14. Application III: The k · p Method for Bandstructure 89 i.e. r) = "(k)u(k. 2 1 = p (ux iuy ). we obtain a equation similar to the Schrodinger equation. The matrix-element is defined to be the constant P ." = uHH.# = p (ux iuy uz ) 3 uLH.." = (14. 6 1 uSO. 14.20) . it does not vanish.Chapter 14. 31 888 (1986) (14. Rev B. (i 6= j). 2 Broido and Sham.17) (14.# = p (ux iuy + 2uz ).19) which in words means that Bloch lattice-functions of opposite spins do not interact.# 1 p (ux + iuy ). Using the two results summarized in the last section. 2m0 "n (0) "m (0) m0 m6=n (14. So the Bloch lattice functions are u(0. Let us look at the eigenvalues at k = 0. "HH (0) = 0. and obtained the various eigenvalues (call then "n (0)) for the corresponding eigenstates (call them |ni). Let us denote the bandgap of the (direct-gap) semiconductor other states far away: neglect by Eg . |C> ~ |s> Conduction band minimum +Eg k other states far away: neglect 0 Three p-like states 1 LH. where Eg is the (direct) bandgap.heavy hole (|HHi). The corresponding eigenvalues for a cubic crystal are given by ("c (0) = +Eg . they are all degenerate. and of the three valence bands . at the point for a direct-gap semicon- ductor. In the absence of spin-orbit interaction. Application III: The k · p Method for Bandstructure 14. We will return to spin- orbit interaction later.. "SO (0) = 0) respectively.Chapter 14. "LH (0) = 0.that of the conduction band (|ci) at k = 0. r). We assume that we have solved the eigenvalue problem for k = 0. r)|k · p|um (0.21) . the three valence bands are degenerate at k = 0. In the absence of spin-orbit interaction.5 90 No spin-orbit interaction Let us first look at k · p theory without spin-orbit interaction. light hole (|LHi) and the split-o↵ band (|SOi). 2 HH states k=0 Bandstructure for small k Figure 14. We look at only four eigenvalues .e. i. Bandstructure for small k E s-like state.3: k · p bandstructure in the absence of spin-orbit coupling. r)i|2 + 2 . we directly obtain that the nth eigenvalue is perturbed to "n (k) ⇡ "n (0) + ~2 k 2 ~2 X |hun (0. it is in the form huc |p|uh i.23) m6=n is the reciprocal e↵ective mass of the nth band. they ‘push’ the energies in the . 2m? 1 1 2 X |hun (0. and a typical lattice constant of a ⇡ 0. we do the following. Since valence band states are lower in energy than the conduction band. (14. Let us look at the conduction band e↵ective mass. it is a very good approximation to assume that 2P 2 /m0 = 20eV . It is given by 1 1 2 1 k2 P 2 1 k2 P 2 1 k2 P 2 = [1 + [ ( ) + ( ) + ( )]. where a is the lattice constant of the crystal. bandgap in eV. r)i|2 = [1 + ] m?n m0 m0 k 2 "n (0) "m (0) (14. Looking at the momentum matrix element. Application III: The k · p Method for Bandstructure 91 which can be written in a more instructive form as "n (k) = "n (0) + where ~2 k 2 . 1 + 20eV EG (14. r)|k · p|um (0. Cancelling k 2 .22) (14. which leads to the relation m?c ⇡ m0 .5nm we find that 2P 2 8⇡ 2 ~2 = ⇡ 24eV. it will extract a value |P | ⇡ ~ · 2⇡/a.25) To get an estimate of the magnitude of the momentum matrix element P . m?c m0 m0 k 2 2 Eg 6 Eg 3 Eg (14. The momentum operator will extract the k value of the state it acts on.24) Here we have used to form of Bloch lattice functions given in Equations (17)-(22). m0 m0 a2 (14. and recasting the equation. we get m?c ⇡ m0 1+ 2P 2 m0 EG .27) which in the limit of narrow-gap semiconductors becomes m?c ⇡ (Eg /20)m0 .Chapter 14. Since the valence (and conduction) band edge states actually occur outside the first Brillouin Zone at |k| = G = 2⇡/a and are folded back in to the -point in the reduced zone scheme. the momentum matrix element of most semiconductors is remarkably constant! In fact.26) In reality. This is a remarkably simple and powerful result! It tells us that the e↵ective mass of electrons in a semiconductor increases as the bandgap increases. We also know exactly why this should happen as well: the conduction band energies have the strongest interactions with the valence bands. Using this fact. the bandstructure for the conduction band is "c (k) ⇡ Eg + ~2 k 2 .12 CdTe 0.18 0. and it does a rather good job for all semiconductors.22 0.p theory 2 2P /m 0 ~ 20eV 0. 2m?c (14. we have to understand that it is a purely relativistic e↵ect (which immediately implies there will be a speed of light c somewhere!).0 2. This directly leads to a lower e↵ective mass. far from the BZ edges.08 InP 0. The linear increase of e↵ective mass with bandgap found from the k · p theory is plotted in Figure 5 with the experimentally measured conduction band e↵ective masses. Putting it in words. the electric field of the nucleus Lorentz-transforms to a magnetic field seen by .4: Conduction band e↵ective masses predicted from k · p theory. Note that the straight line is an approximate version of the result of k · p theory. increasing the curvature of the band.16 0.5 3. Finally.. i.10 0.6 With spin-orbit interaction What is spin-orbit interaction? First.Chapter 14. One has to concede that theory is rather accurate.5 2.28) where the conduction band e↵ective mass is used.04 0.0 1.06 0. 14. Note that this result is derived from perturbation theory.e. and is limited to small regions around the k = 0 point only.5 Bandgap (eV) Figure 14. conduction band upwards.00 0.20 Effective mass ( m 0 ) GaN k. One rule of thumb is that the results from this analysis hold only for |k| ⌧ 2⇡/a. when electrons move around the positively charged nucleus at relativistic speeds. in the absence of spin-orbit interaction.14 0.02 GaAs GaSb InAs Ge 0.0 3. Application III: The k · p Method for Bandstructure 92 0.5 1.0 InSb 0. and does give a very physical meaning to why the e↵ective mass should scale with the bandgap. hold your breath . Spin-orbit splitting occurs in the bandstructure of crystal precisely due to this e↵ect. the spin-orbit splitting energy of semiconductors increases as the fourth power of the atomic number of the constituent elements. I have plotted rough fit to a 4 Zav against average atomic number in Figure 6. just like electrons around the proton in the hydrogen atom. and the consequent magnetic field seen by such an electron (rotating at a radius r0 = 0. Specifically. and should have perceivable e↵ects. refer to the textbooks (Yu and Cardona) mentioned in the end of this chapter. To give you an idea. Application III: The k · p Method for Bandstructure 93 the electrons. Furthermore. It is a well-known result that the spin-orbit splitting for atomic systems goes as Z 4 . the situation is not very di↵erent for semiconductors.12 Tesla! That is a very large field.5x10 -4 x ( Z av ) 4 Spin-orbit splitting 600 InAs 400 GaAs Ge 200 Si InP GaN 0 0 10 20 30 40 Average atomic number Z av 50 60 (amu) Figure 14. and shown a polynomial. consider a Hydrogen atom 1 137 the velocity of electron in the ground state is v ⇡ ↵c where ↵ = is the fine structure constant. That is because the atomic number is equal to the number of protons. because the valence electrons are very close to the nucleus. hence more field! 1000 (meV) CdTe 800 InSb 1. 2 c2 c2 (14. we can make some predictions about the magnitude of splitting . which determines the electric field seen by the valence electrons. .53˚ A) from the nucleus is .Chapter 14. For a detailed account on the spin-orbit splitting e↵ects.in general.since the nuclei have more protons.5: The spin-orbit splitting energy for di↵erent semiconductors plotted against the average atomic number Zav . the splitting should be more for crystals whose constituent atoms have higher atomic number . it occurs in semiconductors in the valence band.29) where the approximation is for v ⌧ c. In fact. The transformation is given by B= 1 (v ⇥ E)/c2 q ⇡ 2 2 1 v 1v⇥E . it is rather easy now to show the following. |C> ~ |s> Conduction band minimum +Eg 0 HH LH other states far away: neglect -D k=0 SO Two p-like states LH. E CB s-like state. and a split-o↵ state separated by the spin-orbit splitting energy . "HH (0) = 0. "LH (0) = 0. "SO (0) = ) respectively.6: k · p bandstructure with spin-orbit splitting. Using the same results as for the case without spin-orbit splitting.30) where the e↵ective mass is now given by 1 1 2 1 k2 P 2 1 k2 P 2 1 k2 P 2 = [1 + [ ( ) + ( ) + ( )]. For the conduction band. These bandgap Eg . 2m?c (14. The bandstructure around the point for the four bands and the corresponding e↵ective masses can be written down. the spin-orbit splitting . Spin-orbit coupling splits the 3 degenerate valence bands at k = 0 into a degenerate HH and LH states. The eigenvalues at k = 0 are thus given by ("c (0) = +Eg .Chapter 14. we have "c (k) ⇡ Eg + ~2 k 2 . the conduction-band e↵ective mass m?c ) evaluated in the last section other states far away: neglect are the inputs to the k · p theory to calculate bandstructure .that is. equivalently. m?c m0 m0 k 2 2 E g 6 Eg 3 Eg + (14. HH band maxima p-like state Split off valence band maximum Figure 14. Application III: The k · p Method for Bandstructure 94 Let us now get back to the business of building in the spin-orbit interaction to the k · p theory. and the momentum matrix element P (or. they are known.31) . 2m?LH (14. L.35) Finally.Chapter 14. 2m?SO "SO (k) = (14. i. 3 E. Kane. Application III: The k · p Method for Bandstructure 95 which can be re-written as m?c ⇡ 1+ m0 2P 2 2 1 3m0 ( Eg + Eg + ) . Solids.33) ~2 k 2 . which is the same as the case without the SO-splitting if one puts 20eV is still valid. J. after Kane’s celebrated paper3 of 1956.32) = 0. We chose not to talk about valence bands in the last section. since the degeneracy prevents us from evaluating the perturbed eigenvalues. (14. so.36) where the split-o↵ hole e↵ective mass is given by m?LH = m0 1+ 2P 2 3m0 (Eg + ) .34) and the light-hole bandstructure is given by "LH (k) = where the light-hole e↵ective mass is given by m?LH = m0 1+ 4P 2 3m0 Eg . k · p is very useful in calculating optical transition probabilities and oscillator strengths. 1995). O. (14.e. with spin-orbit splitting. However. (14. Phys. it is easy to show the following. There is a very good section on the uses of this form of bandstructure calculation in the text by S. Chuang (Physics of Optoelectronic Devices. 1. 2m0 (14. the split-o↵ valence bandstructure is given by ~2 k 2 . 2P 2 /m0 ⇡ Spin-orbit splitting causes changes in the valence bandstructure.. the e↵ective mass is the same as free-electron mass. "HH (k) = ~2 k 2 .37) This model is known as the Kane-model of k · p bandstructure. The HH valence bandstructure is that of a free-electron. Chem. 82 (1956) . and so on. which can be intimidating for beginners.edu/⇠djena . The theory is covered in 80 pages. 1968). in particular. Interscience Publishers. and the shifts of bands can be calculated to a great degree of accuracy. It makes heavy usage of group theory. An old and classic monograph. Chapter 2 in this comprehensive text has one of the best modern treatments of semiconductor bandstructure. and the rest of the book analyzes bandstructures of specific materials. The most popular k · p calculations employ what is called a 8-band k · p formalism. not learnt from book only. they are rarely used. The authors do not assume that you come all prepared with results from group theory . and scale inversely as the energy separation. My personal favorites for bandstructure theory and applications are two books - 1) Fundamentals of Semiconductors (Yu and Cardona. 1999). with a spin degeneracy of 2 for each band. Springer. but by experience. However. it still remains one of the few books entirely devoted to the topic.7 Further reading As Kittel states in his text on Solid State Physics. 14. but nevertheless very rewarding. Application III: The k · p Method for Bandstructure 96 The e↵ects of strain can be incorporated into the k · p theory rather easily.it is the four bands we have been talking about all along. The theory is easily extendable to heterostructures. 2) Energy Bands in Semiconductors (Donald Long. Thus. To make the calculations more accurate. gain in lasers.Chapter 14. the e↵ects of these distant bands are weak.nd. as we have seen. to quantum wells for calculating density of states. Where do the eight bands come from? We have already seen all 8 . Debdeep Jena: www. learning how to calculate bandstructure is an art.they actually have ‘crystallized’ the results that are needed from group theory in the chapter. one can include bands higher than the conduction band and lower than the valence band. One runs into a fundamental problem in dealing with a particle location in real space and its momentum at the same time. So long as the perturbations of the crystal potential is not drastic.1 E↵ective Mass Approximation. it suffices to investigate properties of electrons and holes located very close to the band extrema in the k space. the concept of a wave packet is necessary. let us consider the 1-dimensional case. The sum is over the whole BZ.1) We now make two crucial approximation a) We assume that wavefunctions from only one band play a part in the wavepacket. have a finite spread both in the momentum and real space. To illustrate this. unlike pure Bloch-eigenstates. We construct a wavepacket by taking a linear combination of Bloch eigenstates nk (x) from the nth band with wavevector k. therefore.Chapter 16 The E↵ective Mass Approximation 16. one can re-cast the Schr¨odinger equation in a form that is very useful for discussing transport and device applications. (x) = XX n k X Z dk C(k) C(k) nk (x) = 2⇡ n nk (x) (16. A wave packet is nothing but a linear combination of Bloch eigenstates for small k values around a region of interest in the Brillouin zone. To do that. and creates a wavepacket by taking their linear combinations. For most cases. 98 . one collects Bloch eigenstates around such points. Envelope Functions Before we jump into considering real semiconductors with impurities and corresponding perturbations from perfect periodic potentials. and thus drop the sum over all bands. Wave packets. it is worthwhile to develop a very powerful formalism that greatly simplifies our treatment of transport properties. E k . when carried over to real space. wavevectors from a small region say around k0 = 0 are important (see Figure 16. which is the Fourier transform of the weights C(k). and summing them with weights.2) envelope where the integral term is identified as the Fourier transform of the weights C(k) $ C(x). (16. . and represents the wavefunction of a particle.2./a k0 RECIPROCAL SPACE r~1/ k k atoms /a r REAL SPACE Figure 16. The E↵ective Mass Approximation 99 b) We assume that in the single band we are interested in. | {z } | {z } 2⇡ Bloch ikx . The real space wavefunction is given by the Bloch wavefunction at the k0 point. Bloch functions can be written as Then the wavepacket takes the form (x) ⇡ n0 (x) Z nk (x) = eikx unk (x) ⇡ un0 (x)eikx = n0 (x)e dk C(k)eikx = n0 (x) · C(x) . since the weights C(k) are over a small region in k space. This is illustrated in Figure 16. H0 nk (x) = En (k) nk (x).Chapter 16. It is typically a smooth function spreading over several lattice constants. Then.1). modulated by an envelope function C(r).01 = 100 atoms in real space. The weights C(k) have a small extent k in reciprocal space. the wavepacket spreads over 1/0. If we restrict the sum in reciprocal space to 1% of the BZ. The real-space function C(x) which is a Fourier transform of the weights of the wavepacket is called as the envelope function. the spread is large.1: A wavepacket is constructed by taking Bloch functions from a small region of the reciprocal space. How does the wavepacket behave when we apply the periodic crystal Hamiltonian H0 on it? Since nk (x) and we recover are Bloch-eigenfunctions of this Hamiltonian. since r ⇠ 1/ k. thus the wavepacket has a finite spread in real space. C(x) is spread over real space. if f (k) $ f (x).Chapter 16.4) m and Schr¨ odinger equation becomes X Z dk C(k)am k m eikx .5) We now use a property of Fourier transforms .6) . En (k) = X am k m (16. then kf (k) $ ( id/dx)f (x). k m f (k) $ ( id/dx)m f (x). Thus. H (x) ⇡ n0 (x) 2⇡ m (16.2: Envelope function C(r) modulates the Bloch function the wavefunction of the wavepacket (x). Z dk m d k C(k)eikx $ ( i )m C(x). 2⇡ to produce (16.3) We now write out the energy eigenvalues as a Taylor-series of small wavevectors around k = k0 = 0. 2⇡ dx (16. H0 (x) = Z dk C(k)En (k) 2⇡ nk (x) ⇡ n0 (x) Z n0 (x) dk C(k)En (k)eikx . and in general. The E↵ective Mass Approximation 100 Envelope Function Periodic Part of Bloch Functions Atoms Envelope Function Figure 16. and the operator En ( ir) thus becomes En (k) ⇡ Ec (r) + ~2 k 2 ! En ( ir) ⇡ Ec (r) 2m? ~2 2 r . from which one recovers the real wavefunction of the wavepacket by multiplying with the Bloch function (r) ⇡ n0 (r)C(r) = un0 (r)C(r). if we already know the bandstructure of the semiconductor. no operators act on it.Chapter 16. 2m? (16.the Bloch function part has been pulled out as a coefficient.7) which can be generalized to the 3-D case. moving in a potential Ec (r) + V (r)! All information about the bandstructure and crystal potential has been lumped into the e↵ective mass m? . This step is crucial . Now. (16. it becomes [En ( ir) + V (r)]C(r) = EC(r). Take a moment to note what has been achieved. instead of the periodic potential Hamiltonian. Schr¨odinger equation becomes H0 n0 (r)C(r) + V (r) n0 (r)C(r) =E n0 (r)C(r).7.10) and the Schr¨ odinger equation takes the enormously simplified form [ ~2 2 r + Vimp (r)]C(r) = [E 2m? Ec (r)]C(r). making it an operator that acts on the envelope function only. The Schrodinger equation has been re-cast into a much simpler problem of a particle of mass m? . Thus. The wavefunctions are envelope functions C(r).9) where the Bloch functions do not appear at all! Furthermore. if we have another potential (say a perturbation) V (r) present. (16. (16. we make the substitution k ! i@/@r.11) which is the celebrated “E↵ective Mass Approximation”. then we can write the energy around the point k0 = 0 of interest in terms of the e↵ective mass. (16. The envelope functions C(r) can be actually deter- mined for any potential .8) and using Equation 16. in the energy term.it amounts to solving the Schr¨odinger equation for a particle . The E↵ective Mass Approximation 101 and the Schr¨ odinger equation is recast as H (x) ⇡ n0 (x)En ( ir)C(x). An impurity potential can now be included as a perturbation to the periodic crystal.14) which is identified as the same as the classic problem of a hydrogen atom. consider an ionized impurity. In semiconductor heterostructures. Debdeep Jena: www. The e↵ective mass approximation is a natural point of departure. the new energy levels that appear are given by E Ec = E1 m? . and the dielectric constant is that of the semiconductor.15) and the e↵ective Bohr-radius is given by a?B = aB ✏r m? (16. and the new energy eigenvalues can be found. depending upon composition of the semiconductor. or dots. The E↵ective Mass Approximation 102 in the potential Ec (r) + V (r). where analysis of such low-dimensional structures begins.12) and the corresponding eigenvalues of the Schro¨odinger equation are given by E = Ec (r) + ~2 |k|2 . As an example.nd.Chapter 16. 2m? (16. V (16. (16. the band-edge variation in real space can be varied by applying electric fields. or by doping variations.the mass term is an e↵ective mass instead of the free electron mass. albeit with two modifications . Ec (r) is the spatial variation of the conduction band edge .edu/⇠djena . The e↵ective mass equation takes the form [ ~2 2 r 2m? e2 ]C(r) = (E 4⇡✏r Ec )C(r).exactly what one draws in band diagrams.13) If we consider electrons at the bottom of the conduction band. Then.16) In bulk semiconductors. one can further engineer the variation of the band-edge Ec (r) in space by quasi-electric fields the band edge can behave as quantum-wells. wires. which has a Coulomb potential. ✏2r (16. Note that the envelope function in the absence of any impurity potential V (r) = 0 is given by 1 C(r) = p eik·r . An example is the semiconductor (diode) laser. So.1 Introduction With the explosion of usage of semiconductor heterostructures in a variety of applications. C(r) is the envelope function of carriers in the band under consideration. wires and dots have become important.e. or nanowires / nanotubes / nanocrystals can be grown by bottom-up approaches (by CVD techniques. and have a potential to perform functions that are difficult.these structures can be grown by compositional variations in epitaxially grown semiconductor layers by MBE/MOCVD techniques.1) Here. We have derived the e↵ective mass equation for carriers in bulk semiconductors in the envelope-function approximation. if not impossible to achieve in bulk materials. they were p-n junctions). The first semiconductor lasers were band-engineering by doping (i. The Schr¨ odinger equation is thus re-cast in a form which is identical to that of an electron in 103 .Chapter 17 Electrons in Quantum Heterostructures 17. or by solution chemistry). (17. low-dimensional structures such a quantum wells. understanding bandstructure of these artificially engineered materials is of great interest. Many of these designer materials have niche applications. They come in various avatars . The three-dimensional e↵ective mass equation is [ ~2 2 r + V (r)]C(r) = (E 2m? Ec (r))C(r). The goal of many clever expitaxial/bottom-up techniques to create nanostructures amounts to modifying the bandstructure of the constituent bulk semiconductor material.. V (17. which is solvable. and thus the solution of the e↵ective mass equation yields envelope functions 1 ~ C(r) = p eik·~r . the beauty of the e↵ective mass approximation is that the envelope function is all that is needed to find the bandstructure of the low-dimensional structures1 ! The envelope function concept is a powerful tool..2 Bulk Bandstructure 17. 17. c0 2m? 2 m?xx m?yy m?zz (17. . The real wavefunction of the wavepacket that models the particle-like nature of the electrons is given by (r) ⇡ un0 (r)C(r). k s can be assumed continuous. Electrons in Quantum Heterostructures 104 a total potential V (r) + Ec (r). as is demonstrated in its use in determining bandstructure modifications due to quantum confinement of carriers in low-dimensional structures..Chapter 17. V (r)+Ec (r) = Ec0 is a constant energy (flatband conditions). where un0 (r) is the periodic part of the Bloch eigenstates of the crystal that result from the periodic crystal potential. It has mapped the complex problem of an electron moving through a crystal experiencing very complicated potentials to a textbook-type ‘particle in a well-defined potential’ problem. Since L is a macroscopic length. However. The e↵ective mass contains information about the bulk bandstructure.3) and energies One should not forget that even thought the k s is written as a continuous variable.2) ky2 ~2 k 2 ~2 kx2 kz2 E(k) = Ec0 (r) + = E (r) + ( + + ). The particle mass is renormalized.1 Pure Semiconductors In a bulk semiconductor in the absence of external fields. assuming values kx = ky = kz = 2⇡ m L (17.2. ±1. and for all practical purposes. ±2. determined by the band-edge behavior..4) where m = 0. 1 Note that the bulk bandstructure is assumed to be known. they are actually quantized. the quantization is very fine. . absorbing the details of the crystal potential. .7) Similar results hold for valence bands.) is the Fermi-Dirac integral function.Chapter 17. This is shown schematically in Figure ??.5) from which one gets a carrier concentration in the conduction band n= Z 1 0 dEfF D (E)g3D (E) = NC3D F1/2 ( EC EF ) ⇡ NC3D e kB T EC EF kB T . Here. (17. 2⇡~2 (17.Ec 0 Donor DOS CB Ec0 Gap Ev0 VB Total Acceptor DOS ~ E . The approximation holds only when Fermi-Dirac distribution can be approximated by a Maxwell-Boltzmann form. . Electrons in Quantum Heterostructures 105 The density of states (DOS) is given by g3D (E) = 1 2m? 3/2 p ( ) E 2⇡ 2 ~2 Ec0 .Ev 0 LH HH DOS Moderate Doping Heavy Doping Figure 17. (17.. ~ E .6) where Fj (. where the contributions from the Light and Heavy hole bands add to give the total DOS. it is easily shown that NC3D is a e↵ective band-edge DOS is NC3D = 2( m? kB T 3 )2 .1: Density of States of bulk (undoped). moderately doped and heavily doped semiconductors. the Hydrogenic-model solution from the e↵ective mass equation [ ~2 2 r 2m? e2 ]C(r) = (E 4⇡✏r Ec )C(r) (17. If the donor electron envelope function is spread over 1000 atoms in real space. the donor states are assumed to be “atomic-like”. leaving motion in the x y plane free. A shallow dopant adds states close to the band-edges. this implies that in k-space. for all practical purposes.9) a). the square-well potential (with reference to the conduction band edge Ec0 ) is written as . k ⇠ 1/r.8) showed that the eigenvalues were similar to that of a hydrogen atom. since thermal activation of carriers into the bands is not necessary for transport. many changes can occur. The adjacent radii of electrons associated with adjacent donors can overlap.6 ⇥ (m? )/✏2r is the modified Hydrogenic energy levels. the semiconductor acquires metal-like properties. Thus. For heavy doping however. The e↵ects of moderate and heavy doping on the DOS of bulk semiconductors is shown in Figure ??. The finite extent of the quantum well layer makes the conduction band profile mimic a one-dimensional quantum well in the direction of growth (z direction). Then. Electrons in Quantum Heterostructures 17.2. 17. Thus.3 Quantum Wells Quantum wells are formed upon sandwiching a thin layer of semiconductor between wider bandgap barrier layers.Chapter 17. Energy separations between these individual atomic-like states is very small. given by En = Ec0 Ry ? /n2 .2 106 Doped Semiconductors Doping adds states in the bandgap of the semiconductor. The ground-state envelope functions around the donor atoms C(r) ⇠ e r/r0 is spread over many lattice constants (r0 = aB (✏r /m? ) the donor states are localized to (17. leading to formation of impurity bands. in k-space it will be restricted to ⇠1/1000 of the volume of the Brilloiun zone. where Ry ? = 13. Considering a shallow donor. y. z < 0 (17. The bandstructure is the set of energy eigenvalues is obtained from the e↵ective mass equation. y. z) = (x. . 0 z W. 3. z) = 0. leading to knz = 2⇡/ = (⇡/W )nz . z) = 0. . . W (17. Electrons in Quantum Heterostructures 107 V (x.11) Ec . From simple particle-in-a-box model in quantum mechanics. the normalized z component of the envelope function is nz (z) (17. (17. and DOS of realistic heterostructure quantum wells. given by 2 Only waves that satisfy nz ( /2) = W fit into the well of width W .Chapter 17. . by simple wave-fitting procedure2 the z component of the electron quasi-momentum is quantized to k nz = ⇡ nz .12) V (x.. y) nz (z) 1 = [ p ei(kx x+ky y) ] · [ A nz (z)] (17. y. it is evident that the envelope function should decompose as Cnz (x.15) E E y nz =3 x E3(k) Subbands Ec0 z Ec 2 ⇡nz z = p sin .2: Bandstructure.10) V (x. z) = Using the e↵ective mass equation with this potential. W W E2(k) E1(k) AlGaAs GaAs nz =2 AlGaAs m* Ev 0 W Ev0 2 nz =1 (in x-y plane) k g(E) Figure 17. y. 2.14) where nz = 1.13) If the quantum well is assumed to be infinitely deep. z > W (17. 2 | ⇡~ {z } (17. Electrons in Quantum Heterostructures E(k) = Ec0 + 108 ky2 ~2 kx2 ~2 ⇡nz 2 ( ? + ? )+ ( ) 2 mxx myy 2m?zz W {z } | {z } | E1D (nz ) E2D (kx .2. It is important to note that if the confining potential in the z direction can be engineered almost at will by modern epitaxial techniques by controlling the spatial changes in material composition. . each subband corresponding to an nz is an ideal 2D system. which houses many subbands. a popular quantum well structure has a parabolic potential (V (z) ⇠ z 2 ). ky ). this is shown in Figure 17. given by g2D (E) = m? /⇡~2 . EF is the Fermi level. The bandstructure consists of multiple bands E2D (kx . The DOS of electrons confined in an ideal 2-D plane is a constant.18) 2D NC where E1 is the ground state energy. (17.2). the DOS becomes a sum of each subband (Figure 17. . (Verify the units of each!) For the quantum well.7.2.Chapter 17. The carrier density of an ideal 2D electron system is thus given by n2D = Z 1 dEfF D (E)g2D (E) = 0 EF E1 m? k B T kB T ln(1 + e ).) is the unit step function. In the quantum well. (17.16) y plane and a direction. the 2-dimensional counterpart of NC3D defined in Equation 17. For example. each indexed by the quantum number nz . This is shown schematically in Figure 17.17) z where ✓(. which leads to the Enz values spaced in equal energy intervals - . and NC2D is the e↵ective bandedge DOS. and the total carrier density is thus a sum of 2D-carriers housed in each subband - n2D = X nj = Nc2D j X ln(1 + e EF Ej kB T ). no approximation of the Fermi-Dirac function is necessary to find the carrier density analytically. the DOS of the quantum well is gQW (E) = m? X ✓(E ⇡~2 n Enz ). Thus.ky which evidently decomposes to a free-electron component in the x quantized component in the z (17. and each subband contributes g2D (E) the the total DOS.19) j Note that for a 2D system. 21) and the energy eigenvalues are given by E(nx . and were free to move in the other two. and quantum wells under electric fields. 2m?zz (17.4 Quantum Wires Artificial quantum wires are formed either lithographically (top-down approach). Assume that the length of the wire (total length Lz ) is along the z direction (see Figure 17. In a quantum well. and the corresponding subband separations. and the wire is quantum-confined in the x naturally decomposes into y plane (Lx . the bandstructure and the DOS remain similar to the square well case. . or by direct growth in the form of semiconductor nanowires or nanotubes (bottom-up approach). y. 17. a wave-fitting procedure gives ⇡ nx . 2. the only modification being the Enz values. . the envelope function C(x. .Chapter 17. independently. . In a quantum wire. and the other two degrees of freedom are quantum-confined. Ly ⌧ Lz ). out of the three degrees of freedom for real space motion. electrons are free to move freely in one dimension only (hence the name ‘wire’).22) kny (17. ny . 3. Then. carriers were confined in one. Electrons in Quantum Heterostructures 109 this is a characteristic of a square.23) where nx . Lx ⇡ = ny . Regardless of these details specific to the shape of the potential. then similar to the quantum well situation. ny ) + If the confinement in the x y directions is by infinite potentials (a useful model applicable in many quantum wires).3). HEMTs. Ly knx = (17. ny = 1. or Harmonic Oscillator potential. which appears in MOSFETs. The triangular well leads to Enz values given by Airy funtions. Lz (17. z) = nx (x) · ny (y) 1 · ( p eikx x ).20) ~2 kk2 . Another extremely important quantum well structure is the triangular well potential (V (z) ⇠ z). kz ) = E(nx . 25) E(nx .26) . Electrons in Quantum Heterostructures 110 The eigenfunctions assume the form Cnx .24) and the corresponding bandstructure is given by ~2 ⇡nx 2 ~2 ⇡ny 2 ~2 k 2 E(nx . y.1) (nx .2) z x Subbands (2.3) y Lx Ly (2. The DOS of electrons confined to an ideal 1-D potential is given by 1 g1D (E) = ⇡ r 2m? 1 p .ny (x. Ly Ly Lz (17. A new subband forms at each eigenvalue E(nx . and DOS of realistic quantum wires. similar to the quantum well structure.3: Bandstructure.2) Quantum Wire (1. 2mxx Lx 2myy Ly 2mzz | {z } (17. and each subband has a dispersion E(kz ) = ~2 kz2 /2mzz (Figure 17. 2 ~ E E1 (17.Chapter 17. E E (2.3).ny ) Multiple subbands are formed. ny ) (in z direction) kz g(E) Figure 17.1) (1. z) = [ r 2 ⇡nx sin( x)] · [ Lx Lx s ⇡ny 2 1 sin( y)] · [ p eikx x ]. ny ). ny . kz ) = [ ( ) ]+[ ( ) ] + ?z .1) (1. it does not make sense to talk about “bandstructure” of quantum dots. and the peaks can occur at irregular intervals as opposed to the quantum well case. 17. and are indexed by three quantum numbers (nx . y. therefore there is no plane-wave component of electron wavefunctions.5 Quantum Dots The quantum dot is the ultimate nanostructure. The envelope functions are thus given by C(x. Ly .n E x y 1 . nz ) = ~2 ⇡nx 2 ~2 ⇡ny 2 ~2 ⇡nz 2 ( ) + ( ) + ( ) . y. Lz (see Figure 17.nz ). ny ).27) which is shown schematically in Figure 17. The envelope function for a “quantum box” of sides Lx . written as gQDot = X nx . (17. Lz Lz (17. ny . some eigenvalues can be degenerate. 2mxx Lx 2myy Ly 2mzz Lz (17. ny . nz ). z) = [ r 2 ⇡nx sin( )] · [ Lx Lx s ⇡ny 2 sin( )] · [ Ly Ly r 2 ⇡nz sin( )].30) Note that the the energy eigenvalues are no more quasi-continuous.3. Electrons in Quantum Heterostructures 111 where E1 is the lowest allowed energy (ground state). the DOS is a sum of delta functions. ny ) (17.28) and if the confining potential is infinitely strong. we have kni = (⇡/Li )ni for i = x.31) .nz (E Enx . Thus. The general DOS for a quantum wire can thus be written as 1 gQW ire (E) = ⇡ r 2m? X p ~2 n . E(nx . Since there are two quantum numbers involved.29) and the energy eigenvalues are given by E(nx .4) is thus written as C(x. Due to multiple subbands. All three degrees of freedom are quantum confined.ny . (17. the DOS acquires peaks at every eigenvalue E(nx . y.Chapter 17.ny . z) = nx (x) ny (y) nz (z). z. Debdeep Jena: www.nd. epitaxial techniques can coax quantum dots to self-assemble by cleverly exploiting the strain in lattice-mismatched semiconductors.1) (1. there is no transport within a quantum dot. and there is no quasi-continuous momentum components.edu/⇠djena .1. Fabricating quantum dots by lithographic techniques is pushing the limits of top-down approach to the problem.4.2. Since there is no direction of free motion. On the other hand. bottom-up techniques of growing nanocrystals in solution by chemical synthetic routes is becoming increasingly popular.2) (1.1.1.nz ) Figure 17. Electrons in Quantum Heterostructures E 112 E Quantum Dot z x y Lx Ly Lz (2. On the other hand. ny .Chapter 17.4: Energy levels and DOS of quantum dots. This is shown schematically in Figure 17.1) (1.1) atomic-like levels : artificial atoms g(E) (nx .