Nature of electron Zitterbewegung in crystalline solids

Physics Letters A 374 (2010) 3533–3537 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Nature of electron Zitterbewegung in crystalline solids Wlodek Zawadzki a,∗ , Tomasz M. Rusin b a b Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-688 Warsaw, Poland PTK Centertel Sp. z o.o., ul. Skierniewicka 10A, 01-230 Warsaw, Poland a r t i c l e i n f o Article history: Received 1 April 2010 Received in revised form 5 June 2010 Accepted 14 June 2010 Available online 19 June 2010 Communicated by R. Wu Keywords: Zitterbewegung Periodic potential Instantaneous velocity Kronig–Penney model Two-band model a b s t r a c t We demonstrate both classically and quantum mechanically that the Zitterbewegung (ZB, the trembling motion) of electrons in crystalline solids is nothing else, but oscillations of velocity assuring the energy conservation when the electron moves in a periodic potential. We show that the two-band k.p model of electronic band structure, formally similar to the Dirac equation for electrons in a vacuum, gives a very good description of ZB in solids. Our results unambiguously indicate that the trembling motion is the basic way of electron propagation in periodic potentials and, as such, it is certainly observable. A recent experimental simulation of ZB with the use of trapped ions is in good agreement with our calculations.  2010 Elsevier B.V. All rights reserved. 1. Introduction The phenomenon of electron Zitterbewegung (ZB, the trembling motion) in absence of external fields was predicted by Schrödinger in 1930 as a consequence of the Dirac equation for a free relativistic electron [1,2]. The observability of ZB for electrons in a vacuum was debated ever since (see e.g. [3,4]). Experimental difficulties to observe the ZB in a vacuum are great because the predicted frequency of the trembling is very high: h̄ω Z ≃ 2m0 c 2 ≃ 1 MeV, and its amplitude is very small: λc = h̄/m0 c = 3.86 × 10−3 Å. It was later suggested that a phenomenon analogous to ZB should exist for electrons in semiconductors if they can be described by a two-band model of band structure [5,6]. In particular, an analogy between the behavior of free relativistic electrons in a vacuum and that of non-relativistic electrons in narrow gap semiconductors (NGS) was used to predict that the ZB of electrons in NGS should have the frequency h̄ω Z ≃ E g (where E g is the energy gap), and the amplitude λ Z = h̄/m∗ u, where m∗ is the effective mass and u = ( E g /2m∗ )1/2 ≃ 108 cm/s is a maximum electron velocity according to the two-band k.p model [7]. This results in much more advantageous characteristics of ZB as compared to a vacuum; in particular λ Z ≃ 64 Å for InSb, 37 Å for InAs, and 13 Å for GaAs. After the papers of Zawadzki [8] and Schliemann et al. [9] the ZB of electrons in crystalline solids and other systems became a subject of intensive theoretical studies, see [10–24]. A classical phe- * Corresponding author. E-mail address: zawad@ifpan.edu.pl (W. Zawadzki). 0375-9601/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.06.028 nomenon analogous to the ZB was observed in macroscopic sonic crystals [25]. Very recently, the Zitterbewegung of free relativistic electrons in a vacuum was simulated experimentally by trapped ions [26]. The physical origin of ZB remained mysterious. As to electrons in a vacuum, it was recognized that the ZB is due to an interference of states corresponding to positive and negative electron energies [27]. Since the ZB in solids was treated by a two-band Hamiltonian similar to the Dirac equation, its interpretation was also similar. This did not explain its origin, it only provided a way to describe it. Our Letter treats the fundamentals of electron propagation in a periodic potential and elucidates the nature of electron Zitterbewegung in solids. The physical origin of ZB is essential because it resolves the question of its observability. The second purpose of our work is to decide whether a two-band k.p model of the band structure, used until now to describe the ZB in solids, is adequate. One should keep in mind that the above quoted references describe various kinds of ZB. Every time one deals with two interacting energy bands, an interference of corresponding states results in electron oscillations called ZB. In particular, one deals with ZB related to the Bychkov–Rashba-type spin subbands [9] or to the Luttinger-type light and heavy hole subbands [12,28]. Not entering the question of whether the oscillation effects related to these systems should be called Zitterbewegung, we emphasize that we do not describe systems related to the spin or to degenerate bands. The problem of our interest is the simplest electron propagation in a periodic potential. The trembling motion of this type was first treated in Ref. [8]. We come to this point again in the Discussion. 3534 W. Zawadzki, T.M. Rusin / Physics Letters A 374 (2010) 3533–3537 2. Semiclassical approach It is often stated that an electron moving in a periodic potential behaves like a free particle characterized by an effective mass m∗ . The above picture suggests that, if there are no external forces, the electron moves in a crystal with a constant velocity. This, however, is clearly untrue because the electron velocity operator v̂ i = p̂ i /m0 does not commute with the Hamiltonian Ĥ = p̂2 /2m0 + V (r), so that v̂ i is not a constant of the motion. In reality, as the electron moves in a periodic potential, it accelerates or slows down keeping its total energy constant. This situation is analogous to that of a roller-coaster: as it goes down losing its potential energy, its velocity (i.e. its kinetic energy) increases, and when it goes up its velocity decreases. We demonstrate below that the electron velocity oscillations due to its motion in a periodic potential of a solid are in fact the Zitterbewegung. Thus the electron Zitterbewegung in solids is not an exotic obscure phenomenon − in reality it describes the basic way of electron propagation in periodic potentials. We first consider the trembling frequency ω Z . The latter is easy to determine if we assume, in the first approximation, that the electron moves with a constant average velocity v̄ and the period of the potential is a, so ω Z = 2π v̄ /a. Putting typical values for GaAs: a = 5.66 Å, v̄ = 2.3 × 107 cm/s, one obtains h̄ω Z = 1.68 eV, i.e. the interband frequency since the energy gap is E g ≃ 1.5 eV. The interband frequency is in fact typical for the ZB in solids. Next, we describe the velocity oscillations classically assuming for simplicity a one-dimensional periodic potential of the form V ( z) = V 0 sin(2π z/a). The first integral of the motion expressing the total energy is: E = m0 v 2z /2 + V ( z). Thus the velocity is dz dt =  2E m0  1− V ( z) E 1/2 (1) . One can now separate the variables and integrate each side in the standard way. In a classical approach V 0 must be smaller than E. In general, the integration of Eq. (1) leads to elliptical integrals. However, trying to obtain an analytical result we assume V 0 ( z) ≃ E /2, expand the square root retaining first two terms and solve the remaining equation by iteration taking in the first step a constant velocity v z0 = (2E /m0 )1/2 . This gives z = v z0 t and v z (t ) = v z0 − v z0 V 0 2E  sin 2π v z0 t a  . (2) Thus, as a result of the motion in a periodic potential the electron velocity oscillates with the expected frequency ω Z = 2π v z0 /a around the average value v z0 . Integrating with respect to time we get an amplitude of ZB:  z = V 0 a/(4π E ). Taking again V 0 ≃ E /2, and estimating the lattice constant to be a ≃ h̄ p cv /(m0 E g ) (see Luttinger and Kohn [33]), we have finally  z ≃ h̄ p cv /(8π m0 E g ), where p cv is the interband matrix element of momentum. This should be compared with an estimation obtained previously from the two-band k.p model [8]:  z ≃ λ Z = h̄/m∗ u = h̄(2/m∗ E g )1/2 ≃ 2h̄ p cv /m0 E g . Thus the classical and quantum results depend the same way on the fundamental parameters, although the classical approach makes no use of the energy band structure. We conclude that the Zitterbewegung in solids is simply due to the electron velocity oscillations assuring the energy conservation during motion in a periodic potential. Now we describe the electron velocity oscillations using a somewhat different semiclassical method. We begin with the periodic Hamiltonian Ĥ = p̂ 2 /2m0 + V ( z). The velocity operator is v̂ z = p̂ z /m0 . Using the above Hamiltonian one obtains m0 d v̂ z dt = 1 ih̄ [ p̂ z , Ĥ ] = − ∂ V ( z) , ∂z (3) Fig. 1. Plot of (a) potential, (b) acceleration versus position, and (c) velocity, (d) displacement versus time for the electron motion in a saw-like potential (schematically). which is a quantum analogue of the Newton law of motion in an operator form. In order to integrate Eq. (3) we assume a particularly simple periodic saw-like potential. Thus V ( z) = − gz for 0  z < a/2, a  z < 3a/2, etc., and V ( z) = − V 0 + gz for a/2  z < a, (3a/2)  z  2a, etc., where g is a constant force, see Fig. 1a. In each half-period z is counted from zero. The derivatives are −∂ V /∂ z = g for 0  z < a/2, a  z < 3a/2, etc., and −∂ V /∂ z = − g for a/2  z < a, (3a/2)  z  2a, etc., as illustrated in Fig. 1b. Thus the electron moves initially with a constant acceleration g /m0 from z = 0 to z = a/2, reaches the maximum velocity v̂ m = (ag /m0 )1/2 at a time tm = (m0 a/ g )1/2 , and then slows down reaching v̂ = 0 at a time 2tm . Then the cycle is periodically repeated. We calculate v̂ (t ) = ( g /m0 )t and ẑ(t ) = ( g /2m0 )t 2 in the first, third, fifth time (or distance) intervals, and v̂ (t ) = v m − ( g /m0 )t and ẑ(t ) = v m t − ( g /2m0 )t 2 in the second, fourth, sixth time (or distance) intervals, as illustrated in Figs. 1c and 1d. We assumed for simplicity v̂ (0) = ẑ(0) = 0. It is seen from Fig. 1c that the velocity oscillates in time around the average value v av = v m /2. This is reflected in the oscillations of position  z(t ) around the straight line z = v av t, as seen in Fig. 1d. The latter are easily shown to be  z = ±a/8, which compares well with the classical results. Using the above estimation for a we identify again the motion due to periodicity of the potential with the Zitterbewegung calculated previously with the use of two-band k.p model. Strictly speaking, the physical sense of the above operator reasoning is reached when one calculates average values. Our procedure follows the original approach of Schrödinger [1], who integrated operator equations for v̂ (t ) and ẑ(t ). Similar approach is commonly used for ẑ(t ) and p̂ (t ) operators in the harmonic oscillator problem [29]. The behavior of ZB indicated in Fig. 1d closely resembles the simulated experimental data shown in Fig. 2 of Ref. [26]. Still, the electron motion in a periodic potential is inherently of the quantum nature, so the above semiclassical considerations should be treated as a mere illustration. 3. Quantum treatment Now we describe the ZB using a rigorous quantum approach. We employ the Kronig–Penney delta-like potential since it allows 3535 W. Zawadzki, T.M. Rusin / Physics Letters A 374 (2010) 3533–3537 us to calculate explicitly the eigenenergies and eigenfunctions [30, 31]. In the extended zone scheme the Bloch functions are ψk ( z) = e ikz A k ( z), where A k ( z) = e −ikz C k e ika sin[βk z] + sin βk (a − z) ,   (4) in which √ k is the wave vector, C k is a normalizing constant and βk = 2m0 E /h̄ is a solution of the equation Z sin(βk a) βk a + cos(βk a) = cos(ka), (5) with Z > 0 being an effective strength of the potential. In the extended zone scheme, the energies E (k) are discontinuous functions for k = nπ /a, where n = . . . , −1, 0, 1, . . . . In this convention, if nπ /a  k  (n + 1)π /a, the energies E (k) belong to the n-th band and the Bloch states are characterized by one quantum number k. Because A k ( z) is a periodic function, one may expand it in the Fourier series A k ( z) = n A n exp(ikn z), where kn = 2π n/a. In the Heisenberg picture the time-dependent velocity averaged over a wave packet f ( z) is v̂ (t ) = h̄ m0  dk dk′  f |kk| ∂ i∂ z |k′ k′ | f e i ( E k − E k′ )t /h̄ , (6) where |k are the Bloch states. The matrix elements of momentum k| p̂ |k′  = h̄δk′ ,k+kn K (k, k′ ) are calculated explicitly. The wave packet f ( z) is taken in a Gaussian form of the width d and centered at k0 , and its matrix elements are  f |k = n A n F (k, kn ), where F (k, kn ) = ∞ f ∗ ( z)e iz(k+kn ) dz. (7) −∞  h̄  m0 n,n′ ,l  ′  × K k, k e   dk dk′ A n∗ A n′ F ∗ (k, kn ) F k′ , kn′ i ( E k − E k′ )t /h̄ δk′ ,k+kl . v̂ (t ) =  f | v̂ (t )| f  =  kk′ nn′  f |nknk| v̂ (t )|n′ k′ n′ k′ | f , (9) where the velocity in the Heisenberg picture is v̂ (t ) = (h̄/m0 ) × e i Ĥ kp t /h̄ (∂/i ∂ z)e −i Ĥ kp t /h̄ . Restricting the above summation to the conduction and valence bands: n, n′ = 1, 2 Inserting the above matrix elements to Eq. (6) we obtain v̂ (t ) = Fig. 2. Calculated electron ZB velocities and displacement in a superlattice versus time. The packet width is d = 400 Å, Kronig–Penney parameter is Z = 1.5π , superlattice period is a = 200 Å. (a) Packet centered at k0 = π /a; (b) and (c) packet centered at k0 = 0.75π /a. The dashed lines indicate motions with average velocities. v̂ (t ) ≈ (8) Fig. 2 shows results for the electron ZB, as computed for a superlattice. The electron velocity and position are indicated. A relative narrowness of the wave packet in k space cuts down contributions from k values away from k0 , and one deals effectively with the vicinity of one energy gap. It is seen that for a superlattice with the period a = 200 Å the ZB displacement is about ±50 Å, i.e. a fraction of the period, in agreement with the rough estimations given above. The period of oscillations is of the order of several picoseconds. We emphasize that the above description uses explicitly the Kronig–Penney periodic potential both in the energies and the Bloch functions. Again, the behavior of ZB indicated in Fig. 2c is very similar to that shown in Fig. 2 of Ref. [26]. In particular, the results of Ref. [26] confirm experimentally that the ZB of an electron prepared in the form of a Gaussian wave packet decays in time, see Ref. [32]. The oscillations of the packet velocity calculated directly from the periodic potential have many similarities to those computed on the basis of the two-band k.p model. The question arises: does one deal with the same phenomenon in the two cases? To answer this question we calculate ZB using the two methods for the same periodic potential. We also want to demonstrate that the two-band k.p model, used until present to calculate the Zitterbewegung [23], is adequate for the description of this phenomenon. We calculate the packet velocity near the point k0 = π /a for a one-dimensional Kronig–Penney periodic Hamiltonian using the Luttinger–Kohn (LK) representation [33]. The LK functions χnk ( z) = e ikz unk0 ( z), where unk0 ( z) = unk0 ( z + a), also form a complete set. We have  kk′     f | · · · + |1k1k| + |2k2k| + · · ·  v̂ (t )   × · · · + |1k1k| + |2k2k| + · · · | f , (10) we obtain in the matrix form v̂ (t ) ≈  f1 f2 †  v̂ (t )11 v̂ (t )12 v̂ (t )21 v̂ (t )22  f1 f2  , (11) where f n = nk| f , and v̂ (t )nn′ are the matrix elements of the time-dependent velocity operator between the LK functions. Equation (11) looks like the k.p approach to ZB used previously in the literature. Next, we approximate Ĥ by the k.p Hamiltonian Ĥ kp . To find Ĥ kp we proceed in the standard way [33,34] and assume that ψnk (x) ≈ c 1 (k)e ikx u 1k0 (x) + c 2 (k)e ikx u 2k0 (x) + · · · . (12) Substituting ψnk (x) from Eq. (12) to the Schrödinger equation Ĥ ψnk = E ψnk , and calculating matrix elements of both sides with periodic functions u 10k0 and u 2k0 we obtain Ĥ kp =  h̄2 q2 /2m + E 1 h̄q P 21 /m h̄q P 12 /m h̄2 q2 /2m + E 2  , (13) where E 1 and E 2 are the energies at band extremes, P 12 = h̄/mu 1k0 |∂/i ∂ x|u 2k0 , and q = k − π /a. The velocity matrix at t = 0 is v̂ kp = ∂ Ĥ kp /h̄∂ k. The calculation of velocity in the Heisenberg picture is described in Ref. [14]. Apart from the small free-electron terms on the diagonal, Eq. (13) simulates the 1 + 1 Dirac equation for free relativistic electrons in a vacuum. 3536 W. Zawadzki, T.M. Rusin / Physics Letters A 374 (2010) 3533–3537 Fig. 3. ZB of electron velocity in a periodic lattice versus time. Solid line: the Kronig– Penney model, dashed line: the two-band k.p model. Inset: Calculated bands for the Kronig–Penney (solid line) and the two-level k.p model (dashed line) in the vicinity of k = π /a. The wave packet f (k) centered at k0 = 0.75π /a is also indicated (not normalized). In Fig. 3 we compare the ZB oscillations of velocity calculated using: (a) real E (k) dispersions resulting from the Kronig–Penney model and the corresponding Bloch functions of Eq. (4); (b) twoband E (k) dispersions and the corresponding LK functions. It is seen that, although we take the packet not centered at k = π /a, the two-band k.p model gives an excellent description of ZB for instantaneous velocities. For k0 = π /a the two descriptions are almost identical. This agreement demonstrates that the theories based on (a) the periodic potential and (b) the band structure, describe equivalently the same trembling motion of the electron. Both methods give practically the same results for the packet widths d  a, when the main k contributions to the packet arise from two neighboring energy bands. For narrower packets in real space one needs more k values to construct the packet and a simple two-band model is not adequate. 4. Discussion It is clear that, if one deals with an external potential, the position r will always be a function of time in the Heisenberg picture. Our case of an electron moving in a periodic potential belongs to this category. It has been customary to call Zitterbewegung a motion resulting from two (at least) interacting bands: Dirac-type model, Rashba spin subbands, Luttinger-type light and heavy hole bands, etc., and exhibiting an interband frequency. We have calculated (see Ref. [18]) the motion of electrons in graphene in a magnetic field and found that there is an intraband component of the motion, having the cyclotron frequency, and an interband frequency component which we called Zitterbewegung. We showed in the present Letter that the motion (classical or quantum) in a periodic potential can be modelled by a two-energy-bands structure and it is characterized by the interband frequency, so it should be called Zitterbewegung. The procedure based on the energy band structure appears to be quite versatile since it also includes cases like the Rashbatype spin subbands or the Luttinger-type light and heavy hole subbands which do not exhibit an energy gap and do not seem to have a direct classical interpretation. In fact, every time one deals with interacting energy bands the interference of states related to the different bands occurs and the Zitterbewegung will appear [12,13]. The distinctive character of the situation we considered is that it has a direct spatial interpretation and it is in analogy to the situation first considered by Schrödinger for the vacuum due to its alternative two-band description. In addition, since in our two-band k.p description of the above situation there appears an energy gap, it is possible to define unambiguously the interband frequency component of the motion, i.e. the Zitterbewegung. Such clear distinction is not possible in systems without a gap. The main conclusion of our work is that the electron Zitterbewegung in crystalline solids is not an obscure and marginal phenomenon but the basic way of electron propagation in a periodic potential. The ZB oscillations of electron velocity are simply due to the total energy conservation. The trembling motion can be described either as a mode of propagation in a periodic potential or, equivalently, by the two-band k.p model of band structure. The latter gives very good results because, using the effective mass and the energy gap, it reproduces the main features of the periodic potential. According to the two-band model the ZB is related to the interference of positive and negative energy components while the direct periodic potential approach reflects the real character of this motion. One should bear in mind that the standard conductivity theories use average electron velocities v̄ = h̄k/m∗ , where the pseudo-momentum h̄k is a constant of the motion. On the other hand, the instantaneous velocity is v = p/m0 (where p is the standard momentum) and it is not a constant of the motion. Our work shows that at very short times the electron dynamics is completely different from its average behavior. In particular, it is seen in Fig. 3 that in some time intervals the instantaneous velocity has an opposite sign to the average velocity. The established nature of ZB indicates that the latter should certainly be observable. In presenting our results we insist on the electron velocity since the latter is directly related to observable electric current: j = ev. Both periods and amplitudes of ZB, as shown in Fig. 2, are comparable to those appearing in the Bloch oscillation measurements [35], so the ZB should be observable experimentally by similar methods. Clearly, it is difficult to follow the behavior of a single electron and, in order to observe the trembling motion, one should produce many electrons moving in phase. This can be most readily done using laser pulses, see [23,35]. Recently, it was shown that also very short electric pulses can produce coherent electron beams [36]. We emphasize again that the behavior predicted theoretically in Figs. 1d and 2c agrees with the experimental simulation for free relativistic electrons observed by Gerritsma et al. [26]. This should not be surprising because these authors simulate the 1 + 1 Dirac equation and our calculations are based on the two-band k.p model which also simulates the Dirac equation [see Eq. (13)]. 5. Summary In summary, we considered fundamentals of electron motion in periodic structures and showed that the intensively studied phenomenon of electron Zitterbewegung in crystalline solids is caused by oscillations of velocity assuring the total energy conservation as the electron moves in a periodic potential. Thus the ZB represents the basic way of electron propagation in a periodic potential. This means that, although the ZB in solids was often studied in literature using the two-band k.p model of band structure analogous to the Dirac equation for relativistic electrons in a vacuum, the origins of ZB in a solid and in a vacuum are completely different. We also performed a rigorous quantum calculation of ZB for an electron in the Kronig–Penney potential and showed that the two-band k.p model is adequate for its description. It is emphasized that the two approaches treat the same phenomenon. 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