- Email: [email protected]
Novel surface states in lateral magnetic superlattices
Solid State Communications,
103, No. 3, pp. 161-165, 1997 8 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 003s1098/97 $17.00+.00
Pergamon
Vol.
PII: s0038-1098(97)00171-3
NOVEL SURFACE
STATES IN LATERAL
MAGNETIC
SUPERLATTICES
A. Akjouj* and B. Djafari-Rouhani Equipe de Dynamique des Interfaces, Laboratoire de Dynamique et Structure des Materiaux Moleculaires, URA CNRS 801, Unite de Physique, Universite de Lille I, 59655 Villeneuve d’Ascq Cedex, France (Received 3 February 1997; accepted in revised form 15 April 1997 by D.J. Lockwood)
A terminated Magnetic Kronig-Penney model is proposed for studying electronic localized states associated with the termination (surface) of an effective potential-modulation lateral magnetic superlattice. The lateral magnetic superlattice can be created by submitting a two-dimensional electron gas to a perpendicular magnetic field periodically modulated in one direction. The existence of these localized surface states in the forbidden minigaps of the superlattice is shown explicitly and their behaviors are discussed. 0 1997 Elsevier Science Ltd Keywords: A. nanostructures, A. semiconductors, A. interfaces, D. electronic states, D. quantum localization.
The interest in the electronic states and transport properties of electrons in a two-dimensional electron gas (2DEG) in spatially periodic lateral magnetic fields has been stimulated by the recent experimental availability of such systems [l-5]. In [ 11, the periodic magnetic field was produced by an array of superconducting stripes on the surface of the GaAs/AlGaAs heterojunctions with a 2DEG, while in [2-51 it was produced by deposition of ferromagnetic microstructures on top of the highmobility 2DEG. On the theoretical side, the creation of superlattices by an inhomogeneous magnetic field has been investigated [6-S] as well as electronic states and transport properties of a 2DEG in a weakly and periodically modulated magnetic field [9-171. Motivated by this wealth of experimental and theoretical results we propose in this letter to search the existence and behavior of localized states associated with the surface (termination) of this new type of superlattice. We limit ourselves in this paper to the simplest model of the superlattice and hope that further investigations of these new types of localized states and their influence on transport properties will be stimulated by this preliminary work. More specifically, we shall consider a
* Corresponding lillel .fr).
author (e-mail:
Akjouj @LipSrx.univ-
surfaces
and
magnetic Kronig-Penney (MKP) model [7] for a terminated effective potential-modulation lateral magnetic superlattice (LMSL). Ibrahim and Peeters [7] gave the bulk band structures of the MKP model containing a succession of bands separated by forbidden minigaps. Localized states associated with a perturbation of the perfect LMSL may exist inside these minigaps. The aim of this work is to show the existence of these surface states and calculate analytically their energy and localization properties by using the formalism of interface response theory [ 181. Then, the corresponding expressions are solved numerically. To obtain the electronic band structure of a semiinfinite compositional LMSL we assume a magnetic field profile [Fig. l(a)] modeled by a series of 6 function with alternating sign along the x axis. The vector potential and the effective potential which are depicted schematically in Figs l(b) and (c) consist of alternating wells and barriers of lengths 11 and 12 respectively. The LMSL is terminated at a layer (“cap layer”) which may have a width I, and a potential depth different from those in the interior layers; this layer is in contact with an homogeneous part (“substrate”) delimited by the step effective potential Vs. This system is the analogue of a classical semi-infinite compositional superlattice covered by a cap layer and in contact with an homogeneous substrate [ 191. 161
162
NOVEL SURFACE I
f
Lateral magnetic superlattice
STATES IN LATERAL I
1Cap layer1 Substrate
I (a)
MAGNETIC
SUPERLATTICES
Vol. 103, No. 3
form q&y) = eikvy e(x), where k, is the wave vector of the electron in the y direction. Then, with the above dimensionless quantities, the wave function *Jr(x) satisfies the one-dimensional Schrodinger equation
-$+2E-
I
2V(x, ky) q(x) = 0,
where V(x, k,J = +[k, + A(
(3)
can be interpreted as a k,-dependent effective potential. Note that the effective potential for electron motion in the x direction depends on the electron wave vector for motion in the y direction. The magnetic field can be modeled as [see Fig. l(a)]
F=
Fig. 1. Magnetic Kronig-Penney model of a semi-infinite lateral magnetic superlattice terminated by a different well at the surface and in contact with an homogeneous medium. We show the profiles of (a) the magnetic field B(x), (b) the vector potential A(x) and (c) the effective potential V(x, kJ for k,, = 2.5 (solid) and kY = - 2 (dashed). The motion of an electron in a two-dimensional plane (x, y) under the effect of a magnetic field B oriented along the z direction can be described by the Hamiltonian
H =
&(p+>)‘,
(1)
where m and e are respectively the mass and charge of the electron. The vector potential A will be defined in the Landau gauge A(x) = [0, A(x), 0] which results in B, = B(x) = dA(x)ldx. To express the results, we shall use exactly the same definitions and notations as those introduced by Ibrahim and Peeters [7] in the case of an infinite superlattice. Let Bs be the strength of some typical magnetic field in the problem. Then one can define the cyclotron frequency WC= eB,/mc and the magnetic length lB = [licleBo]“2. We express all the physical quantities in dimensionless units; namely the magnetic field B(x), the vector potential A(x), the energy E and the coordinate r are respectively expressed in units of BO, BdB, hw, and lg. In our numerical illustrations, we also use the same parameters as in [7], namely 1a = 813 A and hw, = 0.17 meV for GaAs and an estimated B0 of 0.1 T. The two-dimensional Schriidinger equation describing an electron with energy E admits solutions of the
I: n=O,-
[6(x+1,
+nz)-6(x+nZ)],
(4)
l,...
where 1= El + E2is the period length of the LMSL. This is the magnetic analogue of the well-known electronic Kronig-Penney model and is called the magnetic Kronig-Penney model [7]. The vector potential is a step function [see Fig. l(b)] and can be taken as - l/2 A(x) = A I- { l/2 Ao
inside the well(i = 2), inside the barrier(i = 1). (5)
The resulting effective potential [equation (3)] is also steplike and is depicted in Fig. l(c) for ky = 2.5 (solid curve) and ky = - 2 (dashed curve). Equation (2) with this potential (3) can be easily solved exactly [19]. The resulting dispersion relation between the electron energy E and its wave vector k = (k, ky) is given by cos(kl) = ClC2 + $,S, where Ci = Ch(aili),
(7a)
Si = sh(aiZi),
(7b)
and ai =
[ky +Ai12-22E,
i = 1,2.
(7c)
For the semi-infinite LMSL with a cap layer and in contact with a homogeneous substrate schematically depicted in Fig. 1, the closed form expressions enabling one to study within the forbidden gaps the existence and the dispersion relation of surface electronic states are
c,s2($-53 +s,s,($3 +C2S,($-$)
=0
(8a)
Vol. 103, No. 3
NOVEL SURFACE STATES IN LATERAL MAGNETIC SUPERLATTICES
together with the condition
with (9) and S,, C,., 01, and (Y,have the same definitions as Si, Ci and oi given by equation (7). Condition (8b) ensures that the wave is decaying when penetrating into the LMSL far from the surface. We now illustrate these theoretical results by a few numerical calculations for some specific geometry. For the sake of simplicity, we shall assume that the potential depth in the cap layer is the same as that in layers 2 of the superlattice (i.e. (Y,= (Ye) and keep as parameters the length I, and the effective potential step at surface V,. First, we consider the case for which I= 6, 1, = l/3, 1, = 12= 2113and V, = [k,, + 2]212. Figure 2 gives the dispersion of bulk and surface states as a function of the wave vector k,. The bulk bands are the shaded regions separated by forbidden minigaps where the surface states exist and are shown by dotted curves. These surface states are localized at the interface between the LMSL and the substrate. Electrons occupying these states will be confined to move parallel to the y axis with vanishing velocity v, component. The heavy line indicates the bottom of the substrate bulk band. One can notice the non reciprocal behavior of the dispersion curves which is
-3
-2
-1
0
1
2
3
kyW4J
Fig. 2. Dispersion relation (energy vs k,,) for the MKP system with period 1 = 6, 1, = l2 = 2113,l1 = l/3 and V, = 1% + 212/2. The shaded regions represent the bulk bands and the dotted curves show the localized surface states inside the minigaps. The heavy line indicates the bottom of the substrate bulk band. Only the lowest five minibands are shown here.
163
a consequence of the presence of a magnetic field. Besides the asymmetry between the bulk bands of the superlattice for positive and negative values of k, (already mentioned in [7]), one can observe, in our example, the absence of surface modes when k, is negative. Notice that when k, is positive in Fig. 2, the outermost layer in the superlattice (which is in contact with the substrate) behaves like a well. On the contrary, for the negative values of k,, the superlattice terminates at an effective potential barrier; moreover, in this latter case, most of the gaps fall inside the bulk band of the substrate and therefore localized states at the LMSLsubstrate boundary cannot exist. Of course, one can also realize the opposite situation where the surface modes exist for negative values of k,, rather than for positive values; for instance, this happens by assuming that the vector potential A2 in the outermost layer is positive and putting the substrate at a negative value of the vector potential A,. To study the localization properties of the surface states, local densities of states (LDOS) have been computed as a function of the space position x. This LDOS reflects the spatial behavior of the square modulus of the wave function l$(x)12.The degree of localization strongly depends on the energy separation between the surface state level and mini-band edges, as well as on the minigap width relative to the width of the neighboring mini-bands. To illustrate the behavior of surface states, we have plotted in Fig. 3 the LDOS as a function of the space position x for a given wave vector ky = 2.5 and for the lowest four surface states in Fig. 2. The energies of these surface states are respectively Qliw,) = 2.223, 2.882, 3.932 and 5.194. The attenuation of the wave function from one period of the LMSL to the next when penetrating deep into the LMSL far from the surface takes, respectively, the values 0.0587,0.0886,0.118 and 0.601. Besides the exponential decrease of the envelope of the LDOS, one can also observe an increasing number of oscillations in each period of the LMSL when going to higher energies. One can also notice that in case (d), where the energy is above the energy of the barrier in the LMSL, the maximum of the density of states occurs below the surface, in the barrier and not in the well. The energies of surface states are very sensitive to the effective potential V, of the substrate as well as to the width 1, of the surface layer in the LMSL. In Fig. 4(a), we show the variation of these energies vs V,, for k,, = 2.5. When V, increases the energies of the existing localized states increase until the corresponding branches merge into the bulk bands and become resonant states; at the same time new localized branches are extracted from the bulk bands and become almost constant for V, 2 12. Figure 4(b) presents the dependence of the energies of surface states with the thickness 1, of the surface layer,
NOVEL SURFACE
STATES IN LATERAL
MAGNETIC
SUPERLATTICES
Vol. 103, No. 3
6
5
-4
-3
-2
-1 Xle
0
1
2
-4
-3 -2
-1
0
1
r//////;y/////“““‘////////////q
2 6
xle
Fig. 3. Spatial dependence of the local density of states for the four lowest surface states in Fig. 2 at k,, = 2.5. The energies of these surface states are respectively E(hw,) = 2.223 (a), 2.882 (b), 3.932 (c) and 5.194 (d). The decay factors of the wave function from one period of the LMSL to the next are respectively: 0.0587,0.0886, 0.118 and 0.601. for kY = 2.5 and V, = [k, + 1]*/2 = 6.125. When 1, increases, the energies of the existing states decrease until the corresponding branches merge into the bulk bands and become resonant states; at the same time new localized branches are extracted from the bulk bands. Let us mention that, for any given energy E in Fig. 4(b), there is a periodic repetition of the localized states as a function of I,. Notice also that both the bulk bands and localized states are very dependent upon the period length 1 of the LMSL as well as upon the ratio 1,/12 (for a fixed k, and Z), even though we have not emphasized this aspect here. In summary, we have stressed for the first time to our knowledge the possibility of localized states in this new type of superlattices where the periodicity is created by the application of an alternating magnetic field to a 2DEG. Using a simple magnetic Kronig-Penney type model, we have investigated the existence and energies of surface states and shown their very dependence upon the effective potential barrier V, at the surface and the width 1, of the surface layer. The MKP model can of course be improved to describe more realistic realizations of the magnetic field. For example, the case of a periodic step magnetic field [7] can be treated analytically by using for instance a transfer matrix method; other realizations of the magnetic field can be modeled as a LMSL containing N different layers in each period (the case of arbitrary Ncan
5
^o 3
Q4
12
3
4
5
6
7
6
Fig. 4. Variations of the energies of the surface states vs: (a) the substrate effective potential V, with 1, = 203; (b) the width 1, of the surface well with V, = 6.125. The other parameters are 1= 6, l2 = 2113, 1, = 113 and k = 2.5. The shaded areas represent the bulk bands of tie LMSL. be handled like in usual compositional superlattice [20]). Our theoretical approach [18, 191 also enables us to investigate other interesting geometries; let us mention the case of a defect layer inside a perfect superlattice [21] or the case of a finite superlattice [22,23], where one can determine transmission and reflection coefficients of an incoming wave. The internal surface states in the electronic band structure of usual superlattices has already been observed by photoluminescence excitation spectra [24]. It would be interesting to have experimental data to test our prediction of localized surface states in the LMSL. In typical electronic transport experiments where the density of states predominates, the effect of surface states, as compared to bulk states, may be emphasized by using a
Vol. 103, No. 3 NOVEL SURFACE STATES IN LATERAL MAGNETIC SUPERLATTICES finite size LMSL, or by using a sample built of a periodical repetition of a finite size superlattice bounded by two wide media. The latter problem again falls in the category of complex basis superlattice [20, 251. REFERENCES 1. Carmona, H.A., Geim, A.K., Nogaret, A., Main, P.C., Foster, T.J., Henini, M., Beaumont, S.P. and Blamine, M.G., Phys. Rev. Left., 74, 1995, 3009. 2. Ye, P.D., Weiss, D., Gerhardts, R.R., Seeger, M., von Klitzing, K., Eberl, K. and Nickel, H., Phys. Rev. L&t., 74, 1995, 3013. 3. Izawa, S., Katsumoto, S., Endo, A. and Iye, Y., J. Phys. Sot. Jpn., 64, 1995, L706.
4. Ye, P.D., Weiss, D., von Klitzing, K., Eberl, K. and Nickel, H., Appl. Phys. Let?., 67, 1995, 1441. 5. Weiss, D., Liitjering, G., Ye, P.D. and Albrecht, C., in Quantum Transport in Semiconductor Submicron Structures (Edited by B. Kramer), p. 185. Kluwer, Netherlands, 1996. 6. Vil’ms, P.P. and Entin, M.V., Fiz. Tekh. Poluprovodn. 22, 1988, 1905, [Sov. Phys. Semicond., 22, 1988, 12091. 7. Ibrahim, I.S. and Peeters, F.M., Am. J. Phys., 63, 1995, 171; Phys. Rev., B52, 1995, 17321. 8. Krakovsky, A., Phys. Rev., B53, 1996, 8469.
9. Matulis, A., Peeters, F.M. and Vasilopoulos, P., Phys. Rev. L&t., 72, 1994, 1518; Vasilopoulos, P. and Peeters, F.M., Superlatt. Microstruc., 7, 1990, 393; Peeters, F.M. and Vasilopoulos, P., Phys. Rev., B47, 1993, 1466; Peeters, F.M. and Matulis, A., Phys. Rev., B48, 1993, 15 166.
10. Muller, J.E., Phys. Rev. Left., 68, 1992, 385. 11. Van Roy, W., De Boeck, J. and Borghs, G., Appl. Phys. L&t., 61, 1992, 3056.
12. Xue, D.P. and Xiao, G., Phys. Rev., B45, 1992, 5986; Wu, X. and Ulloa, E., Solid State Commun., 82, 1992, 945; Phys. Rev., B47, 1993, 7182; Phys. Rev., B47, 1993, 10028.
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13. Yoshika, D. and Iye, Y., J. Phys. Sot. Jpn., 56, 1987, 448; Yagi, R. and lye, Y., J. Phys. Sot. Japan, 62, 1993, 1279.
14. Calvo, M., Phys. Rev., B48, 1993, 2365; J. Phys.: Condens. Matter, 6, 1994, 3329; Phys. Rev., B51, 1995,2268. 15. Foden, C.L., Leadbeater, M.L., Burroughes, J.H. and Pepper, M., J. Phys.: Condens. Matter, 6, 1994, L127.
16. Chang, M.C. and Niu, Q., Phys. Rev., B50, 1994, 10843.
17. You, J.O., Zhang, Lide and Ghosh, P.K., Phys. Rev., B52, 1995, 17243.
18. Dobrzynski, L., Suti Sci. Rep., 11, 1990,139. 19. Bah, M.L., Akjouj, A., El Boudouti, E.H., DjafariRouhani, B . and Dobrzynski, L., J. Phys. : Condens. Matter, 8,1996,4171; Steslicka, M., Kucharczyk, R., El Boudouti, E.H., Djafari-Rouhani, B., Bah, M.L., Akjouj, A. and Dobrzynski, L., Vacuum, 46, 1995, 459; El Boudouti, E.H., Djafari-Rouhani, B., Khourdifi, E.M. and Dobrzynski, L., Phys. Rev., B48, 1993, 10987 (and references therein). 20. El Boudouti, E.H., Djafari-Rouhani, B., Akjouj, A. and Dobrzynski, L., Phys. Rev., B54, 1996, 14728 (and references therein). 21. Tamura, S., Phys. Rev., B38, 1989, 1261; Khourdifi, E.M. and Djafari-Rouhani, B., J. Phys.: Condens. Matter, 1, 1989, 7543. 22. El Boudouti, E.H. and Djafari-Rouhani, B., Phys. Rev., B49, 1994,4586; Mizuno, S. and Tamura, S., Phys. Rev., B53, 1996, 4549.
23. Kucharczyk, R. and Steslicka, M., Solid State Commun., 81, 1992,557;
84,1992,727.
24. Ohno, H., Mendez, E.E., Brum, J.A., Hong, J.M., AgullbRueda, F., Chang, L.L. and Esaki, L., Phys. Rev. Z&t., 64,1990,2555; Ohno, H., Mendez, E.E., Alexandrou, A. and Hong, J.M., Surf? Sci., 267, 1992, 161. 25. Yuh, P.F. and Wang, K.L., Phys. Rev., B38, 1988, 13307.
103, No. 3, pp. 161-165, 1997 8 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 003s1098/97 $17.00+.00
Pergamon
Vol.
PII: s0038-1098(97)00171-3
NOVEL SURFACE
STATES IN LATERAL
MAGNETIC
SUPERLATTICES
A. Akjouj* and B. Djafari-Rouhani Equipe de Dynamique des Interfaces, Laboratoire de Dynamique et Structure des Materiaux Moleculaires, URA CNRS 801, Unite de Physique, Universite de Lille I, 59655 Villeneuve d’Ascq Cedex, France (Received 3 February 1997; accepted in revised form 15 April 1997 by D.J. Lockwood)
A terminated Magnetic Kronig-Penney model is proposed for studying electronic localized states associated with the termination (surface) of an effective potential-modulation lateral magnetic superlattice. The lateral magnetic superlattice can be created by submitting a two-dimensional electron gas to a perpendicular magnetic field periodically modulated in one direction. The existence of these localized surface states in the forbidden minigaps of the superlattice is shown explicitly and their behaviors are discussed. 0 1997 Elsevier Science Ltd Keywords: A. nanostructures, A. semiconductors, A. interfaces, D. electronic states, D. quantum localization.
The interest in the electronic states and transport properties of electrons in a two-dimensional electron gas (2DEG) in spatially periodic lateral magnetic fields has been stimulated by the recent experimental availability of such systems [l-5]. In [ 11, the periodic magnetic field was produced by an array of superconducting stripes on the surface of the GaAs/AlGaAs heterojunctions with a 2DEG, while in [2-51 it was produced by deposition of ferromagnetic microstructures on top of the highmobility 2DEG. On the theoretical side, the creation of superlattices by an inhomogeneous magnetic field has been investigated [6-S] as well as electronic states and transport properties of a 2DEG in a weakly and periodically modulated magnetic field [9-171. Motivated by this wealth of experimental and theoretical results we propose in this letter to search the existence and behavior of localized states associated with the surface (termination) of this new type of superlattice. We limit ourselves in this paper to the simplest model of the superlattice and hope that further investigations of these new types of localized states and their influence on transport properties will be stimulated by this preliminary work. More specifically, we shall consider a
* Corresponding lillel .fr).
author (e-mail:
Akjouj @LipSrx.univ-
surfaces
and
magnetic Kronig-Penney (MKP) model [7] for a terminated effective potential-modulation lateral magnetic superlattice (LMSL). Ibrahim and Peeters [7] gave the bulk band structures of the MKP model containing a succession of bands separated by forbidden minigaps. Localized states associated with a perturbation of the perfect LMSL may exist inside these minigaps. The aim of this work is to show the existence of these surface states and calculate analytically their energy and localization properties by using the formalism of interface response theory [ 181. Then, the corresponding expressions are solved numerically. To obtain the electronic band structure of a semiinfinite compositional LMSL we assume a magnetic field profile [Fig. l(a)] modeled by a series of 6 function with alternating sign along the x axis. The vector potential and the effective potential which are depicted schematically in Figs l(b) and (c) consist of alternating wells and barriers of lengths 11 and 12 respectively. The LMSL is terminated at a layer (“cap layer”) which may have a width I, and a potential depth different from those in the interior layers; this layer is in contact with an homogeneous part (“substrate”) delimited by the step effective potential Vs. This system is the analogue of a classical semi-infinite compositional superlattice covered by a cap layer and in contact with an homogeneous substrate [ 191. 161
162
NOVEL SURFACE I
f
Lateral magnetic superlattice
STATES IN LATERAL I
1Cap layer1 Substrate
I (a)
MAGNETIC
SUPERLATTICES
Vol. 103, No. 3
form q&y) = eikvy e(x), where k, is the wave vector of the electron in the y direction. Then, with the above dimensionless quantities, the wave function *Jr(x) satisfies the one-dimensional Schrodinger equation
-$+2E-
I
2V(x, ky) q(x) = 0,
where V(x, k,J = +[k, + A(
(3)
can be interpreted as a k,-dependent effective potential. Note that the effective potential for electron motion in the x direction depends on the electron wave vector for motion in the y direction. The magnetic field can be modeled as [see Fig. l(a)]
F=
Fig. 1. Magnetic Kronig-Penney model of a semi-infinite lateral magnetic superlattice terminated by a different well at the surface and in contact with an homogeneous medium. We show the profiles of (a) the magnetic field B(x), (b) the vector potential A(x) and (c) the effective potential V(x, kJ for k,, = 2.5 (solid) and kY = - 2 (dashed). The motion of an electron in a two-dimensional plane (x, y) under the effect of a magnetic field B oriented along the z direction can be described by the Hamiltonian
H =
&(p+>)‘,
(1)
where m and e are respectively the mass and charge of the electron. The vector potential A will be defined in the Landau gauge A(x) = [0, A(x), 0] which results in B, = B(x) = dA(x)ldx. To express the results, we shall use exactly the same definitions and notations as those introduced by Ibrahim and Peeters [7] in the case of an infinite superlattice. Let Bs be the strength of some typical magnetic field in the problem. Then one can define the cyclotron frequency WC= eB,/mc and the magnetic length lB = [licleBo]“2. We express all the physical quantities in dimensionless units; namely the magnetic field B(x), the vector potential A(x), the energy E and the coordinate r are respectively expressed in units of BO, BdB, hw, and lg. In our numerical illustrations, we also use the same parameters as in [7], namely 1a = 813 A and hw, = 0.17 meV for GaAs and an estimated B0 of 0.1 T. The two-dimensional Schriidinger equation describing an electron with energy E admits solutions of the
I: n=O,-
[6(x+1,
+nz)-6(x+nZ)],
(4)
l,...
where 1= El + E2is the period length of the LMSL. This is the magnetic analogue of the well-known electronic Kronig-Penney model and is called the magnetic Kronig-Penney model [7]. The vector potential is a step function [see Fig. l(b)] and can be taken as - l/2 A(x) = A I- { l/2 Ao
inside the well(i = 2), inside the barrier(i = 1). (5)
The resulting effective potential [equation (3)] is also steplike and is depicted in Fig. l(c) for ky = 2.5 (solid curve) and ky = - 2 (dashed curve). Equation (2) with this potential (3) can be easily solved exactly [19]. The resulting dispersion relation between the electron energy E and its wave vector k = (k, ky) is given by cos(kl) = ClC2 + $,S, where Ci = Ch(aili),
(7a)
Si = sh(aiZi),
(7b)
and ai =
[ky +Ai12-22E,
i = 1,2.
(7c)
For the semi-infinite LMSL with a cap layer and in contact with a homogeneous substrate schematically depicted in Fig. 1, the closed form expressions enabling one to study within the forbidden gaps the existence and the dispersion relation of surface electronic states are
c,s2($-53 +s,s,($3 +C2S,($-$)
=0
(8a)
Vol. 103, No. 3
NOVEL SURFACE STATES IN LATERAL MAGNETIC SUPERLATTICES
together with the condition
with (9) and S,, C,., 01, and (Y,have the same definitions as Si, Ci and oi given by equation (7). Condition (8b) ensures that the wave is decaying when penetrating into the LMSL far from the surface. We now illustrate these theoretical results by a few numerical calculations for some specific geometry. For the sake of simplicity, we shall assume that the potential depth in the cap layer is the same as that in layers 2 of the superlattice (i.e. (Y,= (Ye) and keep as parameters the length I, and the effective potential step at surface V,. First, we consider the case for which I= 6, 1, = l/3, 1, = 12= 2113and V, = [k,, + 2]212. Figure 2 gives the dispersion of bulk and surface states as a function of the wave vector k,. The bulk bands are the shaded regions separated by forbidden minigaps where the surface states exist and are shown by dotted curves. These surface states are localized at the interface between the LMSL and the substrate. Electrons occupying these states will be confined to move parallel to the y axis with vanishing velocity v, component. The heavy line indicates the bottom of the substrate bulk band. One can notice the non reciprocal behavior of the dispersion curves which is
-3
-2
-1
0
1
2
3
kyW4J
Fig. 2. Dispersion relation (energy vs k,,) for the MKP system with period 1 = 6, 1, = l2 = 2113,l1 = l/3 and V, = 1% + 212/2. The shaded regions represent the bulk bands and the dotted curves show the localized surface states inside the minigaps. The heavy line indicates the bottom of the substrate bulk band. Only the lowest five minibands are shown here.
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a consequence of the presence of a magnetic field. Besides the asymmetry between the bulk bands of the superlattice for positive and negative values of k, (already mentioned in [7]), one can observe, in our example, the absence of surface modes when k, is negative. Notice that when k, is positive in Fig. 2, the outermost layer in the superlattice (which is in contact with the substrate) behaves like a well. On the contrary, for the negative values of k,, the superlattice terminates at an effective potential barrier; moreover, in this latter case, most of the gaps fall inside the bulk band of the substrate and therefore localized states at the LMSLsubstrate boundary cannot exist. Of course, one can also realize the opposite situation where the surface modes exist for negative values of k,, rather than for positive values; for instance, this happens by assuming that the vector potential A2 in the outermost layer is positive and putting the substrate at a negative value of the vector potential A,. To study the localization properties of the surface states, local densities of states (LDOS) have been computed as a function of the space position x. This LDOS reflects the spatial behavior of the square modulus of the wave function l$(x)12.The degree of localization strongly depends on the energy separation between the surface state level and mini-band edges, as well as on the minigap width relative to the width of the neighboring mini-bands. To illustrate the behavior of surface states, we have plotted in Fig. 3 the LDOS as a function of the space position x for a given wave vector ky = 2.5 and for the lowest four surface states in Fig. 2. The energies of these surface states are respectively Qliw,) = 2.223, 2.882, 3.932 and 5.194. The attenuation of the wave function from one period of the LMSL to the next when penetrating deep into the LMSL far from the surface takes, respectively, the values 0.0587,0.0886,0.118 and 0.601. Besides the exponential decrease of the envelope of the LDOS, one can also observe an increasing number of oscillations in each period of the LMSL when going to higher energies. One can also notice that in case (d), where the energy is above the energy of the barrier in the LMSL, the maximum of the density of states occurs below the surface, in the barrier and not in the well. The energies of surface states are very sensitive to the effective potential V, of the substrate as well as to the width 1, of the surface layer in the LMSL. In Fig. 4(a), we show the variation of these energies vs V,, for k,, = 2.5. When V, increases the energies of the existing localized states increase until the corresponding branches merge into the bulk bands and become resonant states; at the same time new localized branches are extracted from the bulk bands and become almost constant for V, 2 12. Figure 4(b) presents the dependence of the energies of surface states with the thickness 1, of the surface layer,
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Fig. 3. Spatial dependence of the local density of states for the four lowest surface states in Fig. 2 at k,, = 2.5. The energies of these surface states are respectively E(hw,) = 2.223 (a), 2.882 (b), 3.932 (c) and 5.194 (d). The decay factors of the wave function from one period of the LMSL to the next are respectively: 0.0587,0.0886, 0.118 and 0.601. for kY = 2.5 and V, = [k, + 1]*/2 = 6.125. When 1, increases, the energies of the existing states decrease until the corresponding branches merge into the bulk bands and become resonant states; at the same time new localized branches are extracted from the bulk bands. Let us mention that, for any given energy E in Fig. 4(b), there is a periodic repetition of the localized states as a function of I,. Notice also that both the bulk bands and localized states are very dependent upon the period length 1 of the LMSL as well as upon the ratio 1,/12 (for a fixed k, and Z), even though we have not emphasized this aspect here. In summary, we have stressed for the first time to our knowledge the possibility of localized states in this new type of superlattices where the periodicity is created by the application of an alternating magnetic field to a 2DEG. Using a simple magnetic Kronig-Penney type model, we have investigated the existence and energies of surface states and shown their very dependence upon the effective potential barrier V, at the surface and the width 1, of the surface layer. The MKP model can of course be improved to describe more realistic realizations of the magnetic field. For example, the case of a periodic step magnetic field [7] can be treated analytically by using for instance a transfer matrix method; other realizations of the magnetic field can be modeled as a LMSL containing N different layers in each period (the case of arbitrary Ncan
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Fig. 4. Variations of the energies of the surface states vs: (a) the substrate effective potential V, with 1, = 203; (b) the width 1, of the surface well with V, = 6.125. The other parameters are 1= 6, l2 = 2113, 1, = 113 and k = 2.5. The shaded areas represent the bulk bands of tie LMSL. be handled like in usual compositional superlattice [20]). Our theoretical approach [18, 191 also enables us to investigate other interesting geometries; let us mention the case of a defect layer inside a perfect superlattice [21] or the case of a finite superlattice [22,23], where one can determine transmission and reflection coefficients of an incoming wave. The internal surface states in the electronic band structure of usual superlattices has already been observed by photoluminescence excitation spectra [24]. It would be interesting to have experimental data to test our prediction of localized surface states in the LMSL. In typical electronic transport experiments where the density of states predominates, the effect of surface states, as compared to bulk states, may be emphasized by using a
Vol. 103, No. 3 NOVEL SURFACE STATES IN LATERAL MAGNETIC SUPERLATTICES finite size LMSL, or by using a sample built of a periodical repetition of a finite size superlattice bounded by two wide media. The latter problem again falls in the category of complex basis superlattice [20, 251. REFERENCES 1. Carmona, H.A., Geim, A.K., Nogaret, A., Main, P.C., Foster, T.J., Henini, M., Beaumont, S.P. and Blamine, M.G., Phys. Rev. Left., 74, 1995, 3009. 2. Ye, P.D., Weiss, D., Gerhardts, R.R., Seeger, M., von Klitzing, K., Eberl, K. and Nickel, H., Phys. Rev. L&t., 74, 1995, 3013. 3. Izawa, S., Katsumoto, S., Endo, A. and Iye, Y., J. Phys. Sot. Jpn., 64, 1995, L706.
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