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Subband-Landau-level coupling in tilted magnetic fields: Exact results for parabolic wells
Solid State Communications, Vol. 64, No. 1, pp. 99-101, 1987. Printed in Great Britain.
0038-1098/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
S U B B A N D - L A N D A U - L E V E L C O U P L I N G IN T I L T E D M A G N E T I C FIELDS: EXACT RESULTS FOR PARABOLIC WELLS R. Merlin* Max-Planck-Institut ffir Festk6rperforschung Heisenbergstrasse 1, D-7000 Stuttgart 80, F R G
(Received 10 April 1987 by M. Cardona) The problem of quasi-two-dimensional electrons in magnetic fields at arbitrary orientations is solved analytically for parabolic wells. The energy spectrum reveals features that are not apparent in results based on perturbation theory. IN QUASI T W O - D I M E N S I O N A L (2D) electron tlon describes an exact solution to the problem of systems, the motions in the confinement plane (x, y) quasi-2D electrons confined to parabolic wells in tilted and perpendicular to it are coupled for magnetic fields fields. The spectral properties at large angles show B at angles 0 v~ 0 with respect to z [1, 2]. This coupl- features that differ significantly from results of pering allows the study of intersubband excitations using turbation methods. The Hamiltonian for electrons confined in the (x, the technique of cyclotron resonance [3-7] and the observation of nominally forbidden inter-Landau y) plane and B = (B~, 0, Bz) is level transitions in Raman spectra [8]. Theoretically, n = (2m) ~[(Px + eB:y/c) 2 + P~ investigations of effects due to tilted fields have been + (Pz -- eBxy/C) 2] + mE~lh 2z2/2, (1) limited to perturbative approaches. First-order perturbation theory predicts a positive diamagnetic shift in the gauge A = y ( - Bz, 0, Bx); Eoj = hfL where f~ for the subbands proportional to IBI2 sin z 0, and subis the frequency of the harmonic motion along z (E01 band-Landau level anti-crossing with a minimum is used to emphasize the fact that hf~ is, in particular, splitting oc 0 [1, 2]. Diamagnetic shifts were first reporthe 0 ~ 1 intersubband energy). The momentum ted by Tsui [9] in tunneling experiments at 0 ~ rr/2. component Px is a constant of motion. In the {y, p.} Recent small-angle infrared [4, 6] and Raman scatterrepresentation, and after a trivial shift of the origin of ing measurements [8] have confirmed the expected coordinates, the Hamiltonian is written as 0-dependence of the cyclotron-intersubband resonant h 2 02 (32 p~ repulsion. 1 j.rT. 2 . 2 1 2 2m@2 + ~,,.%.Y -- ~ - m E d , ~ + 2m The results of perturbation theory for the sub- H band shifts are only valid when the cyclotron energy - e), sin Opzy, (2) hco, = IaelBl/mc is small compared to intersubband separations [1, 2]. In addition, the predictions for the with ~oc = e]B]/mc and sin 0 = Bx/IBI. The eigenstates anticrossing behavior apply for 0 ,~ 1 [2]. These con- are of the form exp (ipxx/h)Z ( y - Y0, P: - p0), where ditions are not always met in experiments; large angles p0 = pxtan 0 and Y0 = - P x (m~occos0) -~. Equation or fields are often required to bring particular levels (2) represents coupled harmonic oscillators. It can be close together [3, 5, 7]. Analysis of the data based on easily diagonalized by an appropriate rotation of perturbative calculations appears to be questionable coordinates y and hp:(mEm) -~. The resulting in these cases. However, the importance of higher- eigenenergies are given by [(n: + 1/2)E: + (np + order corrections to values of fm~,, or intersubband 1/2)E~] where n: and n~ are integers, and energies inferred from large-0 measurements has not as yet been elucidated. In this context, it is of interest E= = [h20)2c0s20{ -{- Eo21sin2~ to investigate the energy spectrum of a model that is h~o,.Eolsin (2~) sin 0] t/2, analytically soluble for arbitrary 0. This CommunicaEt~ = [-f-12o~2sin2c~+ Eo2,COS2~ -
-
+ hm,.Eo, sin (2~) sin 0]Ja; * Permanent address: Department of Physics, The University of Michigan, Ann Arbor, MI 48109I 120, USA. 99
(3)
the angle of rotation ~ is obtained from tan (2~) =
2ha~,.Eo,sin O(E2, - boo2) '.
(4)
100
SUBBAND-LANDAU-LEVEL-COUPLING 400
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Vol. 64, No. 1
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3OO 200 "i E
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Fig. 3. E~ and E~ [equation (3)] vs cos 0 for constant LB]. The dashed line is h~occos 0.
.
1
0
cos 8
///Eo~C°Se-w" .
i
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cyclotron resonance (for h~,, >> E0;). However, the lower branch is ~hog, cos0 for h~, ,~ E0~. The dependence of the coupled modes on Bx, for Fig. 1. (a) Energies of the lowest-lying eigenstates o f the Hamiltonian of equation (1) vs hOgc/Eo,. The levels fixed B:, is shown in Fig. 2. The cyclotron resonance correspond to n, = 0, n~ = 0, 1, 2, 3 and n, = 1, measurements of Brummell et al. [5] are displayed in na = 0. Arrows indicate the coupled cyclotron-inter- this form. For small Bx, both the intersubband and the subband transitions (energies E, and E~) shown in (b). cyclotron resonance exhibit diamagnetic shifts of oppThe dashed line is hog~. osite sign. In general, one finds that the shift of E0~ is The degeneracy of the level defined by n, and n~ is positive for E0, > h~o,. We should note that these eB:/(hc), i.e., the filling factor depends only on the field observations are (of course) consistent with perturbacomponent perpendicular to the confinement plane. tion theory: If h~o, cos 0 is not small compared with This result can be derived by applying boundary con- intersubband energies, the second- order contribution ditions on Px and Y0. of the term e B j m c ) ~p:y in equation (1) becomes Figures 1-3 show specific examples illustrating important leading to shifts that can have either sign. the behavior of the solutions, equations (3) and (4). In the example of Fig. 2, perturbation theory provides Figure 1 corresponds to the case where 0 is fixed, as in a good approximation to the actual shifts for the experiments of Wieck et al. [7]. Two important Bx < 0.4 B:. Beyond this range, higher-order contrifeatures should be noticed. First, the lower branch of butions cannot be neglected. The range where perturthe coupled intersubband- cyclotron-resonance exci- bative calculations apply generally increases with tations tends to E0~cos0 at large fields (Fig. lb). decreasing 7 = (h~o,cos0)/E01. Second, the asymptote of the upper branch is h~o~. In Fig. 3, we finally show the 0-dependence of E~ This indicates that, contrary to what it is commonly and Ee (equation (3)) for constant ]BJ. As expected, the assumed, IBI and not B~ determines the position of the lower branch tends to zero energy when 0 ~ ~/2 while the upper mode is displaced to higher energies. The lower branch is linear in cos 0 when 0 approaches ~/2. The slope can easily be derived using perturbation theory and it is given by hoot(1 + ];2) i/2. '
'
'
'
I
.
.
.
eBz
o
1
.
j
2
B×/ B z Fig. 2. E, and Ea [equation (3)] as a function of Bx/Bz.
Note addedAfter this manuscript was submitted, I learned of a recent work by Maan where the coupling problem for parabolic wells is solved using a differentgauge [10]. I thank C. Tejedor for pointing this referenceout to me.
Acknowledgements - - This work was supported in part by the U.S. Army Research Office under Contract No. DAAG-29-85-K-0175. Further support by the Alexander von Humboldt-Stiftung is gratefully acknowledged.
Vol. 64, No. 1
SUBBAND-LANDAU-LEVEL-COUPLING REFERENCES
1. 2. 3. 4. 5.
F. Stern & W.E. Howard, Phys. Rev. 163, 816 (1967). T. Ando, Phys. Rev. B19, 2106 (1979). See also: T. Ando, A.B. Fowler & F. Stern, Revs. Mod. Phys. 54, 437 (1982). W. Beinvogl & J.F. Koch, Phys. Rev. Lett. 40, 1736 (1978). Z. Schlesinger, J.C.M. Hwang & S.J. Allen, Phys. Rev. Lett. 50, 2098 (1983). M.A. Brummell, M.A. Hopkins, R.J. Nicholas, J.C. Portal, K.Y. Cheng & A.Y. Cho, J. Phys. C19, L107 (1986).
6. 7. 8. 9. 10.
101
G.L.J.A. Rikken, H. Sigg, C.J.G.M. Langerak, H.W. Myron, J.A.A.J. Perenboom & G. Weimann, Phys. Rev. B34, 5590 (1986). A.D. Wieck, J.C. Maan, U. Merkt, J.P. Kotthaus, K. Ploog & G. Weimann, Phys. Rev. B35, 4145 (1987). R. Borroff, R. Merlin, R.L. Greene & J. Comas, unpublished. D.C. Tsui, Solid State Commun. 9, 1789 (1971). J.C. Maan, Two Dimensional Systems, Heterostructures, and Superlattices, 183 (1984), (Edited by G. Bauer, F. Kuchar and H. Heinrich) Springer-Verlag, Berlin.
0038-1098/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
S U B B A N D - L A N D A U - L E V E L C O U P L I N G IN T I L T E D M A G N E T I C FIELDS: EXACT RESULTS FOR PARABOLIC WELLS R. Merlin* Max-Planck-Institut ffir Festk6rperforschung Heisenbergstrasse 1, D-7000 Stuttgart 80, F R G
(Received 10 April 1987 by M. Cardona) The problem of quasi-two-dimensional electrons in magnetic fields at arbitrary orientations is solved analytically for parabolic wells. The energy spectrum reveals features that are not apparent in results based on perturbation theory. IN QUASI T W O - D I M E N S I O N A L (2D) electron tlon describes an exact solution to the problem of systems, the motions in the confinement plane (x, y) quasi-2D electrons confined to parabolic wells in tilted and perpendicular to it are coupled for magnetic fields fields. The spectral properties at large angles show B at angles 0 v~ 0 with respect to z [1, 2]. This coupl- features that differ significantly from results of pering allows the study of intersubband excitations using turbation methods. The Hamiltonian for electrons confined in the (x, the technique of cyclotron resonance [3-7] and the observation of nominally forbidden inter-Landau y) plane and B = (B~, 0, Bz) is level transitions in Raman spectra [8]. Theoretically, n = (2m) ~[(Px + eB:y/c) 2 + P~ investigations of effects due to tilted fields have been + (Pz -- eBxy/C) 2] + mE~lh 2z2/2, (1) limited to perturbative approaches. First-order perturbation theory predicts a positive diamagnetic shift in the gauge A = y ( - Bz, 0, Bx); Eoj = hfL where f~ for the subbands proportional to IBI2 sin z 0, and subis the frequency of the harmonic motion along z (E01 band-Landau level anti-crossing with a minimum is used to emphasize the fact that hf~ is, in particular, splitting oc 0 [1, 2]. Diamagnetic shifts were first reporthe 0 ~ 1 intersubband energy). The momentum ted by Tsui [9] in tunneling experiments at 0 ~ rr/2. component Px is a constant of motion. In the {y, p.} Recent small-angle infrared [4, 6] and Raman scatterrepresentation, and after a trivial shift of the origin of ing measurements [8] have confirmed the expected coordinates, the Hamiltonian is written as 0-dependence of the cyclotron-intersubband resonant h 2 02 (32 p~ repulsion. 1 j.rT. 2 . 2 1 2 2m@2 + ~,,.%.Y -- ~ - m E d , ~ + 2m The results of perturbation theory for the sub- H band shifts are only valid when the cyclotron energy - e), sin Opzy, (2) hco, = IaelBl/mc is small compared to intersubband separations [1, 2]. In addition, the predictions for the with ~oc = e]B]/mc and sin 0 = Bx/IBI. The eigenstates anticrossing behavior apply for 0 ,~ 1 [2]. These con- are of the form exp (ipxx/h)Z ( y - Y0, P: - p0), where ditions are not always met in experiments; large angles p0 = pxtan 0 and Y0 = - P x (m~occos0) -~. Equation or fields are often required to bring particular levels (2) represents coupled harmonic oscillators. It can be close together [3, 5, 7]. Analysis of the data based on easily diagonalized by an appropriate rotation of perturbative calculations appears to be questionable coordinates y and hp:(mEm) -~. The resulting in these cases. However, the importance of higher- eigenenergies are given by [(n: + 1/2)E: + (np + order corrections to values of fm~,, or intersubband 1/2)E~] where n: and n~ are integers, and energies inferred from large-0 measurements has not as yet been elucidated. In this context, it is of interest E= = [h20)2c0s20{ -{- Eo21sin2~ to investigate the energy spectrum of a model that is h~o,.Eolsin (2~) sin 0] t/2, analytically soluble for arbitrary 0. This CommunicaEt~ = [-f-12o~2sin2c~+ Eo2,COS2~ -
-
+ hm,.Eo, sin (2~) sin 0]Ja; * Permanent address: Department of Physics, The University of Michigan, Ann Arbor, MI 48109I 120, USA. 99
(3)
the angle of rotation ~ is obtained from tan (2~) =
2ha~,.Eo,sin O(E2, - boo2) '.
(4)
100
SUBBAND-LANDAU-LEVEL-COUPLING 400
.
.
.
.
I
.
.
.
Vol. 64, No. 1
.
3OO 200 "i E
~oo a i .
J .
.
i
i
.
I
i
I
.
i .
L
.
W 5o z b.I
I
.
n- 200 hi Z b.I
/ /
.
.
.
.
I
i
0.5
lOO
.
.
.
I
o
.
.
.
Fig. 3. E~ and E~ [equation (3)] vs cos 0 for constant LB]. The dashed line is h~occos 0.
.
1
0
cos 8
///Eo~C°Se-w" .
i
2
f~c I Eol
cyclotron resonance (for h~,, >> E0;). However, the lower branch is ~hog, cos0 for h~, ,~ E0~. The dependence of the coupled modes on Bx, for Fig. 1. (a) Energies of the lowest-lying eigenstates o f the Hamiltonian of equation (1) vs hOgc/Eo,. The levels fixed B:, is shown in Fig. 2. The cyclotron resonance correspond to n, = 0, n~ = 0, 1, 2, 3 and n, = 1, measurements of Brummell et al. [5] are displayed in na = 0. Arrows indicate the coupled cyclotron-inter- this form. For small Bx, both the intersubband and the subband transitions (energies E, and E~) shown in (b). cyclotron resonance exhibit diamagnetic shifts of oppThe dashed line is hog~. osite sign. In general, one finds that the shift of E0~ is The degeneracy of the level defined by n, and n~ is positive for E0, > h~o,. We should note that these eB:/(hc), i.e., the filling factor depends only on the field observations are (of course) consistent with perturbacomponent perpendicular to the confinement plane. tion theory: If h~o, cos 0 is not small compared with This result can be derived by applying boundary con- intersubband energies, the second- order contribution ditions on Px and Y0. of the term e B j m c ) ~p:y in equation (1) becomes Figures 1-3 show specific examples illustrating important leading to shifts that can have either sign. the behavior of the solutions, equations (3) and (4). In the example of Fig. 2, perturbation theory provides Figure 1 corresponds to the case where 0 is fixed, as in a good approximation to the actual shifts for the experiments of Wieck et al. [7]. Two important Bx < 0.4 B:. Beyond this range, higher-order contrifeatures should be noticed. First, the lower branch of butions cannot be neglected. The range where perturthe coupled intersubband- cyclotron-resonance exci- bative calculations apply generally increases with tations tends to E0~cos0 at large fields (Fig. lb). decreasing 7 = (h~o,cos0)/E01. Second, the asymptote of the upper branch is h~o~. In Fig. 3, we finally show the 0-dependence of E~ This indicates that, contrary to what it is commonly and Ee (equation (3)) for constant ]BJ. As expected, the assumed, IBI and not B~ determines the position of the lower branch tends to zero energy when 0 ~ ~/2 while the upper mode is displaced to higher energies. The lower branch is linear in cos 0 when 0 approaches ~/2. The slope can easily be derived using perturbation theory and it is given by hoot(1 + ];2) i/2. '
'
'
'
I
.
.
.
eBz
o
1
.
j
2
B×/ B z Fig. 2. E, and Ea [equation (3)] as a function of Bx/Bz.
Note addedAfter this manuscript was submitted, I learned of a recent work by Maan where the coupling problem for parabolic wells is solved using a differentgauge [10]. I thank C. Tejedor for pointing this referenceout to me.
Acknowledgements - - This work was supported in part by the U.S. Army Research Office under Contract No. DAAG-29-85-K-0175. Further support by the Alexander von Humboldt-Stiftung is gratefully acknowledged.
Vol. 64, No. 1
SUBBAND-LANDAU-LEVEL-COUPLING REFERENCES
1. 2. 3. 4. 5.
F. Stern & W.E. Howard, Phys. Rev. 163, 816 (1967). T. Ando, Phys. Rev. B19, 2106 (1979). See also: T. Ando, A.B. Fowler & F. Stern, Revs. Mod. Phys. 54, 437 (1982). W. Beinvogl & J.F. Koch, Phys. Rev. Lett. 40, 1736 (1978). Z. Schlesinger, J.C.M. Hwang & S.J. Allen, Phys. Rev. Lett. 50, 2098 (1983). M.A. Brummell, M.A. Hopkins, R.J. Nicholas, J.C. Portal, K.Y. Cheng & A.Y. Cho, J. Phys. C19, L107 (1986).
6. 7. 8. 9. 10.
101
G.L.J.A. Rikken, H. Sigg, C.J.G.M. Langerak, H.W. Myron, J.A.A.J. Perenboom & G. Weimann, Phys. Rev. B34, 5590 (1986). A.D. Wieck, J.C. Maan, U. Merkt, J.P. Kotthaus, K. Ploog & G. Weimann, Phys. Rev. B35, 4145 (1987). R. Borroff, R. Merlin, R.L. Greene & J. Comas, unpublished. D.C. Tsui, Solid State Commun. 9, 1789 (1971). J.C. Maan, Two Dimensional Systems, Heterostructures, and Superlattices, 183 (1984), (Edited by G. Bauer, F. Kuchar and H. Heinrich) Springer-Verlag, Berlin.