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Nonlocal effects in helicon wave propagation in a superlattice
Superlattices
and Microstructures,
Vol. 3, No. 6, 1987
NONLOCAL EFFECTS IN HELICDN IdAvE PROPPFATIOM Ii'! A SUPERLATTICF R. F!. Narahari bchar Memohis State University, Memohis, Tennessee 38157 USA (Received 15 July 19P7) Dispersion relations have been obtained on the basis of linear response theory for helicon waves propagatina parallel to a maanetic field applied along the axis of a superlattice. Numerical apolications have been made to a Kronig-Penney model and the preliminary results clearly indicate nonlocal effects.
1.
Introduction
Helicon wave propagation in semiconductor superlattices has received some attention in recent years. Since the early suggestions about the existence of helicons in superlattices were made by Maan -et al.' on experi[mental qrounds and by Das Sarma and Quinn' on theoretical grounds,.several theoretical papers have appeared.je6 In all of the theoretical papers, with the exception of Ref. 3, the propagation of helicon waves is treated in the local aoproximation. Even though the theoretical formulation in Ref. 3 is quite general, the numerical applications are made in Ref. 4, and then only in the so-called semiclassical limit. The importance of nonlocal effects in helicon dispersion in homogeneous media is well-known.' It has also been shown that nonlocal effects cause drastic changes in the dispersion of helicons in sinusoidally modulated oeriodic structures.' It is the purpose of this paper to study the importance of nonlocal effects for helicon dispersion in superlattices. An outline of the linear response theory for helicons in a superlattice is given in the next section and the results and discussion of a numerical calculation for a Kronig-Penney model are oresented in section 3. 2.
Theoretical Outline for Helicon Dispersion in a Superlattice
Consider a system with an electronic structure which is free electron like along the x and y directions, but is oeriodic alonq the z direction. The periodic potential in the z direction is represented by a KroniqPenney model. There is a static magnetic field co applied along the z direction.
a periodic structure based on the linear response formalism, given by the author,'>'" can be applied to the present case. As shown in Ref. 9, the frequencies of the helicons are determined from 3, (o&
(2 1)
= C2q2
Here :+(Q,>) arethe
wave vector and frequency
dependent components of the dielectric tensor in the polarization representation and are qiven in terms of the corresponding Cartesian comoonents by ,(9,')=
f i-
(9.. 'xY
:2 i;
and C is the soeed of light. The dielectric tensor is related to the conductivity tensor ti(l,.) accordinn to c(o.,,
j = _I?+
i-
(2.3)
'9 Here ~I is the dielectric
constant of the
lattice and 1 is the unit tensor. The expression for the conductivity tensor on the basis of linear response theorv has been given in Ref. C in terms of the energy einen values E = (n+ 1/2)i$+
!2 A\i
(kz)
associated with the eiaen states ‘i,_z e'kYyUn(xt,;kyj~
of the unperturbed
b,(kz+K)ei(k_
K
- K’z ‘2.5!
Hamiltonian
Ho y (i - e$)'/Zm
'?.F_,
+ V(z)
There is also an electromagnetic disturbance fi that va_yies as exp(iq.r - &t), with the wave
where V(z) is the periodic potential alonn z
vector q also taken to be along the z direction. The medium is assumed to be nonmagnetic. A theory for helicon propaaation in
guage correspondinq
and A0 is the vector potential field EQ.
in the Landau
to the static magnetic
Here!J,,aretheharmonic
oscillator
@1987Academtc
PressLImIted
642
Superlattices
and Microstructures,
Vol. 3, No. 6, 7987
‘o’er-l ,_a_;-&
IKRONIG-PENNY
0.0
qd/2lT
MODEL
0.3
0.1
0.5
qd/2TT
Figure la. Real part of the helicon mode frequency vs wave vector. Fiqure lb. Imaginary part of the helicon
mode frequency vs wave vector. Inset shows the parameters of the Kronig-Penney model.
wave functions and $,= (ft/bC)l'*
as a function of the wave vector have been calculated for a Kronig-Penney model of a superlattice and some preliminary results are shown in Fig. la and Fig. lb resoectively. The values of the model parameters used in this calculation have been taken from Ando.'l In the standard notation, indicated by the inset in Fig. lb, the model parameters have the following values: V = 190 meV, a= b=4.5m, 24 -3 and m*= 0.068m The value no= 1.94 x 10 m 0' of the collision parameter T corresponds to
netic length.
is the mag-
The be(kzf K) are the exuan-
sion coefficients associated with a set of plane wave basis functions as described in Ref. 0. Damping effects can be considered by treating the frequency wto be a complex quanas a result of whichthedielectity O- i/T, tric tensor becomes a complex quantity: i_3+(q,o)=e;(q,t0)f islj(q,o)
(2.7)
We choose to work with a real wave vector q and a complex frequency in the dispersion relation given by eq. (2.1). Furthermore, we focus on the mode associated with the positive sign and henceforth drop the subscript and the arguments (a,o). The real part 13' and the imaginary part Lti'of the helicon mode frequency are then qiven by 2 :,; ’
_ .J
=$q5,(;J+:“2)
21*1’ua z _$,$I’,( 3.
E12+ $)
(2.8) (2.9)
Results and Discussion
The variation of both the real and the imaginary parts of the helicon mode frequency
8x'T C
=
d.
As can be seen, both the real and the imaqinary parts of the helicon mode frequency show an initial sharp increase, a maximum well below oc(~lO1*)
followed by an almost equally
sharp drop. The dispersion does not extend For small all the way to the zone boundary. values of q, the dispersion for U' and ,," have been calculated by using a Taylor expansion of E' and E" and retaining terms up to q2. The dispersion for ~2' so obtained in the long wavelength limit is shown as a solid line in Fig. la and agrees with the dispersion obtained by using a local approximation.4 When
Superlattices
and Microstructures,
q becomes larger, and is real as has been assumed in this work, solutions of eq. (2.8) and (2.9) exist only for a few discrete values of q and these are shown as points in the figures. The dotted lines are drawn only as guidelines. In any case, the drop in the frequency as q increases is clearly due to nonlocal effects and is similar to the known case of nonlocal effects in helicon dispersion in a homogeneous medium' and in a sinusoidal periodic structure.@ In contrast, the calculations using the local approximation" show (,' increasinq monotonically up to w and then C reaching a plateau. Work is still in proqress. The alternative description of helicon dampina by treatinq Ui as real and q as complex is also being considered. The details will be given in a later publication. References 1.
643
Vol. 3, No. 6, 1987
J. C. Maan, A. Altarelli, H. Siqq, P. Wvner. L. I_. Chang, and L. Esaki, Surface Science 113, 347 (1982).
2. 3.
4. 5. 6. 7. 8. 9. IO. 11.
S. Das Sarma and J. J. Quinn, Physical Review e, 7603 (1982). A. Tselis, G. Gonzalez de la Cruz, and il. J. Quinn, Solid State Communications 47, 43 (1983). A. C. Tselis and J. J. Quinn, Physical Review E, 2021 (1984). M. S. Kushwaha, Phvsica Status Solidi (b) 136, 757 (1986). _ L. Wendler and M. I. Kaaanov. Phvrica Status Solidi (b) 3, K33 (I98611 E. A. Kaner and V. G. Skobov, Advances in Physics, 11, 605 (1968). 6. N. Narahari Achar, Physica Status Solidi (b), 140, K37 (1987). B. N. Narahari Achar, Physical Review g, 7334 (1987). B. N.‘Narahari Achar (to be published). T. Ando in Physics in m Magnetic Fields, proceedings of the Ei InternationalSiminar, Hakone, Japan, Sept 10-13, 1980. Editors: S. Chikazumi and M. Miura. (Sorinaer-Verlag, Berlin, 1981), D. 301.
and Microstructures,
Vol. 3, No. 6, 1987
NONLOCAL EFFECTS IN HELICDN IdAvE PROPPFATIOM Ii'! A SUPERLATTICF R. F!. Narahari bchar Memohis State University, Memohis, Tennessee 38157 USA (Received 15 July 19P7) Dispersion relations have been obtained on the basis of linear response theory for helicon waves propagatina parallel to a maanetic field applied along the axis of a superlattice. Numerical apolications have been made to a Kronig-Penney model and the preliminary results clearly indicate nonlocal effects.
1.
Introduction
Helicon wave propagation in semiconductor superlattices has received some attention in recent years. Since the early suggestions about the existence of helicons in superlattices were made by Maan -et al.' on experi[mental qrounds and by Das Sarma and Quinn' on theoretical grounds,.several theoretical papers have appeared.je6 In all of the theoretical papers, with the exception of Ref. 3, the propagation of helicon waves is treated in the local aoproximation. Even though the theoretical formulation in Ref. 3 is quite general, the numerical applications are made in Ref. 4, and then only in the so-called semiclassical limit. The importance of nonlocal effects in helicon dispersion in homogeneous media is well-known.' It has also been shown that nonlocal effects cause drastic changes in the dispersion of helicons in sinusoidally modulated oeriodic structures.' It is the purpose of this paper to study the importance of nonlocal effects for helicon dispersion in superlattices. An outline of the linear response theory for helicons in a superlattice is given in the next section and the results and discussion of a numerical calculation for a Kronig-Penney model are oresented in section 3. 2.
Theoretical Outline for Helicon Dispersion in a Superlattice
Consider a system with an electronic structure which is free electron like along the x and y directions, but is oeriodic alonq the z direction. The periodic potential in the z direction is represented by a KroniqPenney model. There is a static magnetic field co applied along the z direction.
a periodic structure based on the linear response formalism, given by the author,'>'" can be applied to the present case. As shown in Ref. 9, the frequencies of the helicons are determined from 3, (o&
(2 1)
= C2q2
Here :+(Q,>) arethe
wave vector and frequency
dependent components of the dielectric tensor in the polarization representation and are qiven in terms of the corresponding Cartesian comoonents by ,(9,')=
f i-
(9.. 'xY
:2 i;
and C is the soeed of light. The dielectric tensor is related to the conductivity tensor ti(l,.) accordinn to c(o.,,
j = _I?+
i-
(2.3)
'9 Here ~I is the dielectric
constant of the
lattice and 1 is the unit tensor. The expression for the conductivity tensor on the basis of linear response theorv has been given in Ref. C in terms of the energy einen values E = (n+ 1/2)i$+
!2 A\i
(kz)
associated with the eiaen states ‘i,_z e'kYyUn(xt,;kyj~
of the unperturbed
b,(kz+K)ei(k_
K
- K’z ‘2.5!
Hamiltonian
Ho y (i - e$)'/Zm
'?.F_,
+ V(z)
There is also an electromagnetic disturbance fi that va_yies as exp(iq.r - &t), with the wave
where V(z) is the periodic potential alonn z
vector q also taken to be along the z direction. The medium is assumed to be nonmagnetic. A theory for helicon propaaation in
guage correspondinq
and A0 is the vector potential field EQ.
in the Landau
to the static magnetic
Here!J,,aretheharmonic
oscillator
@1987Academtc
PressLImIted
642
Superlattices
and Microstructures,
Vol. 3, No. 6, 7987
‘o’er-l ,_a_;-&
IKRONIG-PENNY
0.0
qd/2lT
MODEL
0.3
0.1
0.5
qd/2TT
Figure la. Real part of the helicon mode frequency vs wave vector. Fiqure lb. Imaginary part of the helicon
mode frequency vs wave vector. Inset shows the parameters of the Kronig-Penney model.
wave functions and $,= (ft/bC)l'*
as a function of the wave vector have been calculated for a Kronig-Penney model of a superlattice and some preliminary results are shown in Fig. la and Fig. lb resoectively. The values of the model parameters used in this calculation have been taken from Ando.'l In the standard notation, indicated by the inset in Fig. lb, the model parameters have the following values: V = 190 meV, a= b=4.5m, 24 -3 and m*= 0.068m The value no= 1.94 x 10 m 0' of the collision parameter T corresponds to
netic length.
is the mag-
The be(kzf K) are the exuan-
sion coefficients associated with a set of plane wave basis functions as described in Ref. 0. Damping effects can be considered by treating the frequency wto be a complex quanas a result of whichthedielectity O- i/T, tric tensor becomes a complex quantity: i_3+(q,o)=e;(q,t0)f islj(q,o)
(2.7)
We choose to work with a real wave vector q and a complex frequency in the dispersion relation given by eq. (2.1). Furthermore, we focus on the mode associated with the positive sign and henceforth drop the subscript and the arguments (a,o). The real part 13' and the imaginary part Lti'of the helicon mode frequency are then qiven by 2 :,; ’
_ .J
=$q5,(;J+:“2)
21*1’ua z _$,$I’,( 3.
E12+ $)
(2.8) (2.9)
Results and Discussion
The variation of both the real and the imaginary parts of the helicon mode frequency
8x'T C
=
d.
As can be seen, both the real and the imaqinary parts of the helicon mode frequency show an initial sharp increase, a maximum well below oc(~lO1*)
followed by an almost equally
sharp drop. The dispersion does not extend For small all the way to the zone boundary. values of q, the dispersion for U' and ,," have been calculated by using a Taylor expansion of E' and E" and retaining terms up to q2. The dispersion for ~2' so obtained in the long wavelength limit is shown as a solid line in Fig. la and agrees with the dispersion obtained by using a local approximation.4 When
Superlattices
and Microstructures,
q becomes larger, and is real as has been assumed in this work, solutions of eq. (2.8) and (2.9) exist only for a few discrete values of q and these are shown as points in the figures. The dotted lines are drawn only as guidelines. In any case, the drop in the frequency as q increases is clearly due to nonlocal effects and is similar to the known case of nonlocal effects in helicon dispersion in a homogeneous medium' and in a sinusoidal periodic structure.@ In contrast, the calculations using the local approximation" show (,' increasinq monotonically up to w and then C reaching a plateau. Work is still in proqress. The alternative description of helicon dampina by treatinq Ui as real and q as complex is also being considered. The details will be given in a later publication. References 1.
643
Vol. 3, No. 6, 1987
J. C. Maan, A. Altarelli, H. Siqq, P. Wvner. L. I_. Chang, and L. Esaki, Surface Science 113, 347 (1982).
2. 3.
4. 5. 6. 7. 8. 9. IO. 11.
S. Das Sarma and J. J. Quinn, Physical Review e, 7603 (1982). A. Tselis, G. Gonzalez de la Cruz, and il. J. Quinn, Solid State Communications 47, 43 (1983). A. C. Tselis and J. J. Quinn, Physical Review E, 2021 (1984). M. S. Kushwaha, Phvsica Status Solidi (b) 136, 757 (1986). _ L. Wendler and M. I. Kaaanov. Phvrica Status Solidi (b) 3, K33 (I98611 E. A. Kaner and V. G. Skobov, Advances in Physics, 11, 605 (1968). 6. N. Narahari Achar, Physica Status Solidi (b), 140, K37 (1987). B. N. Narahari Achar, Physical Review g, 7334 (1987). B. N.‘Narahari Achar (to be published). T. Ando in Physics in m Magnetic Fields, proceedings of the Ei InternationalSiminar, Hakone, Japan, Sept 10-13, 1980. Editors: S. Chikazumi and M. Miura. (Sorinaer-Verlag, Berlin, 1981), D. 301.