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A note on the plasmon dispersion in semiconductor superlattices
0038-1098/87 $3.00 + .00
Solid State Communications, VoI.63,No.12, pp. I081-I082, 1987. Printed in Great Britain.
Pergamon Journals Ltd.
A NOTE ON THE PLASMON DISP~SION IN SEMICONDUCTOR SUPERLATTICF~ Rui-Qing Yang Institute of Solid State Physlcs, Nanjing University, Nanjlng and Chi en-Hua Tsai Department of Physics and Institute of Condensed Matter Physics Jiao-Tong University, Shanghai, The People's Republic of China (Received 30 May, 1987 by W. Y. Kuan)
The plasmon dispersion relation in semiconductor superlattices is derived by taking into account energy band curvature due to charge transfer. The result is in nice agreement with experimental data.
There has been a number of theoretical
not negligible in comparison with V . Thus, the
works I-6 on the plasmon dispersion in semiconductor superlattices. The plasmen excitatlon
shape of the well differs greatly from a square one~ despite the fact that calculation of the plasmon frequencies using a square well model may yield results better than that defined by Equation (I). This is owing to the fact that in any model without electron tunnelling effect and with two-dimensional polarizability# the momentum representation of the Coulomb interaction between electrons, in particular when qA <__I, is not sensitive to the concrete form of the electronic wavefunction. Here, A is the modulation wavelength, and q, the wavevector. Square well wave functions allow electrons to form layers of finite thickness. This and the inclusion of interactions between electrons in the lowest and first excited subbands shift the plesmon frequencies nearer to experimental values than those given by Equation (I). However, in cases of heavy deoing, as explained above, the bending of energy bands cannot be ignored. The elegtrons are then pushed to concentrate towards the edges nf the well. We adopt the following electron density profile in the z-direction
o
spectrum has also been observed 7 by means of light scattering experiments, with results agreeing approximately with the theoretical expectation
, 2 ~ e 2"
si~2~a A
,~
(1)
based on a model of periodically arranged layers of electron gases. In Equation (1), n is the 2-D electronic density, m*, the electron's effective mass, eM' the background dielectric function, and ki! and kl, the inplane and perpendicular wavevectors, respectively. Wasserman and Lee 8 obtained plasmon discurves nearer to measured ones than Equation (I) by taking into account the finite thickness of electron layers. However, they adopted an infinitely deep square well model. For the sample 1 of Reference (7), they gave their comouted value of the energy difference between the first exclted and the ground states as A ~ 23. SmeV whereas the experimental value is A ~ 10.83meV. The carrier densities in the two samples of GaAs/Ga1_xAlxAS mentloned in Reference (7) are, respectively, 7.3.1011cm-2 and 5.5.1011cm"2. In cases of high carrier densitles like these, the bending of energy bands resulting from charge transfer becomes signlficant~ and corrections due to the self-consistent potential 9' 10 of the electrons are necessary. The difference V between the conduction levels of Gal.xAS x and e GaAs is given approximately by the empirical formula Vo=10x , with x expressed in percentage and V
in meV. For the two samples of Reference o (I), Vo equals, respectively, to 200meV and 110
meV, whereas, the potential difference between electrons located at the center and at the edge of the well exceeds 30meV, which is certalnly
I~12(z)=(~12) [ ~ ( z - d ) + ~ ( z ~ ,
(2)
z~__~2, d&a/2 a being the width of the well (normalization: Lupercell l~dz=~)" It can then be easily de_ rived,
vC~,,kz)= ~e21. s i n h ~ + sinh (k''A-2~l d )+c° skAA sInh2k"d ~Mm*
(3)
co sb~:llA-co skAA
and
~=(~e2~,/~ Mm*)½ {[slnhk,k+sinh (kllA-2klld) + co SkJLAsinh2kn d] I [coshkllA- co skLA] } ½
(4)
Setting d=O leads back to Equation (I). Starting with Ando's I0 variational wavefunction, we have worked numerically on a computer and have obtained a plasmon spectrum very close to Equation (4), thus justifying its validity.
1081
PLASMON DISPERSION IN SEMICONDUCTOR SUPERLATTICES
1082
Setting 2d=a and asslgnlng for the varlous parameters, w~lues in conformlty wlth the sampl6s of Reference (7), we obtain the disper-
Vol. 63, No.
szon curves shown in Figures 1 and 2. It can be seen that they fit qulte well the experimental polnts.
/i
7
--
6
--
5
-
4
-
i/I
/
,/
///
j
,/ /
_
/~J7"~
,/
-
/
g
3
¢
,>>"
-
+,
,7" 2
--
,/
I 2 Fig.1
I 4
I
I
I
G
8
I0
z
Comparlson of Equatlon (4) (solid line) and Equation (1) (dashed line) with experimental data (+) for sample 1 of Reference (7).
Fig.2
I
I
1
2
4
6
I 8
I I0
Comparison of Equation (4) (solld line) and Equation (I) (dashed line) with experimental data (+) for sample 2 of Reference (7).
REFEH hq~CES I. 2. 3. 4. 5. 6.
A.L. Fetter, Ann. Phys. (N. Y.) 88, I (1974). M. Apostol, Z. Phys. B22, 13 (1975). S.D. Sarms and J. J. Quinn, Phys. Rev. B2_~ 7603 (1982). W.L. Bloss and E. M. Brody, Solid State Comm. ~ 523 [1982). S.D. Sarma, Phys. Rev. B28, 2240 (1983). A.C. Tselis and J. J. Quinn, Phys. Rev.
B29, 3318 (1984). 7. 8. 9. 10.
D. Olego, A. Plnczuk, A. C. Oossard and W. Wiegmann, Phys. Rev. B25, 7867 (1982). A.L. Wasserman and Y. I. Lee, Solld State Comm. 54, 855 (1985). T. Audo and S. Morl, J. Phys. Soc. Jpn. 47, 1518 (1979). S. Mori and T. Ando, J. Phys. SOc. Jpn. ~8, 865 (198o).
12
Solid State Communications, VoI.63,No.12, pp. I081-I082, 1987. Printed in Great Britain.
Pergamon Journals Ltd.
A NOTE ON THE PLASMON DISP~SION IN SEMICONDUCTOR SUPERLATTICF~ Rui-Qing Yang Institute of Solid State Physlcs, Nanjing University, Nanjlng and Chi en-Hua Tsai Department of Physics and Institute of Condensed Matter Physics Jiao-Tong University, Shanghai, The People's Republic of China (Received 30 May, 1987 by W. Y. Kuan)
The plasmon dispersion relation in semiconductor superlattices is derived by taking into account energy band curvature due to charge transfer. The result is in nice agreement with experimental data.
There has been a number of theoretical
not negligible in comparison with V . Thus, the
works I-6 on the plasmon dispersion in semiconductor superlattices. The plasmen excitatlon
shape of the well differs greatly from a square one~ despite the fact that calculation of the plasmon frequencies using a square well model may yield results better than that defined by Equation (I). This is owing to the fact that in any model without electron tunnelling effect and with two-dimensional polarizability# the momentum representation of the Coulomb interaction between electrons, in particular when qA <__I, is not sensitive to the concrete form of the electronic wavefunction. Here, A is the modulation wavelength, and q, the wavevector. Square well wave functions allow electrons to form layers of finite thickness. This and the inclusion of interactions between electrons in the lowest and first excited subbands shift the plesmon frequencies nearer to experimental values than those given by Equation (I). However, in cases of heavy deoing, as explained above, the bending of energy bands cannot be ignored. The elegtrons are then pushed to concentrate towards the edges nf the well. We adopt the following electron density profile in the z-direction
o
spectrum has also been observed 7 by means of light scattering experiments, with results agreeing approximately with the theoretical expectation
, 2 ~ e 2"
si~2~a A
,~
(1)
based on a model of periodically arranged layers of electron gases. In Equation (1), n is the 2-D electronic density, m*, the electron's effective mass, eM' the background dielectric function, and ki! and kl, the inplane and perpendicular wavevectors, respectively. Wasserman and Lee 8 obtained plasmon discurves nearer to measured ones than Equation (I) by taking into account the finite thickness of electron layers. However, they adopted an infinitely deep square well model. For the sample 1 of Reference (7), they gave their comouted value of the energy difference between the first exclted and the ground states as A ~ 23. SmeV whereas the experimental value is A ~ 10.83meV. The carrier densities in the two samples of GaAs/Ga1_xAlxAS mentloned in Reference (7) are, respectively, 7.3.1011cm-2 and 5.5.1011cm"2. In cases of high carrier densitles like these, the bending of energy bands resulting from charge transfer becomes signlficant~ and corrections due to the self-consistent potential 9' 10 of the electrons are necessary. The difference V between the conduction levels of Gal.xAS x and e GaAs is given approximately by the empirical formula Vo=10x , with x expressed in percentage and V
in meV. For the two samples of Reference o (I), Vo equals, respectively, to 200meV and 110
meV, whereas, the potential difference between electrons located at the center and at the edge of the well exceeds 30meV, which is certalnly
I~12(z)=(~12) [ ~ ( z - d ) + ~ ( z ~ ,
(2)
z~__~2, d&a/2 a being the width of the well (normalization: Lupercell l~dz=~)" It can then be easily de_ rived,
vC~,,kz)= ~e21. s i n h ~ + sinh (k''A-2~l d )+c° skAA sInh2k"d ~Mm*
(3)
co sb~:llA-co skAA
and
~=(~e2~,/~ Mm*)½ {[slnhk,k+sinh (kllA-2klld) + co SkJLAsinh2kn d] I [coshkllA- co skLA] } ½
(4)
Setting d=O leads back to Equation (I). Starting with Ando's I0 variational wavefunction, we have worked numerically on a computer and have obtained a plasmon spectrum very close to Equation (4), thus justifying its validity.
1081
PLASMON DISPERSION IN SEMICONDUCTOR SUPERLATTICES
1082
Setting 2d=a and asslgnlng for the varlous parameters, w~lues in conformlty wlth the sampl6s of Reference (7), we obtain the disper-
Vol. 63, No.
szon curves shown in Figures 1 and 2. It can be seen that they fit qulte well the experimental polnts.
/i
7
--
6
--
5
-
4
-
i/I
/
,/
///
j
,/ /
_
/~J7"~
,/
-
/
g
3
¢
,>>"
-
+,
,7" 2
--
,/
I 2 Fig.1
I 4
I
I
I
G
8
I0
z
Comparlson of Equatlon (4) (solid line) and Equation (1) (dashed line) with experimental data (+) for sample 1 of Reference (7).
Fig.2
I
I
1
2
4
6
I 8
I I0
Comparison of Equation (4) (solld line) and Equation (I) (dashed line) with experimental data (+) for sample 2 of Reference (7).
REFEH hq~CES I. 2. 3. 4. 5. 6.
A.L. Fetter, Ann. Phys. (N. Y.) 88, I (1974). M. Apostol, Z. Phys. B22, 13 (1975). S.D. Sarms and J. J. Quinn, Phys. Rev. B2_~ 7603 (1982). W.L. Bloss and E. M. Brody, Solid State Comm. ~ 523 [1982). S.D. Sarma, Phys. Rev. B28, 2240 (1983). A.C. Tselis and J. J. Quinn, Phys. Rev.
B29, 3318 (1984). 7. 8. 9. 10.
D. Olego, A. Plnczuk, A. C. Oossard and W. Wiegmann, Phys. Rev. B25, 7867 (1982). A.L. Wasserman and Y. I. Lee, Solld State Comm. 54, 855 (1985). T. Audo and S. Morl, J. Phys. Soc. Jpn. 47, 1518 (1979). S. Mori and T. Ando, J. Phys. SOc. Jpn. ~8, 865 (198o).
12