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Inelastic light scattering by laterally modulated 2D electron plasma
Superlattices and Microstructures, Vol. 7, No. 2, 1990
I~ELASTIC
LIGHT SCATTERING
1 61
BY LATERALLY MODULATED 2D ELECTRON PLASNL~
A.V. Chaplik and A.O. Govorov Institute
of Semiconductor Physics,
630090, ~ovosibirsk,
Siberian Div.Ac.
of Sci.,
U.S.S.R.
(Received
30 January
1990)
We consider the Raman spectra of periodically modulated 2D electron plasma in the weak-modulation approximation. Peak positions and relative intensities are found as functions of the light momentwn transfer. The spectral manifestations of the 2D plasmon dispersion peculiarities are discussed.
11~elastic light scattering has been proved to be a very sensitive and informative tool for investigation of the plasma excitations in 2D electronic systems[I]. In the present paper we consider this phenomenon for periodically modulated 2D systems. The modulation can, in principle, affect only the incident and scattered light doing nearly nothing with 2D electrons. In this case we deal with the grating assisted Raman scattering-so was it called and observed by Zettler et al[2]. Diffraction ofelectromagnetic waves on the metallic grating gives a series of spatial harmonics inside the sample(where 2D electrons are located)with the in-plane wave vectors ~I + , n = O, ± q, ± 2 .... , = x o 2~/d, where x o is the ort along the direction of gratlng perlodlelty OX, ~1 Is the In-plane wave vector of the incident light. The main result of such a kind of modulation, is arising additional plasmon peaks for the frequencies ~p(~ + n~), - wave vector transfer,
~p(~)
is the
2D plasmon dispersion law. Two of them at n = ± q were observed in[2]. Same grating structure, acting as a gate electrode, can modulate the areal density of the 2D charge carriers(see e.g.[3]). Such systems were intensively investigated by FIR absorption technique, when the absorbed momentum of the incident radiation is z~ and c a n n o t be continuously tuned for a given sample. In this respect the Raman scattering offers more opportunities allowing one to measu0749 6036/90/020161 4-04 $02.00/0
re the plasmon dispersion for arbitrary direction ~ d value of the plasmon moment~n. We hope that the proposed experiment is quite possible for up-to-date technique because recently the inelastic light scabtering on a single(uniform)2D electron gas has been reported[4]. Consider the 2D electron plasma of the areal density Ns(X ) = Ns(X + d). The depth of modulation is supposed s u f f i o i ently small: N o >> ~n for n = ±I, ±2..., where N n are the Fourier components of the function Ns(X). Let ~s(x) be even in x, then i~n = N_n. To determine the intensity of scattered light we have to calculate the admittance of the system for an external scalar perburbation @ .... = • .exp i(~wt), where r and Z are 2D vectors. This formalism is applicable if we are interested in that contribution to the Raman scattering which comes from plasma density fluctuations. Just this type of fluctuations is responsible for plasmons satellites in the spectrum of scattered radiation. ~he nonuniform plasma is described classically by macroscopic conductivity ~(x) = ieZNs(x)/me(~
+ iv), where e and
m e are the charge ~ d the effective mass of electron, respectively, v is the collision rate, and we suppose >>
v.
The system of equations describing the admittance in self-consistent field approximation consists of the Poisson equ© 1990 Academic Press Limited
Superlattices and Microstructures, VoL 7, No. 2, 1990
162 ation, the continuity local 0~n's law:
equation
~ps(~)
+ div
= - div
o(x) V~ex~(r)
where ps(~)
a(x)~(~)
and the
= ...
bidden gaps are situated at qx = qn: = n%/d and the frequencies of the upper(+) and lower(-) edge of the n-th gap are
(1)
is p e r ~ b a t i o n
~±n : ~ p)((~In.
of the areal
charge density, 9(~) - induced part of the total potential in the electron sheet;
~(~) = 2-K~ Z Pn
expi Knr
~
"'"
(2)
-+ Y n / a ) '
'
Yq>
-
(k x ¢ ng, k ), p~ is ~he Fou~ier
se to the points
amplitu-
~
for the wave
vector Kn, e is the permeaability of surrounding medium. After combining eqs. (q) and(2) we obtain the density response to the scalar perturbation @o exp i ~
Then, according to the fluctuation-dissipation theorem, the scattering cross-section is proportional to Im Fo(~). The Haman scattering experiments are usually carried out under resonant conditions: ~ , ~ m Eg, where are the frequencies of incident and scattered light, respectively, Eg is an optical gap(for instance, ~g = = E o + A o - spin split gap in GaAs). The cross-section of the scaOtering has a form[q]:
n(~)
do
-I" 1 ( _ _ _ ~ )
1%
~, ~
e
0) is not clo-
(qn' O) the spectrum
I~o(W)=zx~P (~) 8(~
and sabellite
Pn = ~oF(Kn ' ~) ~ ~oFn (~)"
-
gm
of scattered radiation consists of ~he main peak at ~ = ~ p ( kx) with intensity Jo g i v e n by eq. (3) with
-
- ~t) in the form:
-
We show qualitatively the dispersion law ~(qx' O) on Fig.(la) together with numbers of the gaps. If the momentmn transfer ~(kw,
y
(~)
2v~e ~Noq Yn :
Here K n is 2D vector with components de of the density p s i )
"'"
peaks
-
~p(kx)) (~)
at ~ = ~ p ( ~
= w n. Their intensities
+ng)=
d n are propor-
tional to k
- l~Fo(~)
-
xYn 4
~3~a l'l_
0
s(w - ~n),
~ / o
2bus, the following dependencies experimentally be verified •
÷
(~) ~lay
nl 3/~
t0
e
d
(3)
× R (el.e)ZlmFo(tO).
Here R = (fl/3)Eg /$g - w l i s of resonant
amplification,
Bose occupation I,~
number,
are the vectors
the factor n(w) -
w = ~
-
of polarizati-
ons. The plasmon dispersion law in periodically modulated 2D plasma has been obtained in re~[5] for a special case qx ~ O, qy = O, where is the plasmon momentum. The for-
For k x in a small vicinity the points qn(Ik x - qnl peaks of nearly equaled appear and the distance peaks in the frequency
of one of
< Yn~/d) two intensities between the scale tends
to n-th gap ~+n - ~n when ~ tends to qn" There are, also, the satellite dublets corresponding to the gaps which lie on ~he same vertical with ~+n (see Fig. 2b). For example,
if k x = ~/d,
163
Super/attices and Microstructures, Vol. 7, No. 2, 1990
(a)
I%.
(b)
f
_2v/_
__vr
d d
0
w
"n "n-4_v
iqx~.._ vr
8 2dSd d5d
vr
0
8
vr ~T 4_vr vr 28:58 d 58
Fig.1 D i s p e r s i o n curves of 2D plasmo~s in a one-dimensional lateral superlattice.(a)qy = O, qx- arbitrary, (b)qx = = nm/d,
qy- arbibrary
= 0 (this gap is invisible i~i FIR Y absorption for a single periodic grating structure) we have for the fre± ± qaencies ~I' ~3 k
(o)
d~
A A AAA
.__
j -+
q .~
OJ
j-*_ ~q~3 4(q )
(b)
d~
8
(Y~ -* Y2)~
p
-
In general case the intensity o±' the m-Oh gap dublet is small as colapared with bhe n-th one for k x = n~/d in accord with the relations:
d S_Sd~
A
(c)
+
oJ ~i~.2 ~lasmoil peaks m,um for k
izl the Raman spec-
= 0 (schematically)
(a) kx~
The evolubion of the peaks system i'or varying k x is schematically depicted on ~ig 2.
Superlattices and Microstructures, Vol. Z No. 2, 1990
164
The case kx ~ 0 has to be considered separately. Atlk~I << yn~/d the main p@ak is described by eq(5)i.e. Jo ~ Ikx 13/z" In the satellite peaks intensities of the high-frequent(+) and low-frequent(-) components are essentially different: j+ - ~ ( ~ d / ~ ) ZJo << Jm+ bern ~ Ymzj o' Jm cause of the condition Ikxl << ym~/d. Now consider the scattering with k ~ O . The plasmon dispersion for qy % O, to our kxlowledge, has not been investigated so far in analitical form. The calculations in the perturbation bheory give for the n-th gap (qx = n~/d, qy - a~bitrary) z
~(qy) = ~p(qn,qy)(1 ± -¥n % - q~ )...(7) 2 q~+q~
and then remove each from other in acco~ ding with Fig.lb(the cttrves I, 2, 3). For k x = 0 (the curve 0) the light scattering process is symmetric(even) in x. The plasmons of the frequencies ~± are described by the density oscillations p+ exp (iqyy) ~ sin qn x and p_ exp (iqyy) x cos qnx, respectively. That is why only ~n(ky) - branches are visible in sattelites of the mai~ peak ~p(ky) for the gaps with even numbers (2, 4, 6 ... ). The contribution of p+ vanishes because of oddness. In conclusion we calculate -the positions and relative intensities of plasmons peaks in the Raman spectrum of a periodically modulated 2D electronic system. The inelastic light scattering gives more detailed information on the 2D plasmon dispersion law than FIR absorption.
Hence in this approximation a crossing over of the ~+ and ~- branches occurs at qy = qn = ~n/d and the n-th gap is closed up. (Fig.lb). £he crossing over is not forbidden by general theorems because ~+ and g~- branches are of different symmetry with respect to transformation x * - x. Eq.(7) is valid as long as qy is not too large, namely, the condition qy << q n / v ~ n must be fulfilled. Otherwise the br~iches ~± are not sufficiently far removed from others branches and the problem can not be treated in two-state approximation that was used in deriving Eq.(7). At fixed kx -- n~/d with n / 0 a~d by tuning ky one can observe how peaks become closer when ky tends to n~/d
kefei-enoe s Q. G. kbs~reiter, :~i. Cardona, k. Pinczuk: In Light Scattering in Solids IV, ed.by M. Cardona, G. Gt6utherodt, Topics. Appl. Phys., V. 5@(Springer, Berlin, Heidelberg 1984) P. 5. 2. T. Zetbler, C. Peters, J.P. Kobbmaus, K. Ploog: Phys. ~{ev. B39, 3931 (1989). " 3. D. Heitma~i.Surf.Sci. 170, 932(1986) 4. B. Jusserand, D.R. i{ichards, G. Fasol, G. Weinmann, W. Schlapp: In Proc. 8tn Conf.~lectronic Properties of Two-Dimensional Systems, Grenoble. Sept @-8. 1989., P. 396. 5. M.V. Krasheninnikov, A.V. Chaplik, Soviet Phys.-Semicond. I_5, 19(1981 )
I~ELASTIC
LIGHT SCATTERING
1 61
BY LATERALLY MODULATED 2D ELECTRON PLASNL~
A.V. Chaplik and A.O. Govorov Institute
of Semiconductor Physics,
630090, ~ovosibirsk,
Siberian Div.Ac.
of Sci.,
U.S.S.R.
(Received
30 January
1990)
We consider the Raman spectra of periodically modulated 2D electron plasma in the weak-modulation approximation. Peak positions and relative intensities are found as functions of the light momentwn transfer. The spectral manifestations of the 2D plasmon dispersion peculiarities are discussed.
11~elastic light scattering has been proved to be a very sensitive and informative tool for investigation of the plasma excitations in 2D electronic systems[I]. In the present paper we consider this phenomenon for periodically modulated 2D systems. The modulation can, in principle, affect only the incident and scattered light doing nearly nothing with 2D electrons. In this case we deal with the grating assisted Raman scattering-so was it called and observed by Zettler et al[2]. Diffraction ofelectromagnetic waves on the metallic grating gives a series of spatial harmonics inside the sample(where 2D electrons are located)with the in-plane wave vectors ~I + , n = O, ± q, ± 2 .... , = x o 2~/d, where x o is the ort along the direction of gratlng perlodlelty OX, ~1 Is the In-plane wave vector of the incident light. The main result of such a kind of modulation, is arising additional plasmon peaks for the frequencies ~p(~ + n~), - wave vector transfer,
~p(~)
is the
2D plasmon dispersion law. Two of them at n = ± q were observed in[2]. Same grating structure, acting as a gate electrode, can modulate the areal density of the 2D charge carriers(see e.g.[3]). Such systems were intensively investigated by FIR absorption technique, when the absorbed momentum of the incident radiation is z~ and c a n n o t be continuously tuned for a given sample. In this respect the Raman scattering offers more opportunities allowing one to measu0749 6036/90/020161 4-04 $02.00/0
re the plasmon dispersion for arbitrary direction ~ d value of the plasmon moment~n. We hope that the proposed experiment is quite possible for up-to-date technique because recently the inelastic light scabtering on a single(uniform)2D electron gas has been reported[4]. Consider the 2D electron plasma of the areal density Ns(X ) = Ns(X + d). The depth of modulation is supposed s u f f i o i ently small: N o >> ~n for n = ±I, ±2..., where N n are the Fourier components of the function Ns(X). Let ~s(x) be even in x, then i~n = N_n. To determine the intensity of scattered light we have to calculate the admittance of the system for an external scalar perburbation @ .... = • .exp i(~wt), where r and Z are 2D vectors. This formalism is applicable if we are interested in that contribution to the Raman scattering which comes from plasma density fluctuations. Just this type of fluctuations is responsible for plasmons satellites in the spectrum of scattered radiation. ~he nonuniform plasma is described classically by macroscopic conductivity ~(x) = ieZNs(x)/me(~
+ iv), where e and
m e are the charge ~ d the effective mass of electron, respectively, v is the collision rate, and we suppose >>
v.
The system of equations describing the admittance in self-consistent field approximation consists of the Poisson equ© 1990 Academic Press Limited
Superlattices and Microstructures, VoL 7, No. 2, 1990
162 ation, the continuity local 0~n's law:
equation
~ps(~)
+ div
= - div
o(x) V~ex~(r)
where ps(~)
a(x)~(~)
and the
= ...
bidden gaps are situated at qx = qn: = n%/d and the frequencies of the upper(+) and lower(-) edge of the n-th gap are
(1)
is p e r ~ b a t i o n
~±n : ~ p)((~In.
of the areal
charge density, 9(~) - induced part of the total potential in the electron sheet;
~(~) = 2-K~ Z Pn
expi Knr
~
"'"
(2)
-+ Y n / a ) '
'
Yq>
-
(k x ¢ ng, k ), p~ is ~he Fou~ier
se to the points
amplitu-
~
for the wave
vector Kn, e is the permeaability of surrounding medium. After combining eqs. (q) and(2) we obtain the density response to the scalar perturbation @o exp i ~
Then, according to the fluctuation-dissipation theorem, the scattering cross-section is proportional to Im Fo(~). The Haman scattering experiments are usually carried out under resonant conditions: ~ , ~ m Eg, where are the frequencies of incident and scattered light, respectively, Eg is an optical gap(for instance, ~g = = E o + A o - spin split gap in GaAs). The cross-section of the scaOtering has a form[q]:
n(~)
do
-I" 1 ( _ _ _ ~ )
1%
~, ~
e
0) is not clo-
(qn' O) the spectrum
I~o(W)=zx~P (~) 8(~
and sabellite
Pn = ~oF(Kn ' ~) ~ ~oFn (~)"
-
gm
of scattered radiation consists of ~he main peak at ~ = ~ p ( kx) with intensity Jo g i v e n by eq. (3) with
-
- ~t) in the form:
-
We show qualitatively the dispersion law ~(qx' O) on Fig.(la) together with numbers of the gaps. If the momentmn transfer ~(kw,
y
(~)
2v~e ~Noq Yn :
Here K n is 2D vector with components de of the density p s i )
"'"
peaks
-
~p(kx)) (~)
at ~ = ~ p ( ~
= w n. Their intensities
+ng)=
d n are propor-
tional to k
- l~Fo(~)
-
xYn 4
~3~a l'l_
0
s(w - ~n),
~ / o
2bus, the following dependencies experimentally be verified •
÷
(~) ~lay
nl 3/~
t0
e
d
(3)
× R (el.e)ZlmFo(tO).
Here R = (fl/3)Eg /$g - w l i s of resonant
amplification,
Bose occupation I,~
number,
are the vectors
the factor n(w) -
w = ~
-
of polarizati-
ons. The plasmon dispersion law in periodically modulated 2D plasma has been obtained in re~[5] for a special case qx ~ O, qy = O, where is the plasmon momentum. The for-
For k x in a small vicinity the points qn(Ik x - qnl peaks of nearly equaled appear and the distance peaks in the frequency
of one of
< Yn~/d) two intensities between the scale tends
to n-th gap ~+n - ~n when ~ tends to qn" There are, also, the satellite dublets corresponding to the gaps which lie on ~he same vertical with ~+n (see Fig. 2b). For example,
if k x = ~/d,
163
Super/attices and Microstructures, Vol. 7, No. 2, 1990
(a)
I%.
(b)
f
_2v/_
__vr
d d
0
w
"n "n-4_v
iqx~.._ vr
8 2dSd d5d
vr
0
8
vr ~T 4_vr vr 28:58 d 58
Fig.1 D i s p e r s i o n curves of 2D plasmo~s in a one-dimensional lateral superlattice.(a)qy = O, qx- arbitrary, (b)qx = = nm/d,
qy- arbibrary
= 0 (this gap is invisible i~i FIR Y absorption for a single periodic grating structure) we have for the fre± ± qaencies ~I' ~3 k
(o)
d~
A A AAA
.__
j -+
q .~
OJ
j-*_ ~q~3 4(q )
(b)
d~
8
(Y~ -* Y2)~
p
-
In general case the intensity o±' the m-Oh gap dublet is small as colapared with bhe n-th one for k x = n~/d in accord with the relations:
d S_Sd~
A
(c)
+
oJ ~i~.2 ~lasmoil peaks m,um for k
izl the Raman spec-
= 0 (schematically)
(a) kx~
The evolubion of the peaks system i'or varying k x is schematically depicted on ~ig 2.
Superlattices and Microstructures, Vol. Z No. 2, 1990
164
The case kx ~ 0 has to be considered separately. Atlk~I << yn~/d the main p@ak is described by eq(5)i.e. Jo ~ Ikx 13/z" In the satellite peaks intensities of the high-frequent(+) and low-frequent(-) components are essentially different: j+ - ~ ( ~ d / ~ ) ZJo << Jm+ bern ~ Ymzj o' Jm cause of the condition Ikxl << ym~/d. Now consider the scattering with k ~ O . The plasmon dispersion for qy % O, to our kxlowledge, has not been investigated so far in analitical form. The calculations in the perturbation bheory give for the n-th gap (qx = n~/d, qy - a~bitrary) z
~(qy) = ~p(qn,qy)(1 ± -¥n % - q~ )...(7) 2 q~+q~
and then remove each from other in acco~ ding with Fig.lb(the cttrves I, 2, 3). For k x = 0 (the curve 0) the light scattering process is symmetric(even) in x. The plasmons of the frequencies ~± are described by the density oscillations p+ exp (iqyy) ~ sin qn x and p_ exp (iqyy) x cos qnx, respectively. That is why only ~n(ky) - branches are visible in sattelites of the mai~ peak ~p(ky) for the gaps with even numbers (2, 4, 6 ... ). The contribution of p+ vanishes because of oddness. In conclusion we calculate -the positions and relative intensities of plasmons peaks in the Raman spectrum of a periodically modulated 2D electronic system. The inelastic light scattering gives more detailed information on the 2D plasmon dispersion law than FIR absorption.
Hence in this approximation a crossing over of the ~+ and ~- branches occurs at qy = qn = ~n/d and the n-th gap is closed up. (Fig.lb). £he crossing over is not forbidden by general theorems because ~+ and g~- branches are of different symmetry with respect to transformation x * - x. Eq.(7) is valid as long as qy is not too large, namely, the condition qy << q n / v ~ n must be fulfilled. Otherwise the br~iches ~± are not sufficiently far removed from others branches and the problem can not be treated in two-state approximation that was used in deriving Eq.(7). At fixed kx -- n~/d with n / 0 a~d by tuning ky one can observe how peaks become closer when ky tends to n~/d
kefei-enoe s Q. G. kbs~reiter, :~i. Cardona, k. Pinczuk: In Light Scattering in Solids IV, ed.by M. Cardona, G. Gt6utherodt, Topics. Appl. Phys., V. 5@(Springer, Berlin, Heidelberg 1984) P. 5. 2. T. Zetbler, C. Peters, J.P. Kobbmaus, K. Ploog: Phys. ~{ev. B39, 3931 (1989). " 3. D. Heitma~i.Surf.Sci. 170, 932(1986) 4. B. Jusserand, D.R. i{ichards, G. Fasol, G. Weinmann, W. Schlapp: In Proc. 8tn Conf.~lectronic Properties of Two-Dimensional Systems, Grenoble. Sept @-8. 1989., P. 396. 5. M.V. Krasheninnikov, A.V. Chaplik, Soviet Phys.-Semicond. I_5, 19(1981 )