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Screening of plasmons in modulated 2DEGs by grating gates
SOlid Sta~ Communications, Vol. 83, No. 8, pp. 627-633, 1992. Printed in Great Britain.
0038-1098/92 $5.00 + .00 Pergamon Press Ltd
SCREENING OF PLASMONS IN MODULATED 2DEGs BY GRATING GATES C.D. Ager and H.P. Hughes Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.
(Received 26 April 1992 by G. Fasol) Simple models for the dispersion relation for the plasmon modes of a two-dimensional electron gas (2DEG) take into account the presence of a coupling grating only by means of a sinusoidally periodic conducting layer, and generally include only a weak sinusoidal modulation of the electron density arising from any grating bias. We present here calculations of the energies of two-dimensional plasmons, including the periodic screening effect of a rectangular perfectly conducting grating coupler, in the case of strong modulation of the 2DEG with both rectangular and sinusoidal profiles. The calculations highlight the importance of the 2DEG modulation in determining the screened plasmon energies, and calculations of the plasmon charge density profiles show that as the modulation is increased, the plasmon oscillation becomes confined to regions of low 2DEG charge density with correspondingly tow frequency.
1. INTRODUCTION THE PLASMON modes of a two-dimensional electron gas (2DEG) formed in an MOS structure or a heterojunction have received widespread attention in recent years [1]. These modes are of interest because their frequencies (determined by the areal number densities of the carriers) lie in the far-infrared (FIR) sub-millimetre range and can be tuned electrically by applying a bias voltage to an overlaid gate; unlike the situation for three-dimensional carriers, the frequencies disperse over a wide range with mode wavevector, and approach zero at infinite mode wavelength. The plasmon frequency for a thin sheet of charge carriers has a square-root dependence on wavevector [2,3] with the explicit dispersion relation WCO)~ .Nse2 k = 2m*c0~ '
(1)
where ¢o(°) is the ~lasmon frequency (here and later the superscript (0)is used to denote the plasmon frequency in the absence of any spatial modulation of the 2DEG density), k is the wavevector of the plasmon in the plane of the 2D system, and Ns is the areal density of carriers of charge e and effective mass m*. g is the "effective dielectric function" for the system, which depends on the thicknesses and the dielectric properties of the layers around the 2D
charge sheet. Such 2D plasmons have been observed in AIGaAs/GaAs heterojunctions [4], where the electrons are confined in a very narrow potential well at the interface between the AIGaAs and GaAs layers, forming a 2DEG [5]. The plasmon wavelength is always longer than that of freely propagating EM radiation of the same frequency; direct optical coupling between the plasmon and freely propagating radiation is therefore not possible, and a coupling structure, such as a nearby metal grating [6], usually overlaid on the layered semiconductor system, is required. The metal fingers of the grating spatially modulate the (generally) normally incident radiation, generating components of wavevector k,-= 2nrc/d= nG in the electromagnetic (EM) field incident on the 2DEG; here d is the period of the grating, G is the fundamental grating wavevector and n is an integer. The n ~ 0 components couple to the fields of the plasmon excitation, allowing FIR radiation to be absorbed; this corresponds to the centre of the Brillouin zone arising from the grating periodicity (Fig. 1). The effective dielectric function ~ used in equation (1) includes the screening effect of the metal grating and the other layers adjacent to the 2DEG. For a standard GaAs/AIGaAs heterostructure, ~ is known exactly both for the case of an open top surface [7], ~op, and for the case where the top surface is
627
628
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES (.,O
._..._........------~, O ) + w O)-
i
f
"--------A _.--------4,
,
,k
2d
d
Fig. 1. Schematic zone folded 2D plasmon dispersion relation. Gaps open up in the dispersion at wavevectors which are multiples of 7rid where d is the period of the overlaid grating and the 2DEG modulation. Optical coupling occurs only at the zone centre, and scattering matrix calculations show that only the lower of the first order modes, w-, is of radiative symmetry and can thus be observed experimentally in a symmetric system. overlaid by a continuous, perfectly conducting top gate [8], gel: ~op =
~Ga + eal tanh (kh) 2
£Ga "+" ~A1 c o t h ,
~cl =
(kh)
2
(2) Here h is the distance between the 2DEG and the top surface and eOa(~,k) and eAl(w,k)are the dielectric functions of the GaAs substrate and the AIGaAs layer respectively. ~ for a metal grating-gate might be expected to be intermediate between these two results, a point discussed fully elsewhere [9]. If the metal grating is overlaid on a thin, fiat layer of NiCr, which acts as a continuous semi-transparent conducting gate, any gate bias modifies the 2DEG number density uniformly. In the absence of such a uniform gate, the bias applied to a metal grating, or a semi-transparent gate overlaid on a structured dielectric layer, spatially modulates the 2DEG density as well as changing its average value; this is illustrated schematically in Fig. 2. This modulation can itself provide coupling to incident radiation, and in circumstances when there is both an overlaid grating and a modulation of the 2DEG, the two coupling effects can interfere [10]. Several previous experimental studies of the 2D plasmon dispersion relation in Si MOS and heterojunction systems with such periodically modulated number densities [10-14] have reported a splitting in the 2D plasmon resonance, arising from the artifi-
Vol. 83, No. 8
cially induced periodicity of the structure. In these systems, the plasmon dispersion curve is broken by forbidden gaps at wavevectors which are multiples of 7r/d and, in the reduced zone scheme, becomes as shown in Fig. 1. The gaps arise, in part because of the periodic modulation of the 2DEG number density induced when the grating is biased, and in part from the periodic screening effect of the grating. The incident radiation couples only to the plasmon modes at the zone centre, so the two first order plasmon modes (k x = 2mr~d, n = 1) correspond to coupling at the second gap in the plasmon dispersion curve; the splitting of the two modes is thus also determined largely by the second harmonic of the periodicity of the system. For systems with grating profiles which are symmetric about the centres of the grating fingers, the grating-induced harmonics of the field of the incident radiation are similarly symmetric. The two first order plasmon modes (labelled w+ and w- in Fig. 1) have charge density profiles which are respectively symmetric and antisymmetric, and only the antisymmetric mode couples to the field harmonics generated by the grating and is observed in optical transmission spectroscopy, i.e. is radiative. Theoretical studies of these effects have been made by several groups, but few have included a realistic model for the coupling grating, which is usually represented by a thin sheet with sinusoidally modulated conductivity [15]; in other work, no grating is included and coupling occurs only through the density modulation of the 2DEG itself [16]. These approaches have the advantage of producing analytical results, but fail to include fully the periodic screening effects of a real metallic grating coupler. One such study is that of Krasheninnikov and Chaplik [17] (KC), who predicted theoretically the magnitude of the splitting at the zone boundaries by considering a weakly modulated 2DEG with first and second order Fourier components of number density modulation. For a 2DEG modulated with the grating period, d, the number density Ns(x) can be expanded as a Fourier series: N s ( X ) -~ A S + N - I e -iGx
+ Nj eiax + N-2 e i2ax + N2 ei2Gx + "" ,
(3)
where A s is the average areal number density and N,, N_, are its nth order Fourier components. Assuming that the number density modulation is symmetric about the centres of the grating fingers (i.e. about x =-0 in Fig. 2), N_, = N, for all n. For a weakly modulated system, including only terms up to Inl = 2 in the expansion of Ns(x) and only the first order
Vol. 83, No. 8
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
629
of the grating-gate which is used to induce the modulation. 80 nrn 70 nm
2. C O M P U T E R M O D E L L I N G
We describe below the results of calculations of the frequencies (wG) and charge density profiles of plasmon modes in AIGaAs/GaAs heterojunctions with overlaid perfectly conducting lamellar gratings which act both as optical couplers and as gate contacts; the modes have wavevector k = G, i.e. are Fig. 2. Schematic diagram of the sample structure first order modes at the centre of the BriUouin zone used for the calculations, and the probable "true" number density modulation, which is intermediate (Fig. 1). Any bias voltage, Vg, applied between the between sinusoidal and rectangular. Pinch-off into an grating and the 2DEG varies both the average areal array of ID wires occurs when the minimum density carrier density, Ns, and the number density modulagoes to zero. The system is modelled using both sinu- tion (Fig. 2); for sufficiently strong negative bias, soidal and rectangular 2DEG modulation profiles. carriers are totally depleted from beneath the grating fingers, effectively forming an array of quasi-lD wires. terms in the electric field, KC calculated the In our systems the grating-gate profile is rectangular, frequencies w+, w-, of the two first order plasmon and is symmetric about the centres of the fingers (i.e. modes at the centre of the zone (Fig. 1): about x = 0) so the 2DEG number density distribution is also symmetric and the Fourier coefficients in w± = a~(°)( 1 ±[N2l'~l/2--~-ffs] ' (4) equation (3) satisfy Nm = N_m. The modelling uses a scattering matrix technique [23, 24] for calculating the where IN21<< As. Only one of the branches is of optical response of vertically layered structures which radiative symmetry - the upper mode if N2 > 0, the can include lateral periodicities; unlike the approach lower mode if N2 < 0. This model does not include a of KC, this is not restricted to weak or sinusoidal metal grating explicitly, but assumes that its average modulation of the 2DEG number density, so that screening effect can be modelled using the effective the effects of accumulation and depletion under the dielectric function for a heterojunction with a closed biassed grating fingers can be included using surface, ~d, [equation (2)]; it thus ignores any splitting physically more reasonable charge density profiles. of the 2D plasmon resonance caused by the periodic The properties of the grating are modelled using the screening effect of the grating, and overestimates the eigenmodes of a perfectly conducting lamellar grating average screening effect of the grating. The calculated. [251 rather than a Fourier series of conductivity terms splitting was therefore consistently smaller than that as used by other authors [15,26], and thus more measured experimentally [18]; note also that equation accurately model the screening effects of real gratings (4) predicts no change in the average frequency of the in the far-infrared. Calculations of the mode frequencies for a two modes with increasing modulation, IN2[. It has modulated 2DEG under a metallic grating of period been demonstrated [19] that when higher order d (= 1/~m) and mark fraction r (= 0.33 and 0.5) were components of the electric field are included in the made assuming the 2DEG number density profile was calculation, the plasmon becomes localised in the (i) rectangular, and (ii) sinusoidal, in a sample regions of low density, resulting in a reduction in the with layers as shown in Fig. 2. In both cases, the plasmon frequency. The effects of modulation strong average number density ~rs was held constant (at enough to break up the 2DEG into an array of stripes 3.5 x 1011cm-2), so the bias-induced modulation have been studied theoretically by several groups of produces charge accumulation and depletion reworkers [20], some of whom also included the effects gions, a situation not to be expected in real grating of a magnetic field perpendicular to the plane of the gate samples; we adopt this model charge distribution 2DEG I21,22]; these calculations, however, did not include the effects of the periodic screening of the for calculational purposes in order to examine the metallic grating gate present in the sample structure. shifts in the plasmon frequencies arising, not from Here we present the results of scattering matrix any change in the overall carrier density of the calculations of the plasmon frequency in modulated 2DEG, but from the modulation itself. The results systems for sinusoidal and rectangular depletion are compared with (iii) calculations made for the profiles including in addition the screening effects structure investigated by KC - a sinusoidally
63O
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
modulated 2DEG under a uniformly conducting layer, i.e. ~ = eel [equation (2)].
r
We first model the 2DEG modulation assuming the density under the grating fingers is uniformly depleted (or accumulated, depending on the sign of the modulation), while that under the gaps is accumulated (or depleted) in order to keep the average density hrs constant. The number density distribution, Ns(x ) has a rectangular profile with mark fraction, r, and period, d, equal to those of the metal grating, and with N e the areal 2DEG number density under the gap and Nf that under the fingers themselves. This distribution is then approximated by a Fourier series [equation (2)], here terminated at InI = 8, with coefficients given by sin (nTrr) Ng - Nf mr Ns '
-
-
,
-
-
,
"-
I - ........ . ,
/]i ....
~= 20
.....
,
15
.
10
.
.
.
.
.
.
;,.,,,,, .
.
.
.
c
-2.0
- 1.0
0.0
1.0
2.0
3.0
An
(a) r
=
-
-
40
(1.5 ,
b k
"
'
.
.
.
35~
.
.....
25 (6) ~
15
:'7:.. ::.......
*J 0
-2.0
....
i
....
..... •
................ •
"'ii
: ~
. . . .
- 1.0
', 0.0
',
'
1.0
i "
~
2.0
3.0
An
(b)
N~)
~rs
(7)
for constant hrs ( - 3.5 × 10 ll cm-2). The results are plotted for two values of r in Fig. 3 (solid lines); for increasingly positive values of An, charge is depleted under the grating fingers and accumulated under the gaps. From equations (6) and (7) we have: Ug--- N s ( 1 - r A n )
-,
"7
The first order plasmon mode should be split at the zone boundaries in a periodically modulated system, as discussed above (Fig. 1), and the frequencies of these split modes have been calculated for samples with parameters given in Fig. 2 as functions of the normalised modulation amplitude: (Ns
-
3O
Ns = rUf+ (1 - r)Ug.
=
-,
30 >,25
(5)
where the average number density, Ns, is:
an
0.33
.
3s
2.1. Rectangular modulation
N. _ lqs
=
40
Vol. 83, No. 8
and
N f = ~rs[l + (l - r ) A n ]
(8) so that "pinch-off" occurs, with Ng or Nf ~ O, when A n = 1/r or - 1 / ( 1 - r ) ; near these extremes, the algorithms used for finding the roots of the dispersion relations do not converge reliably, so some of the curves for rectangular modulation do not extend over the full range of An. 2.2. Sinusoidal modulation Assuming a sinusoidal depletion model for the charge density: Ns(X) = Ns + N_ 1 e -iGx + g I e i G x ,
(9)
where N_I = Nx, the external number densities under the grating fingers and under the gaps are respectively
Fig. 3. The frequencies wo of the first order plasmon modes calculated as a function of number density modulation amplitude, An ~ ( N / - N , ) / N s at fixed average areal carrier density, Ns (= 3.g x 10 'l cm-2), for the sample illustrated in Fig. 2. The long-dashed curves labelled " K C " are calculated for the KC systems (see text), and the solid curves and shortdashed curves are calculated for rectangular and sinusoidal number density profiles respectively under a perfectly conducting lamellar grating, which has mark fraction (a) 0.33 and (b) 0.5. Nf = A s - 2Nl and N~ = Ns + 2Nl, or:
Ng = grs(l + An/2)
and
NT = .~s(1 - An/Z).
(10) Calculations were amplitude with Ns "pinch-off" occurs zero, in this case at
made for varying modulation held constant, and as before, when either Ng or iV/ become An = i 2 .
2.3. The KC model Also plotted in Fig. 3 are the scattering matrix calculations for the plasmon frequencies for the
Vol. 83, No. 8
PLASMONS IN MODULATED 2DEGs BY GRATING GATES
631
symmetric with no phase steps. The reduction of the plasmon frequencies at large An can be understood in terms of the spatial shifts of the oscillating charge density; for An----0, the radiative mode is approximately sinusoidal in profile, with the maxima slightly drawn under the edges of the grating fingers to increase the screening effect, while for An -----+1.5 (charge depletion in the gaps) and An ----- 1.5 (charge depletion under the fingers) the maxima are drawn towards the regions of reduced charge density. The improved screening by the gate fingers observable for the case of An = -1.5 accounts in part for the generally lower frequencies for negative values of An [Fig. 3(b)], but it is clear that the reduction in 3. RESULTS AND DISCUSSION plasmon frequencies is dominated by the modulation The long-dash curves in Fig. 3 show the results for itself and the corresponding spatial shift of the the structure adopted by KC, a sinusoidally modu- oscillation to regions of low charge density as noted lated 2DEG under a continuous metal gate. F o r by Wulf et al. [22] rather than the improved sinusoidal modulation, N2---0, so the KC model screening. Figure 3 also shows that the rectangular and does not produce split first order plasmon modes, sinusoidal modulation profiles give similar values for even for An ~ 0 [equation (3)]; the scattering matrix the slopes for both the radiative and non-radiative calculation confirms this, and shows no splitting even at large values of An outside the range of the KC modes close to An = 0, (Nf = Ng), and in this sense algebraic result. The effect of increasing modulation there is qualitative agreement between the models for An is to reduce the plasmon frequencies, by up to small modulation amplitude. For rectangular modu30% at the pinch-off extremes; the frequency lation with r = 0.5, the profile is closer to a sinusoidal reduction is symmetric in An because the gate is profile of the same period than is the case for r = 0 . 3 3 , so it is also not surprising that the continuous. The frequency shifts in the presence of the lamel- agreement between the two profile models is better lar grating gates are more complex in detail because [Fig. 3(b)]. The quantitative differences in the values of w6 of the periodic screening effect of the grating, but show the same overall behaviour as the KC structure. can be understood in physical terms as follows. The Note first that, for the same average 2DEG number KC model assumes the presence of a continuous density and An = 0, the plasmon frequencies are perfectly conducting gate, whereas the scattering generally markedly higher for the case of the lamellar matrix model explicitly includes the less efficient grating gates (and higher for smaller mark fraction r) screening effect of a grating gate, resulting in because of the reduced screening compared with that higher values for ~o6. The scattering matrix of the continuous gate of the KC model. But most model also explicitly includes the periodicity of the striking is the fact that the plasmon modes are split grating-gate screening effect, which introduces a by the periodic screening effect, and the shifts are no further splitting, apparent even at zero modulation, longer symmetric in An. The corresponding calcu- between the radiative and non-radiative modes lated transmission spectra show that only the lower because of their different spatial distributions of energy mode of the split pair is radiative and hence oscillating charge; the radiative and non-radiative experimentally observable. The symmetries of these modes have nodes and antinodes respectively at the modes are different, as shown in Fig. 4, where the centres of the gaps between the fingers. The KC amplitude and phase of the oscillating charge density, model also includes only the lowest two Fourier calculated using the scattering matrix approach, is coefficients for Ns(x) [equation (2)], whereas the shown as a function of position under the grating scattering matrix calculation includes terms up structure (r = 0.5) for several values of An for the to the eighth harmonic, so it is clear that the rectangular depletion model; the radiative modes scattering matrix approach is more appropriate for [Fig. 4(a)] are antisymmetric with respect to the describing our experimental systems which include centres of the fingers (and the gaps between them) lamellar metal grating gates and strong, essentially and have nodes and phase steps of 7r at these points, rectangular, modulation of the 2DEG number while the non-radiative modes [Fig. 4(b)] are density.
sample structure used in the simple KC model, with the 2DEG sinusoidally modualted and a continuous metal gate; here, though, the electric field throughout the structure is represented by a Fourier series with terms up to InI = 8, and the charge density modulation, described as in Section 2.2 above with normalised amplitude An, is not restricted to weak modulation amplitude as in KC's calculations. Because this calculation does not explicitly include any overlaid grating, it is used for comparison with the calcuations for both grating mark fractions r [Figs 3(a) and (b)].
632
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
Vol. 83, No. 8
An=0
An = 0
v
e-.
0f
.................
...................... . . . . . . . . . . .
An = - 1 . 5
An = -1.5
I
v
0
0
An =
1.5
~n = 1.5
v
0
0
0
0.25
0.5
0.75
x/d
(a)
0
0.25
0.5
0.75
x/d
(b)
Fig. 4. The square Ip2l of the oscillating charge density, and its phase, in the 2DEG under a grating with r = 0.5 for An = 0,-1.5 and 1.5, for plasmons of (a) radiative and (b) non-radiative symmetry. The rectangles at the top of the figures indicate the extent of the metal fingers of the grating, and under each figure, the relative phase of the charge density oscillation is shown. Note that modes of radiative symmetry are antisymmetric about x --- 0 and x = d/2(Ip2l has a node and the relative phases of the charge density either side of a node differ by 7r), whereas the non-radiative modes are symmetric (Ip2l antinode, phase continuous and symmetric). Note that for both symmetries at An = +1.5, the amplitude of the charge density oscillation is enhanced in the regions of low charge density, indicating plasmon localisation in these regions resulting in a reduction in plasmon frequency. 4. C O N C L U S I O N We have investigated the optical response and natural mode frequencies of modulated 2DEGs
under a metallic grating using scattering matrix calculations of the resonant frequencies and the charge density amplitudes as functions of the modulation profile and amplitude. Our results are
Vol. 83, No. 8
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
in general agreement with those of Cataudella and Marigliano Ramaglia [20] and Wulf et aL [22], in that the plasmon resonance becomes localised in the region of low charge density resulting in a decrease in plasmon frequency from that expected from the average value of the density; the reduction in plasmon frequency is generally larger when the depletion profile is more rectangular [19]. In addition to these effects, we have also demonstrated that an overlaid grating-gate splits the plasmon frequencies of the radiative and non-radiative symmetry plasmon modes. Furthermore, the shift in plasmon frequency depends on whether depletion occurs under the fingers or in the regions between them: if the low density regions of the 2DEG are under the grating fingers, then the gate screening acts to lower the localised plasmon frequency even further; if on the other hand, the 2DEG density is lowest under the regions between grating fingers, the plasmon is not as well screened and the frequency is higher. REFERENCES 1. 2. 3. 4. 5. 6. 7.
T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54, 437 (1982). R.H. Ritchie, Phys. Rev. 106, 874 (1957). F. Stern, Phys. Rev. Lett. 18, 546 (1967). R. Hopfel, G. Lindemann, E. Gornik, G. Stangl, A.C. Gossard & W. Wiegmann, Surf. Sci. 113, 118 (1982). H.L. Stormer, R. Dingle, A.C. Gossard, W. Wiegrnann & M.D. Sturge, Solid State Commun. 29, 705 (1979). S.J. Allen, D.C. Tsui, R.A. Logan, Phys. Rev. Lett. 38, 980 (1977). N. Okisu, Y. Sambe & T. Kobayashi, Elec. Lett. 22, 877 (1986).
8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
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A.V. Chaplik, Soviet Phys.-JETP 35, 395 (1972). C.D. Ager, R.J. Wilkinson & H.P. Hughes, J. Appl. Phys. 71, 1322 (1992). R.J. Wilkinson, C.D. Ager, T. Duffield, H.P. Hughes, D.G. Hasko, H. Ahmed, J.E.F. Frost, D.C. Peacock, D.A. Ritchie, G.A.C. Jones, C. Whitehouse & N. Apsley, J. Appl. Phys. 71, 6049 (1992). U. Mackens, D. Heitmann, L. Prager & J.P. Kotthaus, Phys. Rev. Lett. 53, 1485 (1984). D. Heitmann, Surf. Sci. 170, 332 (1986). T. Zettler & J.P. Kotthaus, Semi. Sci. Tech. 3, 413 (1988). E. Batke, D. Heitmann & C.W. Tu, Phys. Rev. B34, 6951 (1986). T.N. Theis, Surf. Sci. 98, 515 (1980). C. Dahl, Phys. Rev. B41, 5763 (1990). M.V. Krasheninnikov & A.V. Chaplik, Soy. Phys. Semicond. 15, 19 (1981). T. Zettler & J.P. Kotthaus, Semi. Sci. Tech. 3, 413 (1988). G. Eliasson, P. Hawrylak, Ji-Wei Wu & J.J. Quinn, Solid State Commun. 60, 3 (1986). V. Cautadella & V. Marigliano Ramaglia, Phys. Rev. B38, 1828 (1988). G. Eliasson, Ji-Wei Wu, P. Hawrylak & J.J. Quinn, Solid State Commun. 60, 41 (1986). U. Wulf, E. Zeeb, P. Gies, R.R. Gerhardts & W. Hanke, Phys. Rev. 1342, 7637 (1990). D.Y.K. Ko & J.C. Inkson, Phys. Rev. B38, 9945 (1988). C.D. Ager & H.P. Hughes, Phys. Rev. B 44, 13452 (1991). P. Sheng, R.S. Stepleman & P.N. Sanda, Phys. Rev. B26, 2907 (1982). L. Zheng, W.L. Schaich & A.H. MacDonald, Phys. Rev. 1141, 8493 (1990).
0038-1098/92 $5.00 + .00 Pergamon Press Ltd
SCREENING OF PLASMONS IN MODULATED 2DEGs BY GRATING GATES C.D. Ager and H.P. Hughes Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.
(Received 26 April 1992 by G. Fasol) Simple models for the dispersion relation for the plasmon modes of a two-dimensional electron gas (2DEG) take into account the presence of a coupling grating only by means of a sinusoidally periodic conducting layer, and generally include only a weak sinusoidal modulation of the electron density arising from any grating bias. We present here calculations of the energies of two-dimensional plasmons, including the periodic screening effect of a rectangular perfectly conducting grating coupler, in the case of strong modulation of the 2DEG with both rectangular and sinusoidal profiles. The calculations highlight the importance of the 2DEG modulation in determining the screened plasmon energies, and calculations of the plasmon charge density profiles show that as the modulation is increased, the plasmon oscillation becomes confined to regions of low 2DEG charge density with correspondingly tow frequency.
1. INTRODUCTION THE PLASMON modes of a two-dimensional electron gas (2DEG) formed in an MOS structure or a heterojunction have received widespread attention in recent years [1]. These modes are of interest because their frequencies (determined by the areal number densities of the carriers) lie in the far-infrared (FIR) sub-millimetre range and can be tuned electrically by applying a bias voltage to an overlaid gate; unlike the situation for three-dimensional carriers, the frequencies disperse over a wide range with mode wavevector, and approach zero at infinite mode wavelength. The plasmon frequency for a thin sheet of charge carriers has a square-root dependence on wavevector [2,3] with the explicit dispersion relation WCO)~ .Nse2 k = 2m*c0~ '
(1)
where ¢o(°) is the ~lasmon frequency (here and later the superscript (0)is used to denote the plasmon frequency in the absence of any spatial modulation of the 2DEG density), k is the wavevector of the plasmon in the plane of the 2D system, and Ns is the areal density of carriers of charge e and effective mass m*. g is the "effective dielectric function" for the system, which depends on the thicknesses and the dielectric properties of the layers around the 2D
charge sheet. Such 2D plasmons have been observed in AIGaAs/GaAs heterojunctions [4], where the electrons are confined in a very narrow potential well at the interface between the AIGaAs and GaAs layers, forming a 2DEG [5]. The plasmon wavelength is always longer than that of freely propagating EM radiation of the same frequency; direct optical coupling between the plasmon and freely propagating radiation is therefore not possible, and a coupling structure, such as a nearby metal grating [6], usually overlaid on the layered semiconductor system, is required. The metal fingers of the grating spatially modulate the (generally) normally incident radiation, generating components of wavevector k,-= 2nrc/d= nG in the electromagnetic (EM) field incident on the 2DEG; here d is the period of the grating, G is the fundamental grating wavevector and n is an integer. The n ~ 0 components couple to the fields of the plasmon excitation, allowing FIR radiation to be absorbed; this corresponds to the centre of the Brillouin zone arising from the grating periodicity (Fig. 1). The effective dielectric function ~ used in equation (1) includes the screening effect of the metal grating and the other layers adjacent to the 2DEG. For a standard GaAs/AIGaAs heterostructure, ~ is known exactly both for the case of an open top surface [7], ~op, and for the case where the top surface is
627
628
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES (.,O
._..._........------~, O ) + w O)-
i
f
"--------A _.--------4,
,
,k
2d
d
Fig. 1. Schematic zone folded 2D plasmon dispersion relation. Gaps open up in the dispersion at wavevectors which are multiples of 7rid where d is the period of the overlaid grating and the 2DEG modulation. Optical coupling occurs only at the zone centre, and scattering matrix calculations show that only the lower of the first order modes, w-, is of radiative symmetry and can thus be observed experimentally in a symmetric system. overlaid by a continuous, perfectly conducting top gate [8], gel: ~op =
~Ga + eal tanh (kh) 2
£Ga "+" ~A1 c o t h ,
~cl =
(kh)
2
(2) Here h is the distance between the 2DEG and the top surface and eOa(~,k) and eAl(w,k)are the dielectric functions of the GaAs substrate and the AIGaAs layer respectively. ~ for a metal grating-gate might be expected to be intermediate between these two results, a point discussed fully elsewhere [9]. If the metal grating is overlaid on a thin, fiat layer of NiCr, which acts as a continuous semi-transparent conducting gate, any gate bias modifies the 2DEG number density uniformly. In the absence of such a uniform gate, the bias applied to a metal grating, or a semi-transparent gate overlaid on a structured dielectric layer, spatially modulates the 2DEG density as well as changing its average value; this is illustrated schematically in Fig. 2. This modulation can itself provide coupling to incident radiation, and in circumstances when there is both an overlaid grating and a modulation of the 2DEG, the two coupling effects can interfere [10]. Several previous experimental studies of the 2D plasmon dispersion relation in Si MOS and heterojunction systems with such periodically modulated number densities [10-14] have reported a splitting in the 2D plasmon resonance, arising from the artifi-
Vol. 83, No. 8
cially induced periodicity of the structure. In these systems, the plasmon dispersion curve is broken by forbidden gaps at wavevectors which are multiples of 7r/d and, in the reduced zone scheme, becomes as shown in Fig. 1. The gaps arise, in part because of the periodic modulation of the 2DEG number density induced when the grating is biased, and in part from the periodic screening effect of the grating. The incident radiation couples only to the plasmon modes at the zone centre, so the two first order plasmon modes (k x = 2mr~d, n = 1) correspond to coupling at the second gap in the plasmon dispersion curve; the splitting of the two modes is thus also determined largely by the second harmonic of the periodicity of the system. For systems with grating profiles which are symmetric about the centres of the grating fingers, the grating-induced harmonics of the field of the incident radiation are similarly symmetric. The two first order plasmon modes (labelled w+ and w- in Fig. 1) have charge density profiles which are respectively symmetric and antisymmetric, and only the antisymmetric mode couples to the field harmonics generated by the grating and is observed in optical transmission spectroscopy, i.e. is radiative. Theoretical studies of these effects have been made by several groups, but few have included a realistic model for the coupling grating, which is usually represented by a thin sheet with sinusoidally modulated conductivity [15]; in other work, no grating is included and coupling occurs only through the density modulation of the 2DEG itself [16]. These approaches have the advantage of producing analytical results, but fail to include fully the periodic screening effects of a real metallic grating coupler. One such study is that of Krasheninnikov and Chaplik [17] (KC), who predicted theoretically the magnitude of the splitting at the zone boundaries by considering a weakly modulated 2DEG with first and second order Fourier components of number density modulation. For a 2DEG modulated with the grating period, d, the number density Ns(x) can be expanded as a Fourier series: N s ( X ) -~ A S + N - I e -iGx
+ Nj eiax + N-2 e i2ax + N2 ei2Gx + "" ,
(3)
where A s is the average areal number density and N,, N_, are its nth order Fourier components. Assuming that the number density modulation is symmetric about the centres of the grating fingers (i.e. about x =-0 in Fig. 2), N_, = N, for all n. For a weakly modulated system, including only terms up to Inl = 2 in the expansion of Ns(x) and only the first order
Vol. 83, No. 8
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
629
of the grating-gate which is used to induce the modulation. 80 nrn 70 nm
2. C O M P U T E R M O D E L L I N G
We describe below the results of calculations of the frequencies (wG) and charge density profiles of plasmon modes in AIGaAs/GaAs heterojunctions with overlaid perfectly conducting lamellar gratings which act both as optical couplers and as gate contacts; the modes have wavevector k = G, i.e. are Fig. 2. Schematic diagram of the sample structure first order modes at the centre of the BriUouin zone used for the calculations, and the probable "true" number density modulation, which is intermediate (Fig. 1). Any bias voltage, Vg, applied between the between sinusoidal and rectangular. Pinch-off into an grating and the 2DEG varies both the average areal array of ID wires occurs when the minimum density carrier density, Ns, and the number density modulagoes to zero. The system is modelled using both sinu- tion (Fig. 2); for sufficiently strong negative bias, soidal and rectangular 2DEG modulation profiles. carriers are totally depleted from beneath the grating fingers, effectively forming an array of quasi-lD wires. terms in the electric field, KC calculated the In our systems the grating-gate profile is rectangular, frequencies w+, w-, of the two first order plasmon and is symmetric about the centres of the fingers (i.e. modes at the centre of the zone (Fig. 1): about x = 0) so the 2DEG number density distribution is also symmetric and the Fourier coefficients in w± = a~(°)( 1 ±[N2l'~l/2--~-ffs] ' (4) equation (3) satisfy Nm = N_m. The modelling uses a scattering matrix technique [23, 24] for calculating the where IN21<< As. Only one of the branches is of optical response of vertically layered structures which radiative symmetry - the upper mode if N2 > 0, the can include lateral periodicities; unlike the approach lower mode if N2 < 0. This model does not include a of KC, this is not restricted to weak or sinusoidal metal grating explicitly, but assumes that its average modulation of the 2DEG number density, so that screening effect can be modelled using the effective the effects of accumulation and depletion under the dielectric function for a heterojunction with a closed biassed grating fingers can be included using surface, ~d, [equation (2)]; it thus ignores any splitting physically more reasonable charge density profiles. of the 2D plasmon resonance caused by the periodic The properties of the grating are modelled using the screening effect of the grating, and overestimates the eigenmodes of a perfectly conducting lamellar grating average screening effect of the grating. The calculated. [251 rather than a Fourier series of conductivity terms splitting was therefore consistently smaller than that as used by other authors [15,26], and thus more measured experimentally [18]; note also that equation accurately model the screening effects of real gratings (4) predicts no change in the average frequency of the in the far-infrared. Calculations of the mode frequencies for a two modes with increasing modulation, IN2[. It has modulated 2DEG under a metallic grating of period been demonstrated [19] that when higher order d (= 1/~m) and mark fraction r (= 0.33 and 0.5) were components of the electric field are included in the made assuming the 2DEG number density profile was calculation, the plasmon becomes localised in the (i) rectangular, and (ii) sinusoidal, in a sample regions of low density, resulting in a reduction in the with layers as shown in Fig. 2. In both cases, the plasmon frequency. The effects of modulation strong average number density ~rs was held constant (at enough to break up the 2DEG into an array of stripes 3.5 x 1011cm-2), so the bias-induced modulation have been studied theoretically by several groups of produces charge accumulation and depletion reworkers [20], some of whom also included the effects gions, a situation not to be expected in real grating of a magnetic field perpendicular to the plane of the gate samples; we adopt this model charge distribution 2DEG I21,22]; these calculations, however, did not include the effects of the periodic screening of the for calculational purposes in order to examine the metallic grating gate present in the sample structure. shifts in the plasmon frequencies arising, not from Here we present the results of scattering matrix any change in the overall carrier density of the calculations of the plasmon frequency in modulated 2DEG, but from the modulation itself. The results systems for sinusoidal and rectangular depletion are compared with (iii) calculations made for the profiles including in addition the screening effects structure investigated by KC - a sinusoidally
63O
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
modulated 2DEG under a uniformly conducting layer, i.e. ~ = eel [equation (2)].
r
We first model the 2DEG modulation assuming the density under the grating fingers is uniformly depleted (or accumulated, depending on the sign of the modulation), while that under the gaps is accumulated (or depleted) in order to keep the average density hrs constant. The number density distribution, Ns(x ) has a rectangular profile with mark fraction, r, and period, d, equal to those of the metal grating, and with N e the areal 2DEG number density under the gap and Nf that under the fingers themselves. This distribution is then approximated by a Fourier series [equation (2)], here terminated at InI = 8, with coefficients given by sin (nTrr) Ng - Nf mr Ns '
-
-
,
-
-
,
"-
I - ........ . ,
/]i ....
~= 20
.....
,
15
.
10
.
.
.
.
.
.
;,.,,,,, .
.
.
.
c
-2.0
- 1.0
0.0
1.0
2.0
3.0
An
(a) r
=
-
-
40
(1.5 ,
b k
"
'
.
.
.
35~
.
.....
25 (6) ~
15
:'7:.. ::.......
*J 0
-2.0
....
i
....
..... •
................ •
"'ii
: ~
. . . .
- 1.0
', 0.0
',
'
1.0
i "
~
2.0
3.0
An
(b)
N~)
~rs
(7)
for constant hrs ( - 3.5 × 10 ll cm-2). The results are plotted for two values of r in Fig. 3 (solid lines); for increasingly positive values of An, charge is depleted under the grating fingers and accumulated under the gaps. From equations (6) and (7) we have: Ug--- N s ( 1 - r A n )
-,
"7
The first order plasmon mode should be split at the zone boundaries in a periodically modulated system, as discussed above (Fig. 1), and the frequencies of these split modes have been calculated for samples with parameters given in Fig. 2 as functions of the normalised modulation amplitude: (Ns
-
3O
Ns = rUf+ (1 - r)Ug.
=
-,
30 >,25
(5)
where the average number density, Ns, is:
an
0.33
.
3s
2.1. Rectangular modulation
N. _ lqs
=
40
Vol. 83, No. 8
and
N f = ~rs[l + (l - r ) A n ]
(8) so that "pinch-off" occurs, with Ng or Nf ~ O, when A n = 1/r or - 1 / ( 1 - r ) ; near these extremes, the algorithms used for finding the roots of the dispersion relations do not converge reliably, so some of the curves for rectangular modulation do not extend over the full range of An. 2.2. Sinusoidal modulation Assuming a sinusoidal depletion model for the charge density: Ns(X) = Ns + N_ 1 e -iGx + g I e i G x ,
(9)
where N_I = Nx, the external number densities under the grating fingers and under the gaps are respectively
Fig. 3. The frequencies wo of the first order plasmon modes calculated as a function of number density modulation amplitude, An ~ ( N / - N , ) / N s at fixed average areal carrier density, Ns (= 3.g x 10 'l cm-2), for the sample illustrated in Fig. 2. The long-dashed curves labelled " K C " are calculated for the KC systems (see text), and the solid curves and shortdashed curves are calculated for rectangular and sinusoidal number density profiles respectively under a perfectly conducting lamellar grating, which has mark fraction (a) 0.33 and (b) 0.5. Nf = A s - 2Nl and N~ = Ns + 2Nl, or:
Ng = grs(l + An/2)
and
NT = .~s(1 - An/Z).
(10) Calculations were amplitude with Ns "pinch-off" occurs zero, in this case at
made for varying modulation held constant, and as before, when either Ng or iV/ become An = i 2 .
2.3. The KC model Also plotted in Fig. 3 are the scattering matrix calculations for the plasmon frequencies for the
Vol. 83, No. 8
PLASMONS IN MODULATED 2DEGs BY GRATING GATES
631
symmetric with no phase steps. The reduction of the plasmon frequencies at large An can be understood in terms of the spatial shifts of the oscillating charge density; for An----0, the radiative mode is approximately sinusoidal in profile, with the maxima slightly drawn under the edges of the grating fingers to increase the screening effect, while for An -----+1.5 (charge depletion in the gaps) and An ----- 1.5 (charge depletion under the fingers) the maxima are drawn towards the regions of reduced charge density. The improved screening by the gate fingers observable for the case of An = -1.5 accounts in part for the generally lower frequencies for negative values of An [Fig. 3(b)], but it is clear that the reduction in 3. RESULTS AND DISCUSSION plasmon frequencies is dominated by the modulation The long-dash curves in Fig. 3 show the results for itself and the corresponding spatial shift of the the structure adopted by KC, a sinusoidally modu- oscillation to regions of low charge density as noted lated 2DEG under a continuous metal gate. F o r by Wulf et al. [22] rather than the improved sinusoidal modulation, N2---0, so the KC model screening. Figure 3 also shows that the rectangular and does not produce split first order plasmon modes, sinusoidal modulation profiles give similar values for even for An ~ 0 [equation (3)]; the scattering matrix the slopes for both the radiative and non-radiative calculation confirms this, and shows no splitting even at large values of An outside the range of the KC modes close to An = 0, (Nf = Ng), and in this sense algebraic result. The effect of increasing modulation there is qualitative agreement between the models for An is to reduce the plasmon frequencies, by up to small modulation amplitude. For rectangular modu30% at the pinch-off extremes; the frequency lation with r = 0.5, the profile is closer to a sinusoidal reduction is symmetric in An because the gate is profile of the same period than is the case for r = 0 . 3 3 , so it is also not surprising that the continuous. The frequency shifts in the presence of the lamel- agreement between the two profile models is better lar grating gates are more complex in detail because [Fig. 3(b)]. The quantitative differences in the values of w6 of the periodic screening effect of the grating, but show the same overall behaviour as the KC structure. can be understood in physical terms as follows. The Note first that, for the same average 2DEG number KC model assumes the presence of a continuous density and An = 0, the plasmon frequencies are perfectly conducting gate, whereas the scattering generally markedly higher for the case of the lamellar matrix model explicitly includes the less efficient grating gates (and higher for smaller mark fraction r) screening effect of a grating gate, resulting in because of the reduced screening compared with that higher values for ~o6. The scattering matrix of the continuous gate of the KC model. But most model also explicitly includes the periodicity of the striking is the fact that the plasmon modes are split grating-gate screening effect, which introduces a by the periodic screening effect, and the shifts are no further splitting, apparent even at zero modulation, longer symmetric in An. The corresponding calcu- between the radiative and non-radiative modes lated transmission spectra show that only the lower because of their different spatial distributions of energy mode of the split pair is radiative and hence oscillating charge; the radiative and non-radiative experimentally observable. The symmetries of these modes have nodes and antinodes respectively at the modes are different, as shown in Fig. 4, where the centres of the gaps between the fingers. The KC amplitude and phase of the oscillating charge density, model also includes only the lowest two Fourier calculated using the scattering matrix approach, is coefficients for Ns(x) [equation (2)], whereas the shown as a function of position under the grating scattering matrix calculation includes terms up structure (r = 0.5) for several values of An for the to the eighth harmonic, so it is clear that the rectangular depletion model; the radiative modes scattering matrix approach is more appropriate for [Fig. 4(a)] are antisymmetric with respect to the describing our experimental systems which include centres of the fingers (and the gaps between them) lamellar metal grating gates and strong, essentially and have nodes and phase steps of 7r at these points, rectangular, modulation of the 2DEG number while the non-radiative modes [Fig. 4(b)] are density.
sample structure used in the simple KC model, with the 2DEG sinusoidally modualted and a continuous metal gate; here, though, the electric field throughout the structure is represented by a Fourier series with terms up to InI = 8, and the charge density modulation, described as in Section 2.2 above with normalised amplitude An, is not restricted to weak modulation amplitude as in KC's calculations. Because this calculation does not explicitly include any overlaid grating, it is used for comparison with the calcuations for both grating mark fractions r [Figs 3(a) and (b)].
632
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
Vol. 83, No. 8
An=0
An = 0
v
e-.
0f
.................
...................... . . . . . . . . . . .
An = - 1 . 5
An = -1.5
I
v
0
0
An =
1.5
~n = 1.5
v
0
0
0
0.25
0.5
0.75
x/d
(a)
0
0.25
0.5
0.75
x/d
(b)
Fig. 4. The square Ip2l of the oscillating charge density, and its phase, in the 2DEG under a grating with r = 0.5 for An = 0,-1.5 and 1.5, for plasmons of (a) radiative and (b) non-radiative symmetry. The rectangles at the top of the figures indicate the extent of the metal fingers of the grating, and under each figure, the relative phase of the charge density oscillation is shown. Note that modes of radiative symmetry are antisymmetric about x --- 0 and x = d/2(Ip2l has a node and the relative phases of the charge density either side of a node differ by 7r), whereas the non-radiative modes are symmetric (Ip2l antinode, phase continuous and symmetric). Note that for both symmetries at An = +1.5, the amplitude of the charge density oscillation is enhanced in the regions of low charge density, indicating plasmon localisation in these regions resulting in a reduction in plasmon frequency. 4. C O N C L U S I O N We have investigated the optical response and natural mode frequencies of modulated 2DEGs
under a metallic grating using scattering matrix calculations of the resonant frequencies and the charge density amplitudes as functions of the modulation profile and amplitude. Our results are
Vol. 83, No. 8
PLASMONS IN M O D U L A T E D 2DEGs BY G R A T I N G GATES
in general agreement with those of Cataudella and Marigliano Ramaglia [20] and Wulf et aL [22], in that the plasmon resonance becomes localised in the region of low charge density resulting in a decrease in plasmon frequency from that expected from the average value of the density; the reduction in plasmon frequency is generally larger when the depletion profile is more rectangular [19]. In addition to these effects, we have also demonstrated that an overlaid grating-gate splits the plasmon frequencies of the radiative and non-radiative symmetry plasmon modes. Furthermore, the shift in plasmon frequency depends on whether depletion occurs under the fingers or in the regions between them: if the low density regions of the 2DEG are under the grating fingers, then the gate screening acts to lower the localised plasmon frequency even further; if on the other hand, the 2DEG density is lowest under the regions between grating fingers, the plasmon is not as well screened and the frequency is higher. REFERENCES 1. 2. 3. 4. 5. 6. 7.
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