Two-dimensional plasmons in hole space charge layers on silicon

Solid State Communications, Vol. 46, No. 3, pp. 269-271, 1983. Printed in Great Britain.

0038-1098/83 / 150269-03 $03.00/0 Pergamon Press Ltd.

TWO-DIMENSIONAL PLASMONS IN HOLE SPACE CHARGE LAYERS ON SILICON E. Batke, D. Heitmann, A.D. Wieck and J.P. Kotthaus Institut for Angewandte Physik, Jungiusstrafie 11,2000 Hamburg 36, West Germany

(Received 3 January 1983 by M. Cardona) We have observed two-dimensional plasmons in hole space charge layers of silicon. On Si(1 10) surfaces the plasmon mass depends on the charge density and shows a significant anisotropy for different directions of the plasmon wavevector in the surface. The determination of the plasmon mass allows detailed informations on the anisotropic and nonparabolic 2-D bandstructure of hole space charge layers. TWO-DIMENSIONAL (2-D) plasmons in space charge 5 layers of semiconductor surfaces have been observed so far only in electron inversion layers [1--4]. With im(% \ qll[O01] proved sample preparation we have observed 2-D ! plasmons of holes. The plasmon excitation is investiI Ns= gated in a hole accumulation layer on p-type Si sub3 ! 10.~ strates with (1 10)-surface orientation. The plasmon resonance experiments allow an additional and more ! detailed analysis of the complex surface bandstructure 2 of holes than Shubnikov-de Haas (SdH)- and cyclotron resonance (CR)-measurements [ 5 - 7 ] . Experiments are performed by measuring the trans57 mission of far infrared radiation through metal--oxidesilicon (MOS) capacitors using rapid scan Fourier trans3.~ 2.6,× 1012cm'2 form- and laser-spectroscopy. The radiation is coupled to the plasmon excitation of well defined wavevector 020 6'0 1;0 q = 21r/a via periodically structured gates on the MOSv (cm"1) capacitors of periodicity a. The experimental techniques Fig. 1. Absorptance P of p-Si(l 1O) accumulation layers and fabrication of the periodical gates are described in vs wavenumbers v for different surface charge densities [4]. The samples are p-type Si wafers (20 ~2 cm) with N s. Hole plasmon resonance positions are marked with black triangles. The plasmon wavevector Iql is 1.0 x l0 s 45 nm thermally grown oxide. All experiments are cm -1 and parallel to the [0 01 ]-direction. carried out at low temperatures (4.2 to 12 K). Figure 1 shows spectra obtained with Fourier transform spectroscopy of the absorptance P = - [ T ( V a ) Experimental hole plasmon masses for typical T(Vr)]/T(Vr), where T(Vo) and T(Vr) are the transsamples are shown in Fig. 3. There is a significant anisomission through the MOS-capacitor with gate voltage Vo tropy of the plasmon mass for plasmon wavevectors in [0 01 ]- and [ i" 1 0]-directions, respectively, with a and threshold voltage VT, respectively. The plasmon excitation is clearly resolved from the 2-D-Drude backheavier mass in [0 0 1]-direction. At a fixed value of q and density N 8 the experimental plasmon mass for ground of the hole gas. The resonance position shifts directions ~ (with ~ being the angle of the plasmon with increasing charge density to higher wavenumbers. wavevector with respect to the [0 01 ]-direction) follows Figure 2 shows the differential absorptance dP/dN8 in a sweep of the charge density N s at a fixed laser frequency within the experimental error of 5% the relation rn,~l(~) = mp~ool] cos 2 ¢ + m;,~Tlo] sin 2 ~. With decreas84.3 cm -1 . The resonance position varies with different ing N s and fixed q and ~, there is a slight decrease of the directions of the plasmon wavevector. To extract inforplasmon mass mp(Ns, q, ¢) which is more pronounced mation on the band structure from the dispersion of the for the plasmon mass in [ i"1 0]-direction. Anisotropy 2-D hole plasmons we have evaluated from the peak and Ns-dependence are caused by the anisotropic and position of the plasmon resonance an effective plasmon nonparabolic surface band-structure of holes, which mass mp by applying the classical plasmon dispersion will be described in more detail below. Experiments formula (e.g. formula 27 of [3]). 269

270

TWO-DIMENSIONAL PLASMONS IN SILICON

Vol. 46, No. 3

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Fig. 2. Differential absorptance of a hole accumulation layer on p-Si(1 1 0) at fixed laser wavenumber vL = 84.3 cm -1 and wavevector Iql = 1.0 x l0 s cm -l in a sweep of the surface charge density N 8 for different directions of the plasmon wavevector. Plasmon resonance positions, corresponding to a maximum in P, are marked by arrows. Additional heavy-light (h-1 ISR and lightheavy ( 1 - h ISR) intersubband transitions are indicated. have been performed on many samples with grating periodicities ranging from 660 to 400 nm. All samples show a similar plasmon dependence on N s. The reproducibility for a given q-vector is about 5% and for different q-vectors there is a variation of the plasmon mass of about 10%. Within this 10% there is a tendency of higher plasmon masses for samples with smaller periodicities and corresponding larger wavevectors. From surface effective masses information on the surface band structure can be extracted. SdH- [5,6] and CR [7] -experiments have been applied to determine the CR-mass of holes. The experimental masses of [6, 7] agree reasonably with self-consistent subband calculation of holes in [8, 9]. In Fig. 3 it is shown that the experimental CR-mass of [7] is situated between our experimental plasmon masses for both principal surface directions. The reciprocal plasmon mass for a general (nonparabolic, nonelliptical) surface band structure is in lowest order perturbation theory:

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detailed analysis of the surface band structure since it reflects the anisotropy of the contours of constant energy. In comparison to SdH- and CR-experiments plasmon experiments also have the advantage, that they can be performed without magnetic field. Present surface band structure calculations do not include the effect of a magnetic field and it is not known, how strong its influence on the band structure of holes is. To compare our experimental results with band structure calculations we have graphically evaluated a plasmon mass [formula (1)] from the theoretical surface energy contours in Fig. 6(b) of [9]. As well as can be expected from this rather rough evaluation (which is limited by the graphical resolution and number of data points) the absolute mass and degree of anisotropy agree with our experimental values. However it would be desirable to extract the plasmon masses directly from mp(¢)l h2N1 io~: __ { cos= ¢ a2E'°(K)aK~---~ I- sin 2 ¢ a2E'°(K)t~. -/ theoretical data. Also in order to make direct comparison with our results the subband structure should be (1) calculated for accumulation conditions (calculations in ~bis the angle of the plasmon wavevector with respect to [8, 9] are performed for hole-inversion). An influence the kx-axis. The sum has to be taken for all N occupied of the onset of the occupation of the light hole subband, states K in all subbands i and for both spin orientations which is calculated [ 11 ] and experimentally observed o. Determination of the plasmon mass thus offers a more [14] for densities about 4 x 1012 cm -2, is not observed -

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TWO-DIMENSIONAL PLASMONS IN SILICON

in our plasmon experiments. Also the lifting of the spin degeneracy of the hole-surface bands is not directly observed, since the reciprocal plasmon mass is the averaged reciprocal plasmon mass of both spin orientations. In the classical plasmon theory the plasmon mass is equal to the conductivity mass m c. Thus in principle information on the mass can also be extracted from d.c. or dynamic conductivity measurements. Plasmon resonance experiments have the advantage, that mp can be evaluated directly without knowing the scattering time r. In contrast, to extract m e from conductivity measurements, both m c and r have to be determined from a fit to the data. d.c.-Conductivity measurements on pchannel FETs of(1 10)-surface orientation in [10] show e.g. f o r N s = 5 × 1012 cm -2 within 10% the same ratio of the mobility for current flow in [0 01 ]- and [T 10]direction, respectively, that we find for the reciprocal plasmon mass ratio. However, in [ 11 ] it is found, that the ratio of the mobility depends on temperature. This effect is explained by nonisotropic scattering processes. We have measured the dynamic conductivity o(eo) on Si(1 1 0) with semitransparent gates without gratings in the frequency regime 25-200 cm -1. On the same sample the ratio of the conductivity for light polarized in [0 01]- and [11 0]-direction, respectively, agrees within the experimental error with the ratio that we find for the reciprocal plasmon mass at the same density. However at certain frequencies and charge densities an influence of localization effects [ 12], intersubband transitions [ 13 ] and possibly anisotropic scattering processes is observed. These processes also limit the accuracy of a classical Drude fit to the experimental conductivity to determine both, me and r. Thus plasmon resonance experiments are a more reliable method for determining the conductivity mass. In Fig. 2 additional structures occur near the plasmort resonance (marked by ISR). These structures are observed for different wavenumbers and are identified as intersubband transitions [ 13]. The resonance positions agree surprisingly well with theoretical subband

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energies of [8]. A possible interaction of these intersubband transitions with the plasmon resonance at the same energy may be the origin for weak structures in the plasmon dispersion, which we observe on some samples. This effect is under further investigation. In conclusion, we have observed 2-D-hole plasmons. The plasmon reflects the anisotropic and nonparabolic surface band structure of the (1 1 0) hole space charge layer.

Acknowledgements - We wish to thank W. Beinvogl of Siemens Munich for supplying the oxidized wafers and the Deutsche Forschungsgemeinschaft for financial support.

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