An attempt is made to study the gate capacitance of MOS structure of p-channel inversion layers on tellurium in the presence of a quantizing magnetic field by considering all types of anisotropies of the valence bands of tellurium within the framework of k^→ · p^→ theory. We have derived an analytical expression of the surface electron concentration in low electric field limit in the presence of a quantizing magnetic field. We have then formulated a model expression of the magneto-gate capacitance with the proper use of the electron concentration. For the purpose of relative comparison, we have also derived the same capacitance by including the broadening effects and without any approximations of low or high electric field limits. It has been observed that the gate capacitance of p-channel inversion layers on tellurium exhibits spiky oscillations with changing magnetic field. The corresponding well-known results for p-channel inversion layers on parabolic energy bands have also been obtained under certain limiting conditions from the generalized expressions.Proučavaju se kapacitancija vrata MOS strukture p–kanalnih inverzijskih slojeva na teluru u magnetskom polju na osnovi k^→ · p^→ teorije. Pokazuje se da kapacitancija vrata mijenja periodički s rastućim magnetskim poljem. Dobiveni rezultati svode se u modelu paraboličkih vrpci, uz određene uvjete, na poznate izraze

Banik, Sambhu N.

Ghatak, Kamakhya P.

English

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ISSN1330–0008CODENFIZAE4A SIMPLE THEORETICAL ANALYSIS OF THE MAGNETO–GATECAPACITANCE IN MOS STRUCTURE OF p–CHANNEL INVERSIONLAYERS ON TELLURIUMKAMAKHYA P. GHATAK and SAMBHU N. BANIK∗Department of Electronic Science, University of Calcutta, University College of Scienceand Technology, 92 Acharya Prafulla Chandra Road, Calcutta 700 009, India∗ Department of Physics, Dandirhat Nagendra Kumar Uchcha Siksha Niketan, P.O.Dandirhat, Dt. North 24 Parganas, West Bengal, IndiaReceived 24 February 1995UDC 538.915PACS 71.20.CfAn attempt is made to study the gate capacitance of MOS structure of p-channelinversion layers on tellurium in the presence of a quantizing magnetic field byconsidering all types of anisotropies of the valence bands of tellurium within theframework of ~k · ~p theory. We have derived an analytical expression of the surfaceelectron concentration in low electric field limit in the presence of a quantizingmagnetic field. We have then formulated a model expression of the magneto-gatecapacitance with the proper use of the electron concentration. For the purpose ofrelative comparison, we have also derived the same capacitance by including thebroadening effects and without any approximations of low or high electric field lim-its. It has been observed that the gate capacitance of p-channel inversion layers ontellurium exhibits spiky oscillations with changing magnetic field. The correspond-ing well-known results for p-channel inversion layers on parabolic energy bandshave also been obtained under certain limiting conditions from the generalized ex-pressions.FIZIKA A 5 (1996) 1, 31–40 31ghatak and banik: a simple theoretical analysis . . .1. IntroductionIn recent years, there has been considerable interest in the study of semiconduc-tor inversion layers which are formed at the surfaces of semiconductors in devicesunder the influence of a sufficiently strong electric field, applied perpendicularly tothe surface in the presence of a large gate bias [1]. In such layers, the carriers form atwo-dimensional gas and are free to move parallel to the surface while their motionis quantized in a direction perpendicular to it, leading to the formation of electricsub-bands. This quantized motion gives rise to many special features of MOSFETdevices. One such important feature is the capacitance of MOS structures, sinceits dependence on the surface electric field becomes oscillatory in the presence ofa quantizing magnetic field. The fact that the gate capacitance can be easily con-trolled by varying the gate voltage is of much importance from the point of view oftechnical considerations. Incidentally, it may be stated in this context that, thoughthere has been considerable work on the magneto-transport in inversion layers inMOS structure of tellurium [1], the gate capacitances in these structures have yetto be investigated. It would, therefore, be of much interest to study the effect ofmagnetic quantization on the gate capacitance of MOS structures of tellurium byusing the generalized band model, since the above class of materials is increasinglyused for different technical applications [2].In what follows, we shall first derive the model expressions for gate capaci-tance of MOS structures of p-type Te under magnetic quantization. This will makeour study a generalized one, since we can obtain the corresponding expressionsfor p-channel inversion layers on parabolic semiconductors under certain limitingconditions. However, we shall consider the low-electric-field limit since the creationof large surface field is experimentally difficult. Besides, we shall also derive thegate capacitance by including the broadening effects. We shall then investigate,theoretically, the effect of a quantizing magnetic field on the gate capacitance.2. Theoretical backgroundIn MOS structures, the gate capacitance per unit area is given byC−1g = C−1ins + C−1sc , (1)where Cins = ǫins/dins, ǫins and dins are the permittivity and the thickness of theinsulating layer, respectively,Csc = eddV0[Ns] = edpsdVg(1− edinsǫinsdpsdVg)−1. (2)e is the magnitude of the carrier charge, ps is the hole concentration per unitarea, V0 = Vg − (epsdins/ǫins) is the surface potential and Vg is the gate voltage.It appears, therefore, that the derivation of the magneto-gate capacitance, withthe help of Eq. (1), requires an expression of the total hole concentration under32 FIZIKA A 5 (1996) 1, 31–40ghatak and banik: a simple theoretical analysis . . .magnetic quantization as a function of the gate voltage which, in turn, is determinedby the corresponding two-dimensional hole-energy spectra. Incidentally, it may benoted in this context that the detailed structure of the energy bands nearH point ofTe was derived by many authors [2]. All of them used a ~k·~p perturbation consideringthe coupling of the upper two branches of the valence band. Non–parabolic valenceband terms (∼ k4) are given by a second order interaction between the vH4, vH5,vH61, vH62 states.The energy spectra of the holes in valence bands of bulk specimens of Te canbe expressed as [2]E = Ak2⊥ +Bk2z + βk4⊥ + β′k2⊥k2z − 2|∆1|±[(2|∆1|+ ηk2⊥ + ζk4⊥ + ζ ′k2⊥k2z)2 + S2k2z]1/2, (3a)where E is the total hole energy as measured from the top of the upper valence bandin the downward direction in the absence of any quantization, S is the first-order~k · ~p term, |∆1| is the first-order spin–orbit term, A, B, β, β′, η, ζ and ζ ′ containdifferent contributions of first- and second-order ~k · ~p, first-order spin–orbit andsecond-order spin–orbit ~k · ~p interaction with ~k–independent spin–orbit interactionterms, and k2⊥ = k2x + k2y. Thus, in the presence of a quantizing magnetic field B0applied along the crystal axis, the modified hole energy spectra can be written asEn = a(n) + b(n)k2z ±√[c(n) + d(n)k2z ]2+ e1k2z , (3b)where En is the total energy in the presence of magnetic field as measured fromthe top of the upper valence band in the downward direction in the absence of anyquantization,a(n) = −2eB0Ah̄(n+12)− 2(|∆1|) + β[2eB0h̄(n+12)]2.e is the carrier charge, h̄ = h/2π, h is the Planck’s constant, n(= 0, 1, 2, ...) is themagnetic quantum number,b(n) = B − 2eB0(n+ 12)β′h̄,c(n) = 2|∆1| −2ηeB0h̄(n+12)+ ζ[2eB0h̄(n+12)]2,e1 = S2 and d(n) = −2eB0ζ′h̄(n+12).FIZIKA A 5 (1996) 1, 31–40 33ghatak and banik: a simple theoretical analysis . . .Thus, the carrier energy spectra in p–channel inversion layers of the above classof materials in the presence of a uniform DC quantizing magnetic field, B0, alongc–axis can be expressed, in low electric field limit, asα1(n)3[T1(n)± 2α2(n)√T1(n) + α3(n)]3/2±α1(n)α2(n)2[(√T1(n)± α3(n))(T1(n)± 2α2(n)√T1(n) + α3(n))1/2+α4(n)2 log∣∣∣∣(√T1(n)± α2(n)) +(T1(n)± 2α2(n)√T1(n) + α3(n))1/2∣∣∣∣]−α1(n)3[T3(n)± 2α2(n)√T3(n) + α3(n)]3/2∓α1(n)α2(n)2[(√T3(n)± α3(n))(T3(n) + α3(n)±√T3(n)2α2(n))1/2+α4(n)2 log∣∣∣∣(√T3(n)± α2(n)) +(T3(n)± 2α2(n)√T3(n) + α3(n))1/2∣∣∣∣]=2S3/2i3, (4)whereα1(n) =p2(n)T2(n)3/2eFs.Fs is the surface electric field, p2(n) = b(n)[−b2(n) + d2(n)]−1, T2(n) =t2(n)[−4b2(n) + 4d2(n)]−1, t2(n) = 2e1b(n) − 4b(n)c(n)d(n) + 4d2(n)a(n) −4d2(n)Eni, Eni is the hole energy under magnetic quantization as measured fromthe edge of the valence band at the surface in the absence of any quantization,α2(n) =T2(n)2p2(n), T1(n) =t1(n)4b2(n)− 4d2(n) ,t1(n) = [e21 + 4a(n)b(n)e1 − 8a(n)b(n)c(n)d(n)− 4e1c(n)d(n)+4b2(n)Enia(n)− 2e1Enib(n) + 4b(n)c(n)d(n)Eni + 4Eni2d2(n)+4a2(n)d2(n)− 8d2(n)a(n)Eni + 4b2(n)c2(n)],34 FIZIKA A 5 (1996) 1, 31–40ghatak and banik: a simple theoretical analysis . . .α3(n) =p1(n)T2(n)− p2(n)T1(n)p2(n),p1(n) =a(n)b(n) + (e1/2)− c(n)d(n)− b(n)Enib2(n)− d2(n) ,α24(n) = α3(n)− 4α22(n),T3(n) =p21(n)T2(n)− 2p1(n)p2(n)T1(n)T2(n)− 2p1(n)p2(n).i = 0, 1, 2, ... is the electric sub–band index and si are the zeros of Airy function.Under the conditions A = B = h̄/2m∗ (m∗ is the effective mass at the bandedge), β = 0, β′ = 0, |∆1| = 0, η = 0, ζ = 0, ζ ′ = 0 and s = 0, Eq. (4) assumes theformEni =(n+12)h̄ω0 + Si(h̄eFs√2m∗)2/3(5)where ω0 = eB0/m∗. We wish to note that Eq. (5) is the well–known relation forthe carrier energy in inversion layers on parabolic semiconductors under magneticquantization [1].Therefore, the density–of–states function can be written asD(E) =eB0gsgvh∑n,iδ′(ǫ− Eni), (6)where gs is the spin degenerancy, gv is the valley degenerancy and δ′ is the Dirac’sdelta function. Combining Eq. (6) with the Fermi–Dirac occupation probabilityfactor and using the property of delta function, the surface hole concentration perunit area can be expressed asps =eB0gsgvh∑n,i[1 + exp(Eni − E′FBOkBT)]−1, (7)where kB is the Boltzmann constant, T is the temperature, E′FBO is the Fermienergy under magnetic quantization as measured from the edge of the valencebands at the surface in the absence of any quantization and is given byE′FBO = eVg − pse2dinsǫins− EfbB0 , (8)in which EfbB0 represents the energy separation between the Fermi level at the edgeof the valence band in the bulk of the n–type substrate material in the presence ofFIZIKA A 5 (1996) 1, 31–40 35ghatak and banik: a simple theoretical analysis . . .a quantizing magnetic field. Equation (7) can be written asps =eB0gsgvh∑n,iF−1(η), (9)where η = (E′FB0 −Eni)/kBT and Fj(η) is the one–parameter Fermi–Dirac integralof order j [3].Thus, combining Eqs. (1) and (9), the generalized expression for the gate ofMOS structures of p–type Te under magnetic quantization can be written ascg(B0) =e2ζ0∑n,i F−2(η)1 + ζ0∑n,i p(n)F−2(η), (10)whereζ0 =eB0gsgvhkBT,pn = ρ(n) + e2 dinsǫinsandρ(n) =∂∂ps(Eni),which can be obtained by using Eq. (4).It may be noted that under the conditions A = B = h̄2/2m∗, β = 0, β′ = 0,|∆1| = 0, η = 0, ζ = 0, ζ ′ = 0 and S = 0, the form of the magneto-gate capacitancefor p–channel inversion layers of parabolic semiconductors is expressed by the sameEq. (10), whereρ(n) =23p−1/3s(h̄e2ǫsc√2m∗)2/3Si.Incidentally, in the presence of broadening effects the density-of-states function ofthe carriers in 2D inversion layers on semiconductors can, in general, be expressed[1]D(ǫ) =eB0gsgvh√2π∑n,iΓ−1n exp(−2(ǫ− ǫ′)2Γ2n), (11)where Γn is the line width of the broadened Landau levels and ǫ′ is the unperturbedenergy eigenvalue which in the present case can be determined from the equationǫ′ = a(n)± c(n). (12)36 FIZIKA A 5 (1996) 1, 31–40ghatak and banik: a simple theoretical analysis . . .Thus, combining Eq. (11) with the Fermi–Dirac occupation probability factor, thesurface hole concentration can be expressed asps = K0∑niX0Γn, (13)whereK0 =eB0gsgvh√2πkBT,X0 =∞∑t=0Y+(t) +∞∑t=1Y−(t)−γ√π2(1− erf(U)) ,Y±(t) =√2πγ2exp(Y ′±)L(1∓ erf(Y 2′±)),L = (−1)t exp(−U2), U = η′γ, γ =ΓnkBT√2,η′ =E′FBO − ǫ′kBT, Y ′± =γ2(2Uγ−1 ± t)and erf denotes the error function [4]. Thus, using the appropriate equation, themagneto-gate capacitance for p–channel inversion layers on Te can be expressedincluding broadening effects ascg(B0) =α0∑n,i β0(Γn)−11 + (eα0dins/ǫins)∑n,i β0(Γn)−1, (14)whereα0 =eK0kBT, β0 = exp(−U2) +∞∑t=0Q1,+ +∞∑t=1Q1,−andQ1,± = Lγ2√π(1∓ erf(Y ′±))Y ± ∓ Lγ2 − UL exp(Y2′±)(1∓ erf(Y ′±)).It may also be noted that the general expression for the carrier statistics and thegate capacitance in inversion layers on parabolic semiconductors under magneticquantization, including broadening effects, will be given by Eqs. (13) and (14),whereǫ′ =(n+12)h̄ω0. (15)FIZIKA A 5 (1996) 1, 31–40 37ghatak and banik: a simple theoretical analysis . . .3. Results and discussionUsing Eqs. (9) and (10) and taking the parameters [2] A = −2.57× 1019 eVm2,B = −4.09 × 10−19 eVm2, β′ = 0, gs = 2, gv = 2, β = 1.1 × 10−37 eVm4,2|∆|| = 6.315 × 10−2 eV, η = −1.18 × 10−19 eVm2, ζ = 4 × 10−32 eVm, ζ ′ = 0,|S| = 2.62×10−10 eVm, ǫsc = 30 ǫ0, dins = 15 µm as appropriate for p-Te togetherwith Fs = 5.6×104 V/m and ǫins = 2.8 ǫ0 (the permittivity of Mylar, for example,which is commonly used as the equivalent of the oxide layer in non-parabolic MOSstructures), we have plotted -Cg(B0)/Cg(0) versus 1/B0 at 4.2 K as shown in plotA of Fig. 1. Plot B shows the same dependence in accordance with the parabolicband model.Using the same parameters as used in obtaining Fig. 1 together with Γn =10−3√B0(T) [1], and using Eqs. (13) and (14), we have plotted Cg(B0)/Cg(0)by including broadening effects, shown in plot A of Fig. 2. The plot B in Fig. 2exhibits the same dependence for the parabolic model. The depths of the spikesincrease with increasing magnetic field. With varying magnetic field, each time aLandau level crosses the Fermi level, a change occurs in the capacitance throughredistribution of the carriers among the Landau levels. It may be noted that the3D quantization, in the absence of broadening of Landau levels, leads to the dis-crete energy levels, somewhat like atomic energy levels in steps produce very sharpchanges. This follows from the inherent feature of the 3D quantization of the 2Dhole gas delt with here. Under such quantization, there remains no free hole statein between any two successive Landau levels, unlike that found for 3D hole gasesof semiconductors under 2D quantization in the ~k space in the presence of a quan-tizing magnetic field. Consequently, the crossing of the Fermi level by the Landaulevel under 3D quantization would have much greater impact on the redistribu-tion of the holes amongst the available states, as compared to that found for 2Dquantization. It is basically this impact which results in the increased sharpness ofthe oscillatory spikes for both models of Te. Incidentally, the collision broadeningeffects which have been neglected in the theoretical formulation shown in Fig. 1,would, really, reduce the sharpness of the oscillatory spikes. Thus, to assess theinfluence of broadening, we have plotted the same dependence in Fig. 2. In thepresence of broadening, the basic physics behind 3D quantization is not applicable.Hole motion becomes possible in broadened sub-bands and thus the broadening pa-rameter changes the analytical expressions of the hole statistics and magneto-gatecapacitance, respectively.It may be noted that Eqs. (8), (9), (13) and (14) are the generalized expressionsfor the hole statistics and the magneto-gate capacitance, both in the presence andabsence of broadening effects, respectively, in p-channel inversion layers on Te un-der magnetic quantization. For inversion layers on parabolic semiconductors, thegeneral forms of the above equations will be unaltered. The above facts are trueonly for 2D hole gases in inversion layers under magnetic quantization. Since thebasic physics behind the broadening of Landau levels in inversion layers under mag-netic quantization is radically different as compared to 3D quantization, we can notobtain the results valid for 3D quantization from the corresponding expressions for38 FIZIKA A 5 (1996) 1, 31–40ghatak and banik: a simple theoretical analysis . . .broadening under the condition Γn → 0. That the above conclusion is not true forbulk semiconductors under magnetic quantization has been shown in Ref. 5. Thus,the results which are valid for Γn /=0 are not valid for Γn = 0, and are true onlyfor 2D carrier gases in inversion layers under magnetic quantization. It appearsfrom Fig. 2 that even in the presence of broadening, the numerical magnitude ofthe magneto-gate capacitance in p-channel inversion layers of Te are larger as com-pared to parabolic model, and the nature of oscillations is found to be somewhatsimilar to that observed experimentally in Te MOS structures having p-channelinversion layers [2].Fig. 1. Plot A shows the magnetic-field dependence of the normalized gate capac-itance of MOS structures having p-channel inversion layers under magnetic quan-tization in the absence of broadening. The plot B corresponds to the parabolicmodel.Fig. 2. Plot A shows the magnetic field dependence of the normalized gate ca-pacitance of MOS structures of Te having p-channel inversion layers under mag-netic quantization in the presence of broadening effects. The plot B corresponds toparabolic energy bands (right).It may also be noted that, as far as the determination of the effective massunder degenerate carrier distribution at the surface is concerned, the measurementof magneto-gate capacitance, as compared to that of the conductivity or cyclotronresonance, would not be more advantageous regarding the experimental facilitiesrequired or accuracies achieved. Nevertheless, it is felt that the theoretical in-vestigation presented here would be of much significance, as the interest on gatecapacitance has been growing very much in recent years from the point of view oftechnical applications and of exploration of fundamental aspects of semiconductorsurfaces in MOS structures. It may be noted that since the many-body effects,the carrier-spin, the arbitrary orientation of the quantizing magnetic field, the for-FIZIKA A 5 (1996) 1, 31–40 39ghatak and banik: a simple theoretical analysis . . .mation of band-tails, the hot-electron effects and the dependence of Γn on othervarious physical parameters have not been considered in obtaining both plots, there-fore, the essential meaning of comparison of the gate-capacitance between the twomodels would not be meaningless, and this simplified analysis exhibits the basicqualitative features of the oscillatory magneto-gate capacitance in p-channel MOSstructures of Te. For a more accurate derivation, the modifications as mentionedabove should be taken into account, which present formidable problems due to thelack of analytical techniques. Finally, it may be noted that the basic purpose ofthe present work is not solely to demonstrate the effects of magnetic quantizationon the gate capacitance, but also to formulate the carrier statistics for p-channelinversion layers of Te, since the various transport phenomena and the derivationof the expressions for many important physical parameters of 2D semiconductorsdevices are based on the carrier statistics in such materials.References1) T. Ando, A. B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437;2) M. Lutz, Optical Activity - de Haas-van Alphen Oscillations and Conduction Band Pa-rameters of Tellurium in the Physics of Selenium and Tellurium, edited by G. Gerlachand P. Gosse, Springer-Series in Solid State Sciences 1979, p. 86;3) K. P. Ghatak and B. Mitra, Internat. J. Electronics 72 (1992) 541;4) M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publi-cations, New York, 1965;5) M. Mondal and K. P. Ghatak, Phys. Stat. Sol.(b) 123 (1984) K143.JEDNOSTAVNA TEORIJSKA ANALIZA KAPACITANCIJE MAGNETSKIHVRATA U MOS STRUKTURI p-KANALNOG INVERZIJSKOG SLOJA NATELURUProučavaju se kapacitancija vrata MOS strukture p–kanalnih inverzijskih slojevana teluru u magnetskom polju na osnovi ~k · ~p teorije. Pokazuje se da kapacitancijavrata mijenja periodički s rastućim magnetskim poljem. Dobiveni rezultati svodese u modelu paraboličkih vrpci, uz odredene uvjete, na poznate izraze.40 FIZIKA A 5 (1996) 1, 31–40

Jednostavna teorijska analiza kapacitancije magnetskih vrata u MOS strukturi p-kanalnog inverzijskog sloja na teluru

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