Bandstructure Effects in Ultra-Thin-Body DGFET: A Fullband Analysis
BBandstructure Effects in Ultra-Thin-Body DGFET: A Fullband Analysis
Kausik Majumdar and Navakanta BhatDepartment of Electrical Communication EngineeringIndian Institute of Science, Bangalore-560012, IndiaEmail: { kausik,navakant } @ece.iisc.ernet.in Abstract
This paper discusses a few unique effects of ultra-thin-body double-gate NMOSFET that are arising from thebandstructure of the thin film Si channel. The bandstruc-ture has been calculated using 10-orbital sp d s ∗ tight-binding method. A number of intrinsic properties includ-ing band gap, density of states, intrinsic carrier concentra-tion and parabolic effective mass have been derived fromthe calculated bandstructure. The spatial distributionsof intrinsic carrier concentration and < > effectivemass, arising from the wavefunction of different contribut-ing subbands are analyzed. A self-consistent solution ofPoisson-Schrodinger coupled equation is obtained takingthe full bandstructure into account, which is then appliedto an insightful analysis of volume inversion. The spatialdistribution of carriers over the channel of a DGFET hasbeen calculated and its effects on effective mass and chan-nel capacitance are discussed. The interest in Ultra-Thin-Body (UTB) Double-GateFET (DGFET) has grown in the recent past because ofits superior properties compared to bulk MOSFET andis being considered as one of the future alternatives ofpresent day bulk devices [1, 2]. Apart from superior gatecontrol from both top and bottom, the intrinsic quantumconfinement provided by its unique geometric structureaffects the characteristics of UTB DGFET [3]. The dif-ferent aspects of classical modeling of DGFETs have beendiscussed in [4, 5, 6]. There are a number of reports onthe quantum mechanical effects on DGFET, both ana-lytical as well as numerical [3, 7, 8, 9]. To solve nu-merically, one has to solve coupled Poisson-Schrodingerequations self-consistently [10]. This is done either by as-suming some analytical E − ¯ k relationship, or by takingfull bandstructure into account. There has been consid-erable amount of work on calculation of bandstructure of materials [11, 12, 13, 14, 15, 16, 17] which can be pluggedinto the self-consistent Poisson-Schrodinger equation nu-merically [18].DGFET has a unique property of volume inversionwhich improves the transport characteristics enormously[1], [3]. This can be explained with the help of quantumeffects. Another important aspect is a substantial changein transport properties depending on the crystallographicorientation [19, 20, 21] which again can be analyzed fromdetailed bandstructure calculation. Also, the total num-ber of intrinsic carriers reduces with the thinning of thechannel material. This has an effect on the total gatecapacitance of DGFET [22, 23, 24].The aim of this paper is to focus on detailed analysisof some effects in UTB-DGFET which arise entirely be-cause of bandstructure of the channel material and arenot very apparent. Only Silicon has been considered asthe thin channel material in this work, but this can beeasily extended to other channel materials as well. Thefull-band structure calculation that has been used here isbased on sp d s ∗ tight-binding method [14, 15, 16]. Thecalculated bandstructure has then been used to predictsome intrinsic transport properties of thin film Si includ-ing band gap, density of states, intrinsic carrier concen-tration and effective mass. From this analysis, a numberof features are explained which are unique to ultra-thinfilm semiconductors. Following this, Poisson-Schrodingercoupled equation is solved self-consistently taking care ofthe full bandstructure. With the help of this, volumeinversion phenomenon has been critically analyzed withphysical insights. This in turn throws some light on thespatial distribution of carriers inside the DGFET chan-nel. Taking this into account, total channel capacitanceand evolution of effective mass from source end to drainend along the channel have been analyzed.The rest of the paper is organized as follows: Sec. 2 dis-cusses on the details of sp d s ∗ tight-binding method ofbandstructure calculation. The different intrinsic trans-port properties of ultra-thin film Silicon have been dis-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r ussed in sec. 3. Poisson-Schrodinger coupled equa-tion has been solved self-consistently and different relatedanalyses have been performed in sec. 4. Finally the paperis concluded in sec. 5. Tight-binding method of bandstructure calculation hasbeen extensively studied by many researchers [11, 12, 13,14, 15, 16, 17]. In this work, a 10-orbital sp d s ∗ tight-binding method [14, 15, 16] has been used to find band-structure of the thin film Si, being used as channel ma-terial. Only the onsite energies and two-center overlapintegrals of nearest neighbors have been taken into ac-count. Spin orbit interaction has been neglected, andthus each k point in the Brillouin zone is assumed to bedegenerate with two spin states. Infinite crystal period-icity has been assumed along channel length and widthdirections and thus Bloch’s theorem is assumed to holdgood in those directions. However, along the thickness ofthe channel, the crystal is truncated to a few monolayers,thus crystal periodicity can not be assumed in this direc-tion. Suppose, the thickness contains N atomic mono-layers. Then, the truncated crystal can be formed bytaking a basis of N atoms along the thickness directionand spanning them over the whole 2-D space. Fig. 1shows a 7 monolayer thick channel with the basis atomsshown as black dots. The channel region can be formedby spanning the basis atoms along x and y . The tight-binding fitting parameters for Si, used in this work, havebeen taken from [15, 16]. An N monolayers thick film willproduce a 10 N × N tight-binding Hamiltonian [18]. Toget rid of the huge number of surface states (whose energyeigen values often fall inside the semiconductor band gap)caused from dangling bonds, it has been assumed that thesurfaces are completely passivated by Hydrogen. This hasbeen achieved by artificially increasing the onsite energiesof s and p orbitals of the surface atoms, as described in[17].The assumption of this method is that the electronicwave function is strictly guided in the x − y plane. Thus,the Brillouin zone will comprise of a 2-D k space, as op-posed to a 3-D one in bulk case. k z has been assumed tobe zero throughout this paper. The whole 2-D Brillouinzone has been discretized using step size of 0 . × πa forboth k x and k y where a is lattice constant (=5 . A forSi). In this paper, the film thickness has been referencedto the number of monolayers (AL) in the film. An N AL thick Si film translates to a thickness of a ( N − / (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) xz y z=0z=t Insulator2Insulator1Gate2Gate1 Figure 1: A 7-monolayer thick film with basis atoms(black dots). The basis atoms can be spanned in whole2 − D along x and y to construct the thin film.thick ( ∼ . Si film over the whole 2 − D Brillouinzone. Only the top most valence subband and bottommost conduction subband have been included for clarity.Throughout this paper, the valleys occurring at Γ pointand at ∼ . πa along X direction are termed as Γ valleyand X valley respectively. −101−1 −0.5 0 0.5 1−4−2024 k x k y E ne r g y ( e V ) Figure 2: E − k relationship of top most valence subbandand bottom most conduction subband over the whole 2 − D Brillouin zone of a 17 monolayer thick Si film. Theconduction band minimum occurs at Γ point. The X valley is 4 fold degenerate.2 Intrinsic Properties of ThinFilm Si
In this section, the variations of different intrinsic electri-cal properties of thin film as a function of film thicknesshave been derived from the bandstructure calculation, de-scribed in the previous section.
It has been well established in literature, by both theoryand experiments, that in the nano-scale, bandgap of semi-conductors is a function of size of the material. As sizereduces, the bandgap of the material increases. Fig. 3shows how the Γ gap and X gap of a Si film are changingas a function of the film thickness. One should note that, B and G ap ( e V ) G GapX Gap
Figure 3: Calculated Γ and X gap of Silicon thin film as afunction of film thickness. For sufficiently small thickness,direct Γ gap is quite larger compared to the next gapoccurring at X valley.for a sufficiently thin film, as opposed to the bulk case, theconduction band minimum occurs at direct Γ point, andnot in the X direction. Thus the electrons will first popu-late the Γ valley and hence, one can expect to see drasticchange in transport properties for a thin film Si channelcompared to bulk. As the film thickness increases, the en-ergy difference ∆ E Γ X between Γ and X valleys decreases,and electron will start populating both the valleys. Fi-nally, at sufficiently large film thickness, at the bulk limit, X valley is of less energy compared to Γ valley, and thus,electrons will start populating only X valley. In Fig. 4,the 2-D density of states has been plotted as a functionof electron energy in the conduction band, for 4 differ- Energy (eV) − D D en s i t y o f s t a t e s ( / e V − c m ) Figure 4: 2-D DOS as a function of electronic energy fordifferent thickness values of the Si film. The energy spaceis discretized by steps of 0 . . As it has been discussed already that bandgap increasesas the size goes from bulk to nano-scale, one expects lowerintrinsic carrier concentration as the film thickness re-duces. The per unit area intrinsic electron concentrationat temperature T is given by n A = (cid:88) j (cid:88) ¯ k f ( E ¯ kj ) (1)where the first sum is over different subband indices j ofconduction band and the second sum is over all ¯ k pointsin the first Brillouin zone. E ¯ kj represents the energy eigen-value at k th point of j th subband index. The Fermi-Diracprobability f ( E ¯ kj ) is given by f ( E ¯ kj ) = 11 + e ( E ¯ kj − µ ) /k B T (2)3 B is the Boltzmann constant and µ is the chemical po-tential. Fig. 5 shows that the intrinsic carrier concen- −2 −1 Film thickness (nm) I n t r i n s i c c a rr i e r c on c ( pe r c m ) Figure 5: Intrinsic carrier concentration per unit area (i.e.total number of carriers contained in the film with unityarea) as a function of Si film thickness.tration per unit area increases with film thickness. How-ever, apart from the reduction in carrier concentration,another important observation is that the carriers have adistribution along the film thickness, which peaks at thecenter of the film. This is due to the spatial distributionof the wave function of the electronic states contributingto the carrier concentration. If the film of thickness t has N monolayers, then the film can be assumed to be dis-cretized by N points, at each of which, per unit volumecarrier density is given by n ( z ) = Nt (cid:88) j (cid:88) ¯ k f ( E ¯ kj ) | ψ ¯ kj ( z ) | (3)where ψ ¯ kj ( z ) is the wave function of the electronic state( j, ¯ k ) at z . Fig. 6 plots the fractional contribution of dif-ferent subbands to total electron concentration for a thinfilm of Si. Γ i and X j represent the i th subband of Γ valleyand j th subband of X valley, respectively. It is clear that,for very small thickness, only Γ and Γ subbands con-tribute, but as thickness increases, other subbands alsostart contributing. In Fig. 7 and 8, the spatial distribu-tion of intrinsic carrier concentration, coming from differ-ent subbands, have been shown, for 9 and 33 monolayerthick Si film, respectively. Since for a 9 monolayer thick P e r c en t age c on t r i bu t i on G G G G X Figure 6: Percentage contribution to per unit area intrin-sic carrier concentration from different Si subbands lyingin Γ and X valleys. For very small thickness only Γ and Γ contribute, but at larger thickness, electrons startpopulating other valleys as well. Z (nm) E l e c t r on c on c . ( pe r cc ) TotalX G G G G Figure 7: Carrier distribution along channel thickness ofdifferent subbands lying in Γ and X valleys for a 9 atomiclayer thick Si film.4 Z (nm) E l e c t r on c on c ( pe r cc ) TotalX G G G G Figure 8: Carrier distribution along channel thickness ofdifferent subbands lying in Γ and X valleys for a 33 atomiclayer thick Si film.film ( ∼ . and Γ contribute, the spatialdistribution of the total electron concentration is dictatedby wavefunction distribution of only these two subands.The peak concentration comes at the middle of the film,and reduces as it approaches the surface. However, fora 33 monolayer thick film ( ∼ . X subband contribute, and this in turnaffects the total carrier distribution, as shown in Fig. 8. A simple parabolic effective mass has been derived in thissection at the minima of different subbands to show someinteresting transport properties of thin film. Paraboliceffective mass m ∗ ( i, j ) for the i th valley and j th subbandis defined as m ∗ ( i, j ) = (cid:126) ∂ E ( i, j ) /∂ ¯ k ( i, j ) (4)Fig. 9 and 10 show how parabolic effective mass ( m ∗ ),normalized to electron rest mass ( m ), calculated at theminima of two bottom most subbands Γ and Γ varyalong different crystal direction, for four different filmthicknesses. One should note that, in both cases, for 9monolayer thick film, effective mass is highly anisotropic.For Γ , effective mass increases as one moves from [10] [10][01][1’0] [01’] [11][11’][1’1’][1’1] Figure 9: Variation of parabolic effective mass at the min-imum of first Γ subband with crystal direction and Si filmthickness. Anisotropy is observed at small film thickness. [10][01][01’][1’0] [11][1’1’][1’1] [11’]
Figure 10: Variation of parabolic effective mass at theminimum of second Γ subband with crystal direction andSi film thickness. For very small thickness ( ∼ < > effective mass is smaller than < > effectivemass.5irection to [11] direction, whereas, it reduces for Γ val-ley. However, for larger thickness, effective mass in boththe valleys, becomes fairly isotropic. Since the effective M * ( z ) / m Figure 11: Distribution of < > M ∗ ( z ) along filmthickness for various values of Si film thickness. Increas-ing thickness increases M ∗ ( z ) and reduces spatial unifor-mity.masses vary with subbands, and electron concentration indifferent subbands has different spatial distributions, it isexpected that effective mass should also have a spatialdistribution. A ‘distributed effective mass’ , say M ∗ ( z ), afunction of the depth z along the thickness of the film,has been defined as: M ∗ ( z ) = 1 (cid:80) i,j W ij ( z ) m ∗ ( i,j ) (5)where W ij ( z ) represents the fractional contribution ofelectron concentration at depth z from j th subband of i th valley. This way of defining < > M ∗ ( z ) has theunderlying assumption that all the electrons (more gen-erally, an equal fraction of electrons from each subbandof every valley) are moving along < > direction. Fig.11 shows that for thinner films, < > M ∗ ( z ) is more orless uniform (which is because all the electrons are in Γ and Γ subbands possessing almost same < > effec-tive mass), but as thickness increases, the effective massat position closer to surface becomes larger than that ofthe central part of the film. Thus, for larger thickness,carriers closer to center of the film are expected to bemore mobile than those which are closer to the surface.Note that, this effect is inherent to the intrinsic film, com-ing from spatial distribution of wave functions associated with different subbands. Another interesting observationis that, a 9 monolayer thick film has (merginally) larger < > effective mass than a 17 monolayer thick one atall z . This is because of the fact that the Γ valley haslarger effective mass for 9 monolayer thick film (Fig. 10). p /a) E ne r g y ( e V ) X parabolic fit G parabolic fits D E G D E G D E X G X Figure 12: Original tight-binding bandstructure data andcorresponding parabolic fits at four bottom most Γ sub-bands and bottom most X subband for < > Si. Thefits are reasonable for energy values less than ∼ . E − ¯ k relationship along the < > directionfor the four bottom most conduction subbands, calcu-lated from tight-binding method, as described in sec. 2.The dotted curves show the fitted parabolic bands withsame effective masses, as calculated from eqn. (4). ∆ E s represents the electronic energy range in the s th subbandbetween which the parabolic E − ¯ k tracks tight-binding E − ¯ k fairly well. From Fig. 12, it is clearly visible thatparabolic bands fail to track the actual bands for elec-tron energies in excess of ∼ . Self-consistent Solution ofPoisson-Schrodinger Equation
To study the effect of gate voltage on this structure, amodel device has been assumed consisting of a top gateand a bottom gate separated from the film by thin insu-lator layers, as shown in Fig. 1. The Si film is assumedto be undoped. Thus channel charge corresponds to onlymobile charge. If φ ( z ) is the potential at z , then one canwrite the 1-D Poisson equation as [4, 10] ∂ φ ( z ) ∂z = qN(cid:15) (cid:15) rs t (cid:88) j (cid:88) ¯ k f ( E ¯ kj ) | ψ ¯ kj ( φ, z ) | e qφ ( z ) kBT (6)where q is electronic charge, (cid:15) is permittivity of vacuumand (cid:15) rs is the relative permittivity of the channel mate-rial. The boundary conditions are derived from the factthat the normal component of the displacement vectorsinside Si and insulators will be the same at z = 0 and z = t . Thus, at the boundaries, one can have (cid:15) rs ∂φ ( z ) ∂z | z =0 = (cid:15) r V g − V fb − φ (0) t ox (7)and (cid:15) rs ∂φ ( z ) ∂z | z = t = (cid:15) r V g − V fb − φ ( t ) t ox (8) (cid:15) r and (cid:15) r are the relative permittivities of insulator1and insulator2 respectively, and, t ox and t ox are corre-sponding thickness of the insulators. V fbi is the flatbandvoltage between the i th gate and channel. Eqn. (6) canbe solved iteratively to find φ ( z ). To do this, first oneassumes an initial potential profile φ ( z ), and then calcu-lates bandstructure, which in turn provides the correctionto φ ( z ). This is iterated until it converges. However, oneshould note that, in every iteration, one needs to calculatebandstructure. This is because the potential φ ( z ) adds a z dependent perturbation to the crystal potential, andthus both E ¯ kj and ψ ¯ kj are function of φ ( z ). This makesthe problem computationally intensive.To reduce computation, the following approximationhas been made. Note that, the potential φ ( z ) does notchange drastically along z (which is shown later), andthus, as far as change in bandstructure is concerned, it’sa fair assumption, that φ ( z ) is constant (= φ c ) along z .Suppose, H is the original unperturbed 10 N × N tight-binding Hamiltonian, E and ψ are the unperturbedeigen values and eigen functions respectively. If one as-sumes that external potential only changes the on-siteenergies, and not overlap integrals, then the perturbation∆ H can be written as∆ H = − qφ c I (9) Thickness φ error (%) n error (%)(AL) Mean SD Mean SD9 -0.14 0.03 0.51 2.0933 0.65 0.31 2.98 6.28Table 1: Mean and Standard deviation values of percent-age error in φ ( z ) and n ( z ) for 9 and 33 monolayer thickfilms with V g = 1 . V .where I is 10 N × N diagonal unity matrix. Then, Hψ = ( H + ∆ H ) ψ = ( E − qφ c ) ψ (10)which means that all the energy eigenvalues will be shiftedby same energy (in other words, no relative change inenergy eigenvalues), and the wave functions remain in theunperturbed state. Thus, it is sufficient to calculate thebandstructure only once, and the same E ¯ kj and ψ ¯ kj canbe used through all the iterations. This reduces the totalruntime by nearly same number of times as the numberof iterations it takes to solve the Poisson equation (whichvaries roughly from 20 to 200 for different cases).To validate the approximation, the amount of error be-ing incurred in the worst case (maximum gate voltagewhere band bending is maximum) has been examined andthe results are tabulated in Table 1, for both 9 and 33monolayer thick films. The error here has been defined as P error = P approx − P exact P exact × P approx is the ap-proximate value of parameter P and P exact is the exactvalue of the parameter.To simplify the analysis, in the following, the metalsused as gate electrodes, have been assumed to have mid-gap work-function, and charge trapping inside insulatorsis taken to be zero. This essentially means that the flat-band voltages V fb and V fb are taken to be zero. Forsimulation, it has been assumed that t ox = t ox = 1nm, V g = V g and (cid:15) r = (cid:15) r = 3 . Under the above mentioned assumption that bandstruc-ture remains fairly the same under application of gatevoltage, one can write the carrier density distribution n ( z )as n ( z ) = n ( z ) e qφ ( z ) kBT (11)where n ( z ) is the intrinsic carrier density at z and isgiven by eqn. (3) and φ ( z ) is the potential profile ob-tained by solving the eqn. (6). Fig. 13 and 14 show thedistribution of carrier density over different subbands and7he total density, for 9 and 33 atomic layer thick films re-spectively, when a 1V supply has been applied to both thegates. One should note the difference in shape of the car-rier distribution as compared to intrinsic case. Also, the 9atomic layer thick film shows a higher peak carrier densitycompared to the 33 atomic layer thick one, although thetotal integrated carrier concentration is larger for thickerfilm. This can be explained with ‘potential pinning’ ef-fect, as discussed later. However, at any z , the fractionalcontribution from a subband to the total carrier densityremains the same at any applied gate voltage. Fig. 15 z (nm) E l e c t r on c on c ( pe r cc ) TotalX G G G G Figure 13: Total electron density distribution along thethickness and contribution from different subbands for a9 atomic layer thick Si film with V g = 1 . z at different gate voltages, for 9 and 33 atomiclayer thick films. The corresponding potential distribu-tion plot is shown in Fig. 17. As gate voltage increases,the carriers of the ‘central-peaked’ channel start spreadingtoward the channel-insulator interface, and beyond a par-ticular threshold gate voltage V vt , the carrier distributionwill start showing two ‘humps’ representing separation ofpeaks of carrier concentration inside channel. V vt is de-fined by the condition ∂ n ( z = t ) ∂z | V g = V vt = 0 (12) V vt for a thinner film is expected to be larger than athicker one. In other words, carriers try to stay closer tothe center for thinner film. With increase in gate volt-age, the shifting of carrier density peaks toward the gatescan be thought of as a ‘carrier pulling effect’ of the gate z (nm) E l e c t r on c on c . ( pe r cc ) TotalX1 G G G G Figure 14: Total electron density distribution along thethickness and contribution from different subbands for a33 atomic layer thick Si film with V g = 1 . φ ( z ) which is smaller at points closer to the centerof the film. From eqn. (11), one notices that the overallcarrier density profile arises from the individual contri-bution of the two terms n ( z ) and e qφ ( z ) kBT . n ( z ) peaksat the center of the film, and reduces toward the surfacewhereas e qφ ( z ) kBT follows the opposite trend. Thus, at suffi-ciently large voltage, it is possible that the peak of carrierdensity occurs at the surface, which is qualitatively sameas the classical picture, where the exponential term dom-inates so much that the effect of ‘quantum mechanical’distribution of intrinsic carrier density gets nullified. Asevident from Fig. 17, the potential profile for thinnerfilms is more uniform, and actually ‘pinned’ at a highervalue compared to the films of larger thickness leadingto higher peak carrier concentration in thinner films asshown in Fig. 13 and 14.The above explanation becomes even more clarifiedfrom Fig. 18. Initially, for small gate voltage, both thesurface potential φ s and film center potential φ increasesimultaneously at the same rate, and thus there will beonly a single channel whose peak is at the center of thefilm. Beyond a certain gate voltage, φ s and φ bifur-cate, and φ saturates very quickly. However, φ s keepsincreasing, though at a much slower rate than earlier,causing higher carrier concentration at points closer tothe surface, finally destroying the single peaked channeland creating two channels of ‘double hump’ shape. Notethat, for 9 atomic layer thick film, the ‘pinning’ voltages8 N o r m a li z ed c a rr i e r den s i t y Figure 15: Normalized total electron density distributionalong the film thickness for a 9 atomic layer thick Si filmwith V g = 0 . .
0V and 1 . N o r m a li z ed c a rr i e r den s i t y Figure 16: Normalized total electron density distributionalong the film thickness for a 33 atomic layer thick Si filmwith V g = 0 . .
0V and 1 . C hanne l P o t en t i a l ( V ) V g =1.5V V g =1.0VV g =0.5V Figure 17: Potential distribution φ ( z ) along the thicknessof thin Si film with three different gate voltages: 0 . .
0V and 1 . g (V) f , f s ( V ) Figure 18: Variation of surface potential φ s (solid lines)and film center potential φ (dotted lines) with appliedgate voltage V g , for two different film thicknesses. The ‘carrier pulling effect’ explained in the previous sec-tion can have immense impact on the spatial distributionof total channel charge and hence device performance.9onsider a DGFET where the source end is groundedand the drain is connected to V DD which is same as ap-plied gate voltage V g . At any point x along the channel(with x = 0 being taken as the source end), suppose thequasi-Fermi level is V qf ( x ). V qf ( x ) can be assumed asindependent of z . Then, the Poisson equation in eqn. (6)gets modified as [6] ∂ φ ( x, z ) ∂z = qN(cid:15) (cid:15) rs t (cid:88) j (cid:88) ¯ k f ( E ¯ kj ) | ψ ¯ kj ( φ, z ) | × e q ( φ ( x,z ) − Vqf ( x )) kBT (13)where all references have been made from groundedsource chemical potential. The carrier density n ( x, z )should now be a function of V qf ( x ) as well, which in turndepends on the drain voltage. Deriving the exact draincurrent for a nano-MOSFET needs proper attention ontransport model and the details will be communicatedseparately. However, to get a quantitative estimate, along channel device ( L = 1 µm , W = 1 µm ) with con-stant mobility has been assumed. The drain current and V qf ( x ) have been calculated using a similar procedure asin [6], by self-consistently solving eqn. (13) with draincurrent continuity equation. The extracted normalizedcarrier concentration n (cid:48) ( x, z ) is plotted over the wholechannel region in Fig. 19 for different cases. At any x = x , n (cid:48) ( x , z ) has been defined as n (cid:48) ( x , z ) = n ( x , z ) M AX z { n ( x , z ) } (14)where M AX z { . } represents the maximum value of { . } over z . The normalization is done in such a way whichclearly shows the spatial shape of carrier distribution atdifferent x . It is observed that to the source end ( x = 0),due to larger potential difference between gate and chan-nel, there clearly exist two distinct carrier density peaks.However, as one moves toward the drain end, the po-tential difference between gate and channel reduces, andthe two distinct peaks merge together producing a sin-gle center-peaked channel. Also, The magnitude of thetotal carrier concentration reduces toward the drain end.Putting in another way, near the source, carriers staycloser to the surface, and near the drain, carriers staycloser to the center of the channel. Thus, near the source,one expects more surface scattering and away from it, sur-face scattering is expected to reduce. Similar effects aretrue for gate leakage and gate capacitance, which can nolonger be assumed to be uniform along the channel. It isevident from Fig. 19 that this effect is more pronounced in thicker channel devices, and at higher operating volt-ages. If the channel is thin enough, as is the case of (b2)in Fig. 19, it is possible to have a single peaked channelall over the device. Figure 19: Normalized carrier distribution over the wholeSi channel for four different cases: (a1) 33 monolayer thickchannel, V g = 1 . V DD = 1 .
5V (a2) 33 monolayer thickchannel, V g = 1 . V DD = 1 .
0V (b1) 9 monolayer thickchannel, V g = 1 . V DD = 1 .
5V (b2) 9 monolayer thickchannel, V g = 1 . V DD = 1 . In sec. 3, it has been discussed in detail how < >M ∗ ( z ) varies along the thickness for a thin film Si. Now,when one considers the channel of a DGFET, as has beendiscussed in the previous section, the potential differencebetween gate and channel changes from source end todrain end. Thus total number of carriers at different sub-bands also changes along the channel. Keeping this inmind, one can define an ‘average effective mass’ , M ∗ e ( x ) M ∗ e ( x ) = (cid:82) t n ( x, z ) dz (cid:82) t n ( x,z ) M ∗ ( z ) dz (15)which is basically harmonic average over the carriersalong the thickness at a particular position x along chan-nel length. Physically, this indicates the ‘average’ effec-tive mass of an electron located at distance x from thesource along the channel. Strictly speaking, eqn. (15) isvalid only under the assumption that the vertical field is10airly constant and each electron suffers same scatteringrate. Although this is not a very good approximation, butit gives an idea of how the channel charge distributioncan affect spatial distribution of carrier effective mass.Fig. 20 shows for very small thickness channel (e.g. 9 M e * ( x ) / m g =0.5V V g =1.0VV g =1.5V Figure 20: Variation of ‘average effective mass’ M ∗ e ( x ) ofcarriers with channel along < > direction for differentchannel thickness values and different gate voltages V g with V DD = V g .monolayer thick), M ∗ e ( x ) hardly varies with x as well as V g . However, for higher thickness, a gradual decrease in M ∗ e ( x ) is observed from source end to drain end, and theeffect is more prominent for higher gate voltages. Qualitatively, compared to classical analysis, the chargedistribution in a quantum analysis, has two major dif-ferences: 1) The total charge in the channel reduces and2) The charge distribution peak shifts from the surfacetoward the center of the film. The extent of the shiftdepends on the applied gate voltage. Both these effectscause a change in total gate capacitance [5]. The channelcapacitance per unit volume C si ( z ) at a depth z can bedefined as the rate of change of charge per unit volumewith respect to the potential at that point. Mathemati-cally, C si ( z ) = ∂Q si ( z ) ∂φ ( z ) (16) where Q si ( z ) is given by Q si ( z ) = qNt (cid:88) j (cid:88) ¯ k f ( E ¯ kj ) | ψ ¯ kj ( φ, z ) | e qφ ( z ) kBT (17)Fig. 21 shows the variations of channel capacitance with z/tV g (V) C s i ( F / m ) z/tV g (V) C s i ( F / m ) (a) (b) Figure 21: C si as a function of V g and depth z alongthickness of channel for (a) 9 monolayer thick channeland (b) 33 monolayer thick channel. z and V g . One should note that the position z = z max inside the film, where the coupling with gate is maximum,varies with gate voltage and as gate voltage increases, z max shifts toward the surface. A detailed analysis of generic ultra-thin-body DGFEThas been performed in this work. The channel materialhas been chosen to be Si, but the analysis and methodol-ogy can be readily extended to other promising channelmaterials as well. Ultra thin film of Si has been shownto have larger and direct band gap, as opposed to bulk.It has also been shown that the intrinsic carrier concen-tration is not only less compared to bulk, but also hasa distribution over the channel thickness, peaking at thecenter. The contributions of different subbands from dif-ferent valleys to both intrinsic carrier concentration aswell as effective mass have been analyzed. The spatialdistribution of distributed < > effective mass alongthickness has been studied. It has also been shown thatalong < > direction, parabolic effective mass is fairly11alid till an electronic energy of ∼ . ‘carrier pulling effect’ of gate volt-age, channel charge distribution in a DGFET has beenpredicted. The effects of channel charge distribution oneffective mass and channel capacitance have been ana-lyzed critically. 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