Effect of polysilicon depletion charge on electron mobility in ultrathin oxide MOSFETs

Transcription

1 INSTITUTE OFPHYSICS PUBLISHING Semicond. Sci. Technol. 18 (3) SEMICONDUCTORSCIENCE AND TECHNOLOGY PII: S (03) Effect of polysilicon depletion charge on electron mobility in ultrathin oxide MOSFETs FGámiz, A Godoy, J B Roldán, J E Carceller and P Cartujo Departamento de Electrónica y Tecnología de Computadores, Universidad de Granada, Avd.Fuentenueva s/n, Granada, Spain fgamiz@ugr.es Received 23 April 3, in final form 18 June 3 Published 11 August 3 Online at stacks.iop.org/sst/18/927 Abstract We studied the effect of the depletion charge in the polysilicon gate on electron mobility in ultrathin oxide MOSFETs. An improved theory for remote-charge-scattering-limited mobility in silicon inversion layers is developed. The model takes into account the effects of image charges, screening, inversion layer quantization, the contribution of different subbands, oxide thickness, the actual distribution of charged centres inside the structure, the actual distribution of carriers in the inversion layer, the correlation of charged centres and the charged centre sign. It is shown that if the oxide is thin enough the remote Coulomb scattering due to the depletion charge in the poly-gate becomes an effective scattering mechanism, whose effect is comparable to those of the main scattering mechanisms that control the movement of the carriers in the MOSFET channel. As a consequence, this scattering mechanism must be taken into account in order to satisfactorily explain the experimental results obtained in ultrathin oxide MOSFETs. The model is implemented in a Monte Carlo simulator, where the effects of the ionized impurity charge in the substrate, the interface trapped charge and the contribution of other scattering mechanisms are taken into account simultaneously. Our results show that RCS cannot be neglected for oxide thicknesses below 2 nm, but that its effects for t ox > 5nmarenegligible. Good agreement with experimental results was obtained. 1. Introduction In order to scale CMOS (complementary metal-oxidesemiconductor) devices to smaller dimensions while maintaining good control of the short-channel effects, the gate oxide thickness should be reduced in close proportion to the channel length [1]. Thus, for devices with gate lengths below 0.1 µm, gate oxides below 2 nm could be needed [2]. However, oxide scaling results in several effects that impose serial limitations on MOS devices [1, 3], including the following [4]: (i) A substantial direct tunnelling current flow from the gate to the channel even under low voltage operating conditions. This fact leads to a gate leakage current that increases exponentially with decreasing oxide thickness [5, 6], thus becoming an important fraction of the drain current. This current component is a serious drawback to controlling the power dissipation of the device. (ii) An important long-range Coulomb interaction between the channel and the heavily doped source/drain/gate (electron plasmon interaction) [7] which strongly reduces the electron mobility for oxides thinner than 3 nm. (iii) A decrease in electron mobility due to the gate oxide roughness [8 11]. (iv) Due to the depletion of the polysilicon near the oxide interface, an important degradation of C V characteristics of the MOS structure if the polysilicon gate is not doped enough or the oxide thickness is too small. This fact leads to a reduced inversion charge density in the channel /03/ $ IOP Publishing Ltd Printed in the UK 927

2 FGámiz et al C T /C ox =1x10 19 cm -3 =1x10 20 cm -3 =2.5x10 20 cm -3 =1x10 21 cm Gate Voltage (V) Figure 1. Simulated low frequency C V curves of a p-type MOS capacitor with n + -polysilicon gate doped at different concentrations. The oxide thickness was assumed to be t ox = 1nmand the substrate doping N A-Si = cm 3. and finally to an important degradation of the MOSFET transconductance [12, 13]. (v) An important degree of remote Coulomb scattering due to the poly-gate charge, which also strongly degrades electron mobility [4, 10, 14 18]. (vi) Reliability concerns [4, 19]. The use of highly doped polysilicon gates was a great advance in CMOS technology, since it allowed the source and drain regions to be self-aligned to the gate, thus eliminating parasitic effects from overlay errors [12]. However, not all effects were advantageous since, as mentioned in points (iv) and (v) above, the use of a polysilicon gate can also originate various problems which seriously limit the operation of ultrashort devices, and therefore it is necessary to control the use of polysilicon gate. The first negative effect of poly-depletion is the arising of an additional capacitance in series with the oxide capacitance. This fact limits the total capacitance of the structure in inversion and leads to a reduced inversion charge density and therefore to a degradation of the MOSFET transconductance. Figure 1 shows simulated low frequency C V curves of a p-type MOS capacitor with n + - polysilicon gate doped at different concentrations. The oxide thickness was assumed to be t ox = 1nmandthesubstrate doping N A-Si = cm 3. Poisson and Schrödinger equations have been self-consistently solved in the whole structure, and the total charge, Q S,asafunction of the gate voltage, V G, has been evaluated. The total capacitance of the structure has been calculated according to its definition [20]: C T = d( Q S). (1) dv G When the structure is in accumulation or depletion, C V curves do not depend on the polysilicon doping (as seen in figure 1), but when the substrate becomes inverted, and therefore the poly-gate becomes depleted, the capacitance of the polysilicon in depletion adds in series with the oxide capacitance, according to 1 = (2) C T C ox C Si C p -2 Inversion Charge (cm ) =10 21 cm =10 19 cm Gate Voltage (V) 12-2 Inversion charge (x10 cm ) Figure 2. Electron concentration in the inverted silicon substrate versus the gate voltage for two different polysilicon doping concentrations. The oxide thickness was assumed to be t ox = 1nm and the substrate doping N A-Si = cm 3. where C T is the total capacitance of the MOS structure, C ox is the oxide capacitance, C Si,thesubstrate capacitance, and C p the capacitance of the polysilicon depletion region. C p is a function of the total charge in the polysilicon gate, Q p, and of the polysilicon doping concentration,. The higher Q p,thelower the C p,and according to expression (2), the lower the C T. For the lower values of the polysilicon concentration, C p is lower than C ox,andasaconsequence, the total capacitance of the structure in inversion is smaller than the oxide capacitance, as seen in figure 1 for the curve corresponding to = cm 3. This lower C T value leads to a reduced inversion charge concentration for agiven gate voltage as shown in figure 2, whichshows the electron concentration in the inverted substrate versus the gate voltage for two different polysilicon doping concentrations. Here again, the oxide thickness was assumed to be t ox = 1nm, and the substrate doping N A-Si = cm 3.The reduction of the capacitance of the structure in inversion also leads to a degradation of the transconductance and to a decrease of the drain current. (Note that for the lowest poly-doping concentrations an abrupt rise in the capacitance curve can be observed, corresponding to the onset of inversion at the polysilicon oxide interface [12]). According to figure 1, andfollowing expression (2), in order to make poly-depletion effects negligible, C p must be sufficiently greater than C ox. This can be achieved by increasing the polysilicon doping concentration for a given oxide thickness, t ox (which makes C p increase) or by increasing t ox for a given value, which makes C ox decrease. In conclusion, to avoid the capacitive effects of poly-depletion as the oxide thickness is reduced, the use of high polysilicon doping concentrations is required [13]. On the other hand, the combination of a very small oxide thickness and a high polysilicon doping concentration could result, at least apriori, inanimportant degree of remote Coulomb scattering (RCS) of channel electrons by the poly-depletion charge and a substantial degree of mobility degradation [4, 10, 14 18, 21]. As the whole structure must be electrically neutral, for a given inversion charge concentration, Q inv,the same amount of 928

3 Effect of polysilicon depletion charge on electron mobility in ultrathin oxide MOSFETs Matrix Element (x10-28 V 2 cm 4 ) z' = -0.5 nm z' = -4.5 nm z' = -9.5 nm N inv =1.1x10 12 cm -2 =1 nm Q (cm -1 ) Figure 3. Matrix element for the Coulomb scattering of the electrons in the ground subband of a silicon inversion layer due to a point charge located in the polysilicon gate at 0.5 nm from the poly/oxide interface (solid line) at 4.5 nm from the poly/oxide interface (dashed line) and at 9.5 nm from the poly/oxide interface (dotted line); t ox = 1nm,N A-Si = cm 3. charge with the opposite sign must be located in the polysilicon gate, Q p,regardless of the polysilicon doping concentration. However, although the total amount of charge in the gate is the same, the spatial distribution of this charge strongly depends on the value. The higher the poly-doping concentration the higher the charge concentration near the interface, and although the total Q p is the same for the two values considered, as the Coulomb interaction is inversely proportional to the distance, a higher Coulomb scattering rate, and consequently a mobility decrease, is expected as increases. Figure 3 shows the matrix element for the Coulomb scattering of the electrons in the ground subband of a silicon inversion layer due to a point charge located in the polysilicon gate at 0.5 nm from the poly/oxide interface (solid line) at 4.5 nm from the poly/oxide interface (dashed line) and at 9.5 nm from the poly/oxide interface (dotted line). It can be seen that the farther the charge from the inversion layer, the lower the matrix element as a consequence of a lower Coulomb interaction. On the other hand, one might wonder whether this RCS effect could be partially masked by the Coulomb scattering rate due to the substrate impurity charge, which in such short-channel devices is usually very high. In any case, the actual effect of RCS is not clear theoretically or experimentally, and more research work is needed [18]. In fact, there exists in the literature some controversy about the actual effect of RCS. Krishnan et al concluded, first theoretically [15] and then experimentally [16], that RCS strongly affects electron mobility, which decreases to half the level of the universal mobility curve [22]. On the other hand, Yang et al [17] claimed that RCS has little effect on electron mobility, which has almost the same value regardless of the oxide thickness and the polysilicon doping concentration. The experimental results of Takagi et al [18] (obtained with an improved procedure which takes into account the gate current through very thin oxides) and the calculations of Saito et al [14], are to be found between these two extremes. The latter results show that the effect of RCS on electron mobility, although of some significance, is not so important as Krishnan s results suggested [16]. To shed some light on this topic, different authors have tried to theoretically quantify the RCS effect on electron mobility [14, 16, 17, 21]. Saito et al [14] very recently developed an improved theory for evaluating the RCS rate, taking into account (i) the image charge effect, (ii) the inversion charge distribution due to quantization, (iii) screening, (iv) finite oxide thickness and (v) quantum fluctuation. As pointed out in [14], the calculations of RCS mobility by Yang et al [17] and by Krishnan et al [15] ignore effects which have been proved to be very important. Although the calculations of Saito et al compare much better with the recent experimental data of Takagi et al [18], there is still an important discrepancy between theory and experiment. Saito et al s theory only takes into account one subband in the calculation of electron mobility. Coulomb scattering is known to be more important at low inversion charge concentrations, and therefore at low transverse electric fields, where the electron population of the upper subbands is considerable, especially at room temperature [23]. The contribution of excited subbands in the calculation of RCS-limited mobility could be important for following two reasons: (i) the RCS rate for electrons in the upper subbands is very different from its value for the ground subband due to the different degree of screening and (ii) in a Si () inversion layer, electrons in primed subbands have a higher conduction effective mass than that of electrons in non-primed subbands (such as the ground subband), and therefore a lower mobility [23]. In conclusion, the contribution of upper subbands to total mobility could be significant, and could considerably modify the theoretical results, bringing them closer to the experimental ones. In this paper, we show the importance of the RCS effect on electron mobility. In particular, we find that, depending on the oxide layer thickness and the poly impurity concentration, this scattering mechanism could become as important as the main scattering mechanisms that control the transport properties of carriers in the MOSFET channel. Section 2 describes an RCS model in which the contribution of different subbands is taken into account in order to evaluate electron mobility by using a one-electron Monte Carlo simulator. In addition, the model includes the effects of image charges, screening, inversion layer quantization, the contribution of different subbands, oxide thickness, the actual distribution of charged centres inside the structure (poly charges, interface and oxide trapped charges and silicon bulk impurities), the actual distribution of carriers in the inversion layer and the correlation of charged centres and the charged centre sign [24, 25]. A Monte Carlo method is used to solve the Boltzmann transport equation (BTE) taking into account the effect of RCS mechanism. The contribution of other scattering mechanisms (phonon scattering, surface roughness scattering and Coulomb scattering due to ionized bulk doping impurities and interface charges) is simultaneously taken into account. Prior to this, the one-dimensional Schrödinger and Poisson equations are selfconsistently solved in the whole structure, and the charge in the polysilicon carefully evaluated; thus we take into account the actual distribution of the charge in the polysilicon gate, instead of using the depletion approximation to evaluate the RCS rate. Section 3 discusses the effect of the poly-depletion doping concentration and the role of the oxide thickness. The effect 929

4 FGámiz et al -L 1 0 L 2 L 3 ε poly ε ox ε Si Poly-silicon (Gate) Oxide Silicon t poly Figure 4. Polysilicon Oxide Silicon structure considered in this work to develop a remote charge scattering model. of the substrate doping is also analysed. A comparison with experimental results is provided. Finally, the main conclusions of our work are drawn in section Remote Coulomb scattering model In previous studies of Coulomb scattering of electrons in silicon inversion layers, we have shown that charge centres located in the oxide farther than 10 nm from the interface hardly scatter the channel electrons, that is to say, charges located at such distances do not modify electron mobility curves: the mobility curves obtained taking into account the contribution of these charges or ignoring their effect almost coincide [24]. We have also seen that this threshold distance is reduced to 5 nm or lower if the doping concentration of the substrate increases to N A = cm 3,sincein this case, the stronger contribution of the Coulomb scattering of the substrate doping camouflages the effect of these remote charges. On the other hand, we have also shown that charges located at distances smaller than these values strongly affect mobility curves, and therefore it is necessary to take their effects into account even when highly doped substrates are used. This is the case of the polysilicon depletion charge in ultrathin oxide MOSFETs. However, if we want to accurately study the effect on the electron mobility of the charged centres located in the polysilicon depletion region, we have to improve the previous Coulomb scattering model [24, 25]. This is because oxide and polysilicon have different dielectric constants, and so an image effect is induced in the structure. In the development of the previous model, only two materials with different dielectric constants (oxide and silicon substrate) were assumed. The assumption then was that the oxide thickness was infinite, and consequently that the charges located in the polysilicon depletion region were too far from the channel to significantly affect electron mobility. The situation now is more complicated since we have to consider the finite thickness of the oxide. Therefore, we now have three materials with different dielectric constants. As a consequence, we have to take into account the image effect at the interface between the polysilicon and the oxide. Figure 4 shows the semiconductor structure considered in our study. In the conventional Coulomb scattering model (infinite oxide), charged centres were considered to be only in the oxide t Si t ox or in the silicon layers, and therefore only these two layers were taken into account in the development of the scattering model. In this work, we have improved the previous model by taking into account that the charge centres responsible for Coulomb scattering could be located in the polysilicon gate with an arbitrary distribution. The improved model considers (a) the distribution of electrons in the inversion layers, (b) the geometrical distribution of external charged centres in the silicon bulk, oxide and polysilicon gate, (c) the screening of charged centres by mobile carriers, (d) the charged centre correlation and (e) image charges [24, 26, 27]. Let ρ ext be the external charge density responsible for the Coulomb scattering. As a consequence of this charge density, the electrostatic potential responsible for the confinement of the carriers is spatially modified, and its perturbation, V, obeys the Poisson equation [23], [ɛ(z) V( r,z)] = 2ɛ Si S i g i (z) V( r,z 2 )g i (z 2 ) dz 2 ρ ext (z) (3) i where r is the coordinate parallel to the interface and z is the coordinate perpendicular to it. ɛ(z) is the overall position-dependent permittivity and ɛ sc the permittivity of the semiconductor. The first term on the right-hand side of equation (3) is the induced charge responsible for the screening. g i (z) is the square of the electron envelope-function in the ith subband, ξ i (z),ands i the screening parameter given by [26] S i = e2 N i (4) 2 E d,i where e is the electron charge, N i the population of the subband whose minimum is E i and E d,i = K B T [ 1+e (Ei EF )/KBT ] ln [ 1+e (EF Ei)/KBT ] (5) where E F is the Fermi level, K B the Boltzmann constant and T the temperature. Multiplying equation (3) bye i Q r and integrating over r produces the following equation for the Fourier transform of the electrostatic potential perturbations, V( Q,z 1 ): ( ɛ(z 1 ) ɛ(z 1 )Q 2 z 1 z 1 2ɛ Si S i g i (z 1 ) i ) V( Q,z 1 ) V( Q,z 2 )g i (z 2 ) dz 2 = ρ ext ( Q,z 1 ). (6) By integrating equation (6) with the boundary conditions V( Q, ) = 0 V( Q,L 3 ) = 0 d dz V( Q, ) = 0 d dz V( Q,L 3 ) = 0 d ɛ poly dz V( Q, 0 d ) ɛ ox dz V( Q, 0 + ) = σ ss1 ( Q) d ɛ ox dz V( Q,L 2 ) ɛ d Si dz V ( Q,L + ) 2 = σss2 ( Q) (7) where σ ssi ( Q)(i = 1, 2) is the Fourier transform of the charge density at the polysilicon oxide interface and oxide silicon 930

5 Effect of polysilicon depletion charge on electron mobility in ultrathin oxide MOSFETs interface, respectively; we thus reach the following solution for the Fourier transform of the potential fluctuations, V( Q,z)= 2ɛ Si S i dz 2 V( Q,z 2 )g i (z 2 ) + i dz 1 G Q (z, z 1 )g i (z 1 ) dz 1 G Q (z, z 1 )ρ F ( Q,z 1 ) ρ F being the total external charge responsible for Coulomb scattering, ρ F ( Q,z)= ρ ext ( Q,z)+ σ ss1 ( Q)δ(z) + σ ss2 ( Q)δ(z L 2 ) (9) with G Q (z, z 1 ) the Green functions given by [ ] 1 A G Q (z, z 1 ) = 2ɛ poly Q e Q z z1 + 2Q 1 e Q( z + z1 ) 2ɛ poly Q + A ɛ ox ɛ Si e Q( L2 z1 +L2+ z ) 2Q ɛ ox + ɛ Si for L 1 <z<0 (10) G Q (z, z 1 ) = 1 2ɛ ox Q e Q z z1 + A 4Q + B ( 1 ɛ ) Si e Q( L2 z + L2 z1 ) 4Q ɛ ox + C 4Q e Q[(L2 z)+l2+ z1 ] + C ( 1 ɛ poly ɛ ox 4Q e Q(z+L2+ L2 z1 ) (8) ) e Q( z + z1 ) for 0 <z<l 2 (11) G Q (z, z 1 ) = 1 2ɛ Si Q e Q z z1 [ B + 2Q 1 ] e Q( z L2 + z1 L2 ) 2ɛ Si Q + B 2Q ɛ ox ɛ poly ɛ ox + ɛ Si e Q(z+ z1 ) for L 2 <z<l 3 (12) where the coefficients A, B and C are given by A = B = C = 2(ɛ ox + ɛ Si ) (ɛ ox + ɛ poly )(ɛ ox + ɛ Si ) e 2QL2 (ɛ ox ɛ poly )(ɛ ox ɛ Si ) (13) 2(ɛ ox + ɛ poly ) (ɛ ox + ɛ poly )(ɛ ox + ɛ Si ) e 2QL2 (ɛ ox ɛ poly )(ɛ ox ɛ Si ) (14) 2(ɛ ox ɛ poly )(ɛ ox ɛ Si ) ɛ ox [(ɛ ox + ɛ poly )(ɛ ox + ɛ Si ) e 2QL 2(ɛ ox ɛ poly )(ɛ ox ɛ Si )]. (15) Note that in the case ɛ ox = ɛ poly, and setting the origin at the oxide silicon interface, the Green functions (equations (10 (12)) are reduced to expression (11) of [24], corresponding to the previous scattering model with very thick oxide layers (t ox ). To solve equation (8)wemaketheansatz equal to V( Q,z)= dz ρ F ( Q,z ) e φ( Q,z,z ) (16) and proceed as shown in [24], i.e., V( Q,z) is the solution of equation (8) ifφ( Q,z,z ) is the solution to the following equation: φ( Q,z,z ) = 2ɛ Si S i dz 1 G Q (z, z 1 )g i (z 1 ) i dz 2 φ( Q,z 2,z )g i (z 2 ) + eg Q (z, z ). (17) This is a particular case of equation (8), corresponding to the case ρ F ( Q,z) = eδ(z z ), i.e., a pointed charge located at z. Although expression (17) is significantly simpler than expression (8) (the former does not depend on the external charge density, which greatly simplifies the procedure to achieve the solution) it is still an implicit equation, i.e., the required solution is found in both members of the equation. Therefore we need an iterative procedure to solve equation (17). We developed such a procedure, and numerically solved equation (17). Once the Fourier transform of the potential perturbations due to a pointed charge located at ( 0,z ) has been evaluated for every z,wecan obtain the Fourier transform of the potential perturbations due to the external charge density responsible for Coulomb scattering by evaluating expression (16). The perturbation Hamiltonian is then given by H ( Q,z)= ev ( Q,z)= dz ρ F ( Q,z )φ( Q,z,z ). (18) For simplicity, we have assumed that the external charge distributions responsible for Coulomb scattering are conceptually divided into two-dimensional sublayers parallel to the oxide silicon interface. Let z t be the thickness of the tth sublayer, z t the centre of the sublayer, and σ t ( r,z t ) the charge density per unit area in the tth sublayer and σ t ( Q, z t ) its Fourier transform. Instead of (18)wenowhave poly,ox,si,interface H ( Q,z)= σ t ( Q,z t )φ( Q,z,z t ). (19) t Taking the Fermi golden rule as a starting point, and following a procedure identical to that discussed in [24] we obtained the Coulomb scattering rate for an electron transition from subband i to subband j (expression (33) and following [24]). This expression simultaneously takes into account (i) the screening of charged centres by mobile carriers, (ii) the distribution of charged centres inside the structure (in the polysilicon gate, the oxide and in the silicon substrate, (iii) the actual electron distribution, (iv) the charged centre correlation and (v) the effect of image charges. Expression (33) of [24] is reproduced here for convenience, Ɣ ij ( k) = S [ q 2 2π h t ñ t ( Q, z t )ñ t ( Q, z t ) avg t k (ij) M t ( Q,z t ) 2 ] δ(e E ) d k + S [ qt q u ñ t ( Q, z t )ñ u 2π h ( Q, z u ) avg t u k M (ij) tu ( Q,z t,z u ) 2 ] δ(e E ) d k (20) 931

6 FGámiz et al where q t is the charge of a charged centre in the tth sublayer. To evaluate expression (20) with the extended model developed in this work, the matrix elements, defined as (ij) M t ( Q,z t ) 2 = φ (ij) t ( Q,z t ) [ φ (ij) t ( Q,z t ) ] (21) (ij) M tu ( Q,z t,z u ) 2 = φ (ij) t ( Q,z t ) [ φ (ij) u ( Q,z u ) ] (22) with φ (ij) t ( Q,z t ) = dzξ j (z)φ( Q,z,z t )ξ i (z) (23) have to be evaluated with φ( Q,z,z ) calculated according to expression (17), using the Green functions defined in expressions (10) (12). ñ t ( Q, z t )ñ t ( Q, z t ) avg is the density correlation function of charged centres in the same sublayer, and ñ t ( Q, z t )ñ u ( Q, z u ) avg,t u is the density correlation function of charged centres in different sublayers, defined according to expression (32) of [24]. The remaining terms are defined in [24]. The density correlation function of the charged centres in the same sublayer has been calculated in [26] to be ñ t ( Q, z t )ñ t ( Q, z t ) avg = N ( t 1 2C ) tj 1 (QR 0 ) (24) S QR 0 where N t is the charge density per unit area in the tth sublayer, J 1 the first-order Bessel function and C t = πr 0 N t aparameter which is a measure of the degree of space correlation. C t represents the ratio of the minimum area πr0 2 to the average area Nt 1 occupied by a charge particle. To obtain expression (24) the hard-sphere model [26] has been adopted, according to which the distribution of charged centres is assumed to be random to the extent that no two centres can be found within aradiusofr 0 from each other. Also using the hard-sphere model we have calculated in [28, 29] the density correlation function of charged centres in different sublayers separated by adistance d = z t z u to be d<r 0 ñ t ( Q, z t )ñ u ( Q, z u ) avg = 2C N eff S J 1 (QR 0 ) QR 0 d>r 0 ñ t ( Q, z t )ñ u ( Q, z u ) avg = 0 (25) where N eff = N t N u,r 0 = R0 2 d2 and S is the area of the interface. C is a constant which determines the interaction between the two distributions and is given by C = πr 0 N eff. J 1 is once again the first-order Bessel function. Note that if the two sublayers are separated by a distance greater than R 0 there is no mutual influence between the charges of the two sublayers, and therefore the second term on the right-hand side of expression (20) vanishes. Note also that the scattering rate depends on the sign of the charged centres from the product q t q u. In fact, this second addend represents the correction to the scattering rate due to the mutual interaction among the sublayers into which we have divided the charge distributions responsible for Coulomb scattering. The sign of the correction depends on the relative sign of charges of each sublayer, i.e., if charges in both sublayers have the same sign, the correction is negative and the scattering rate is lower; however, if the charges have different signs, the correction is positive and the scattering rate is higher. This fact is quite important in the system we are considering since, in inversion, the charge in the poly is the opposite sign from that of the charge in the silicon substrate. Until now R 0 wasconsidered as a fitting parameter of the model, i.e., its value is fitted so that the simulation results will coincide with the experimental ones. Following the work of Ridley [30] concerning the average separation of impurities in a three-dimensional distribution, we have obtained an expression for R 0 taking into account that in our model the charge distributions are two-dimensional: ( ) 1 3 R 0 = 2 Ɣ. (26) πn t 2 This expression provides very similar values to those empirically obtained by fitting the simulation results to the experimental ones [28]. Assuming that Coulomb scattering mainly assists intrasubband transitions [7, 24], we finally have ( Nt m [ q 2 t Ɣ i ( k) = π h 3 t M (i) t ( Q,z t ) ] ) 2 dθ ( Neff m π h 3 t u M (i) tu ( Q,z t,z u ) 2 ] dθ ( 1 2C tj 1 (QR 0 ) QR 0 [ q t q u 2C tu J 1 (QR 0 ) ) QR 0 ) (27) The summa is extended to all the charges in the structure, i.e, charges in the poly-gate, in the oxide and in the silicon substrate, that is to say, the model allows us to simultaneously take into account all the charge centres in a MOS structure. The actual distribution of the charge distribution is also considered. Using this scattering model in a Monte Carlo simulator, we made an extensive study of the effect of the polysilicon depletion charge on electron mobility. The simultaneous presence of other Coulomb scattering charges (i.e. silicon doping impurities and interface charges) was also analysed. In addition to Coulomb scattering, phonon scattering and surface roughness scattering were also considered. A detailed description of the Monte Carlo simulator can be found elsewhere [24 29]. 3. Simulation results and discussions 3.1. Coulomb scattering rate We employed the above model to study the effect of the polydepletion charge and oxide thickness on electron mobility. The depletion charge distribution in the poly-gate was calculated by solving the Poisson equation. Thus, the actual distribution of the charge in the poly is taken into account, for each value of the gate voltage and each value of. The particular value of the poly-depletion layer thickness depends on the poly-doping concentration and on the voltage applied to the gate. We observed that for the same value of inversion charge concentration (i.e. the same value of charge in the polydepletion layer) the lower, thewider the thickness of the depletion layer, and the lower the effect of RCS, i.e., the higher the electron mobility. 932

7 Effect of polysilicon depletion charge on electron mobility in ultrathin oxide MOSFETs RCS Coulomb scattering rate (s -1 ) (a) No RCS With RCS Coulomb scattering rate (s -1 ) (a) No RCS (b) Kinetic Energy (ev) Figure 5. Coulomb scattering rate (expression (27)) for the electrons in the ground subband of a silicon inversion layer taking into account the poly-depletion charge (dashed line) and ignoring the effects of poly-depletion charge (solid line). The contribution of the substrate doping charge (N A = cm 3 ) and the interface trapped charge (N it = cm 2 ) was considered when the Coulomb scattering rate was evaluated. The poly-doping concentration was assumed to be = cm 3 ; (a) t ox = 1nm,(b) t ox = 10 nm. Unless otherwise stated, we considered a structure such as that shown in figure 4 with the following parameters: substrate doping concentration N A = cm 3, poly impurity concentration, = cm 3,andaninterface trap concentration of N it = cm 2 at the oxide/silicon and poly/oxide interfaces. Different values of the oxide thickness ranging from t ox = 1nmtot ox = 10 nm are assumed. Figure 5 shows the Coulomb scattering rate (expression (27)) for the electrons in the ground subband taking into account the polydepletion charge (dashed line) and ignoring the effects of the poly-depletion charge (solid line). The contributions of the substrate doping charge (N A = cm 3 ) and of the interface trapped charge (N it = cm 2 ) are considered in evaluating the Coulomb scattering rate. In both cases, the effects of the oxide thickness and of the different dielectric constant between the oxide and the polysilicon are considered, i.e., we used the Green functions (10) (12) to evaluate the scattering rate, even in the case when the RCS contribution was ignored. Two values of the oxide thickness were considered, t ox = 1nmandt ox = 10 nm. Note that, as expected, the contribution of the poly charge Coulomb scattering is more important as the oxide thickness decreases. Figure 5(a) indicates that, for t ox = 1nm,theRCS effect is more important than the Coulomb scattering rate due to substrate doping, even when, as in this case, N A is very large. Figure 6 compares the Coulomb scattering rate for the two values of the oxide thickness considered taking into account the RCS effect (figure 6(a)) and ignoring its effect (figure 6(b)). Figure 6(a) showsthatthe RCS effect is quite important as the silicon thickness is reduced, and thus the Coulomb scattering rate is higher for t ox = 1nmthanfor t ox = 10 nm. However, if we ignore the effect of RCS, (b) Kinetic Energy (ev) Figure 6. Coulomb scattering rate (expression (27)) for the electrons in the ground subband of a silicon inversion layer for two values of the oxide thickness t ox = 1nm(solid line) and t ox = 10 nm (dashed line). The contribution of the substrate doping charge (N A = cm 3 ) and the interface trapped charge (N it = cm 2 ) has been considered to evaluate the Coulomb scattering rate. The poly-doping concentration was assumed to be = cm 3. (a) Theeffect of RCS is considered. (b) The effect of RCS is ignored. and although the same substrate doping concentration and the same interface trap density are used, the Coulomb scattering rate curves for the two oxide thicknesses do not coincide; surprisingly, the rate curve corresponding to the thinnest oxide is lower than that corresponding to the thickest one. This fact is due to the effect of the finite size of the oxide (which is accounted for in the model developed in this work) and the different dielectric constant between the polysilicon and the oxide, which is responsible for an image effect at the interface between the polysilicon and the oxide (as detailed in equations (10) (12). This fact is responsible for the slight dependence of electron mobility on the oxide thickness even when the RCS effects are ignored Electron mobility Oxide thickness. Using a one-electron Monte Carlo simulator, we studied the effect of RCS on electron mobility, for different values of the oxide thickness and different values of the poly-doping concentration. The influence of other scattering mechanisms (phonon and surface roughness) and substrate doping impurities and interface trapped charges is also analysed. The first factor to consider is whether, when the RCS contribution is ignored, mobility curves depend on the oxide thickness. For this purpose, figure 7 shows electron mobility curves versus the transverse effective field (defined as in [31]) for different values of the oxide thickness. Only phonon scattering and surface roughness scattering are taken into account in the curves with open symbols. Surface roughness parameters were assumed to be L sr = 1.5 nm and sr = nm [23, 32]. Coulomb scattering due only to substrate 933

8 FGámiz et al Electron Mobility (cm 2 /Vs) =1x10 20 cm -3 N A =5x10 17 cm -3 =2nm =5nm No Remote Coulomb Scattering Figure 7. Electron mobility curves versus the transverse effective field for different values of oxide thickness, ignoring the effects of RCS. Open symbols: only phonon scattering and surface roughness scattering are considered. Full symbols: Coulomb scattering due to an interface trap concentration of N it = cm 2 at the oxide/silicon interface and silicon bulk impurities (N A = cm 3 ) are also considered. ionized impurities and to the interface traps (considered to be present at a concentration of N it = cm 2 )has been added in the curves with full symbols. In this figure, we see that when Coulomb scattering is fully ignored (i.e. only phonon scattering and surface roughness scattering are taken into account) no dependence of the mobility on the oxide thickness is observed. However, and as previously discussed, when the effect of the Coulomb scattering due to substrate doping concentration is considered (N A = cm 3 ), avery slight increase in electron mobility at low transverse electric fields is observed. Figure 8 shows the same mobility curves as in figure 7,but taking into account the RCS effect. For the sake of comparison, a universal mobility curve (only phonon and surface roughness scattering) is added in solid line (no symbols). The first fact to note is that, as expected, the mobility curve for t ox = 10 nm (downward-pointing triangles) coincides with mobility curves when no RCS effect is taken into account (full-symbol curves in figure 7). However, as the oxide thickness is reduced the effect of RCS becomes more and more important (reaching 30% for t ox = 1nm),mainly, as expected, at low transverse effective fields even for high concentrations of silicon bulk impurities. Mobility curves taking into account only the contribution of the poly-depletion charge to the Coulomb scattering rate are shown in figure 9 as a function of the oxide thickness. For the sake of comparison, a universal mobility curve (only phonon and surface roughness scattering) is added in solid line (no symbols). As can be observed, the effect of the RCS for oxide thicknesses greater than 10 nm is very weak (mobility curves almost coincide with the universal mobility curve in the whole electric field range. To show this more clearly, figure 10 compares the Coulomb contribution to the total mobility for two values of oxide thickness: (a) t ox = 1nm and (b) t ox = 10 nm. In open squares, only phonon scattering and surface roughness scattering are taken into account. When only the contribution of the substrate doping impurities to Electron Mobility (cm 2 /Vs) With RCS =2nm =5nm =1x10 20 cm -3 N A =5x10 17 cm -3 Figure 8. Electron mobility curves versus the transverse effective field for different values of oxide thickness taking into account the effects of RCS ( = cm 2 ).Phonon, surface roughness and Coulomb scattering are taken into account. An interface trap concentration of N it = cm 2 at the oxide/silicon interface and silicon bulk impurities (N A = cm 3 ) are also considered. For the sake of comparison, a universal mobility curve (only phonon and surface roughness scattering) is added in solid line (no symbols). Electron Mobility (cm 2 /Vs) =1x10 20 cm -3 =2nm =5nm N A =5x10 17 cm -3 Only Remote Coulomb Scattering Figure 9. Electron mobility curves versus the transverse effective field for different values of oxide thickness. Only the effects of RCS ( = cm 2 ) are considered as Coulomb scattering source (i.e. substrate doping impurities and interface trapped charges are not taken into account in the evaluation of the Coulomb scattering rate). Phonon, surface roughness and Coulomb scattering are taken into account; (N A = cm 3 ).Forthesake of comparison, a universal mobility curve (only phonon and surface roughness scattering) is added in solid line (no symbols). the Coulomb scattering is added to the previous curve, the mobility curve with open circles is obtained (ph+sr+dcs). If the contribution of the poly-depletion charge is added instead of the substrate doping impurities, mobility curves with open down triangles are obtained (ph+sr+rcs). Finally, if both Coulomb scattering contributions are simultaneously taken into account, the mobility curve with open up triangles 934

9 Effect of polysilicon depletion charge on electron mobility in ultrathin oxide MOSFETs Electron Mobility (cm 2 /Vs) (a) =10 nm RCS Mobility (x10 3 cm 2 /Vs) =2nm =5nm =1x10 20 cm -3 ph. + sr. ph. + sr. +dcs ph. + sr. +rcs ph. + sr. +dcs+rcs (b) Figure 10. Electron mobility versus the transverse effective field in asilicon inversion layer for two values of the oxide thickness: (a) t ox = 1nmand (b) t ox = 10 nm. ( ) only phonon and surface roughness scattering are taken into account. ( ) Phonon, surface roughness scattering and Coulomb scattering. Only the contribution of the substrate doping impurities and the interface trapped charges to the Coulomb scattering is considered. ( ) Phonon, surface roughness scattering and Coulomb scattering. Only the contribution of the poly-depletion charge to the Coulomb scattering is considered. ( ) Phonon, surface roughness scattering and total Coulomb scattering. is obtained (ph+sr+dcs+rcs). From the comparison of figures 10(a)and(b) thefollowing is deduced: For t ox = 1nmthemainCoulomb scattering contribution is due to the poly-depletion charge, which becomes the main scattering event at low transverse effective field. However, for t ox = 10 nm, the RCS effect is much less important than the scattering due to the substrate impurity charges (dcs) which is the main mechanism for this oxide thickness, and for greater ones. In this case, the effect of RCS is almost imperceptible. Finally, we isolated the contribution of RCS mobility by applying Matthiessen s rule to the mobility curves shown in figure 9. Figure 11 shows RCS-limited mobility versus the transverse effective field for different values of the oxide thickness. RCS-limited mobility strongly depends on the oxide thickness. 0 Figure 11. RCS-limited mobility obtained from the mobility curves in figure 9, applying Matthiessen s rule, for different values of the oxide thickness. Electron Mobility (cm 2 /Vs) (a) =10 nm =10 19 cm -3 =10 20 cm -3 =10 21 cm -3 =1 nm (b) Figure 12. Mobility curves versus the transverse effective field for different values of the poly-doping concentration. Two values of the oxide thickness were assumed (a) t ox = 1nmand (b) t ox = 10 nm. Phonon, surface roughness and total Coulomb scattering are assumed Polysilicon doping concentration. We also studied the effect of the concentration of impurities in the polysilicon. Figure 12 shows mobility curves versus the transverse effective field for different values of the poly-doping concentration full squares: = cm 3, closed circles: = cm 3 and closed triangles: = cm 3. Two values of the oxide thickness were assumed (a) t ox = 1nmand(b) t ox = 10 nm. As seen, for the thicker oxide value no influence of the polysilicon doping concentration is observed, even for the highest poly-doping concentration sample. However, in the case of the thinnest oxide (t ox = 1nm),animportant effect on the polydoping concentration is observed: the higher the lower 935

10 FGámiz et al RCS mobility (cm 2 /Vs) 10 6 E EFF =6x105 V/cm Yang et al. (00) This work N inv =2.8x10 12 cm -2 =5x10 19 cm -3 Saito et al. (02) Takagi et al. (02) Krishnan et al. (98) Oxide Thickness (nm) Figure 13. Dependence of RCS-limited mobility on oxide thickness. = cm 2. the electron mobility. Figure 12(a) shows a mobility curve ignoring the effects of RCS (open symbols). The comparison shows that even for the less doped polysilicon sample, the RCS effect is important when very thin oxides are used, and in consequence its effect should be taken into account Comparison with experimental results. We compared our results with those available in the literature. Figure 13 shows the dependence of RCS-limited mobility on gate oxide thickness. RCS-limited mobility was obtained by calculating mobility curves but ignoring the effects of RCS (the polysilicon doping concentration is assumed to be = cm 3 in this figure), and taking into account the RCS. The model developed here provides a better agreement with experimental results than do previous models. Finally, it should be noted that the present model ignores the screening effect from the poly-gate [7], as a consequence of the electrons remaining in the gate due to the incomplete ionization of poly impurities. However, this effect would be more important for higher concentrations than those considered here. If the poly-gate is changed by a metal gate several advantages could be achieved. Gate capacitance degradation due to the depletion of the doped polysilicon will disappear. A further potential benefit of metal gate electrodes is the elimination of carrier mobility degradation due to plasmon scattering from gate electrode. The plasmon frequency of a highly conductive metal electrode is too high to interact with the carriers in the inversion layer [33, 34]. 4. Conclusions In summary, we have shown that when the oxide thickness is reduced, the use of a polysilicon gate can produce important drawbacks in the electrical characteristics of the MOSFET if the polysilicon is not doped sufficiently: an important degradation of the C V characteristics of the MOS structure is produced by the depletion of the polysilicon near the oxide interface, which leads to a reduced inversion charge density in the channel. On the other hand, the combination of an ultrathin oxide and a very highly doped polysilicon gate causes an important degradation of the electron mobility in the channel, which should be taken into account. We have developed an RCS model which considers the effects of image charges, screening, inversion layer quantization, the contribution of different subbands, oxide thickness, the actual distribution of charged centres inside the structure, the actual distribution of carriers in the inversion layer, the correlation of charged centres and the charged centre sign. The model is implemented in a Monte Carlo simulator, where the effects of the ionized impurity charge, the interface trapped charge and the contribution of other scattering mechanisms are taken into account simultaneously. Our results show that RCS cannot be neglected for oxide thicknesses below 2 nm, even when very high substrate doping concentrations or relatively low poly-doping concentrations are used. However, we have seen that RCS effects for t ox > 5nmarenegligible. Finally, we compared simulated results with experimental ones and found that the results obtained with the RCS model developed here compare much better with experimental results than do previous models. Acknowledgments This work has been carried out within the framework of Research Project no TIC supported by the Spanish Government. References [1] 1999 SIA Roadmap [2] Momose H S et al 1996 IEDM Technical Digest (New York: IEEE) 109 [3] Gupta A, Fang P, Song M, Lin M-R, Wollesen D, Chen K and Hu C 1997 Accurate determination of ultrathin gate oxide thickness and effective polysilicon doping of CMOS devices IEEE Electr. Dev. Lett [4] Alam M, Weir B and Silverman P 1 The prospect of using thin oxides for silicon nanotransistors Extend. Abst. International Workshop on Gate Insulators 1 p30 [5] Lo S-H, Buchanan D A, Taur Y and Wang W 1997 Quantum-mechanical modeling of electron tunneling current from the inversion layer of ultra-thin-oxide nmosfets IEEE Electr. Dev. 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