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Surface layer effective density-of-states (SLEDOS) and its applications in MOS devices modeling
Microelectronics Journal Microelectronics Journal 31 (2000) 397–403 www.elsevier.com/locate/mejo
Surface layer effective density-of-states (SLEDOS) and its applications in MOS devices modeling Y. Ma*, L. Liu, Z. Li Institute of Microelectronics, Tsinghua University, Beijing 100084, People’s Republic of China Accepted 2 March 2000
Abstract The concept of surface potential effective density-of-states (SLEDOS) is presented in modeling the quantized inversion layer. Carrier distribution models both in semi-classical and quantum mechanical cases are developed based on SLEDOS. Threshold voltage shift model due to QMEs, a new iteration method to calculate the inversion carrier sheet density and surface potential, as well as a gate capacitance model are built based on the concept of SLEDOS. It is demonstrated that the concept of SLEDOS reveals the physical nature of inversion layer quantization and provides a feasible method to characterize MOS inversion layer. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Surface potential effective density-of-states; Metal-oxide-semiconductor-field-effect-transistor; Maxwell–Boltzmann statistics
1. Introduction VLSI technology, the base of electrical industry is progressing rapidly. The key element in VLSI circuits, MetalOxide-Semiconductor-Field-Effect-Transistor (MOSFET), is scaling down from the sub-micron through the deep submicron to the sub-100 nm regime [1]. Based on the energy band theory [2], the semi-classical theory of semiconductor physics supports the mansion of modern electronics. In semiconductor physics, the conduction band and valence band effective density-of-states ! mdn kB T 3=2 Nc 2 × 2p"2 and
Nv
mdp kB T 2× 2p"2
3=2 !
are key parameters. From Nc and Nv, assisted by Fermi energy level (EF), following the Maxwell–Boltzmann statistics carrier concentration (from now on, we take electron as example, same conclusions are suitable for hole) at an arbitrary point r in a semiconductor is given as: n r Nc ·exp EF 2 Ec =kB T: It is the invention of the concept of the conduction band and valence band effective * Corresponding author. E-mail address: [email protected] (Y. Ma).
density-of-states that make a clear and concise formula for the carrier concentration at any point possible. However, things changed owing to the quantum mechanical nature of the mobile carrier in semiconductors. Carriers in the MOS inversion layer are actually confined in a deep potential well near the Si–SiO2 interface. As a result of the confinement, carrier energy is quantized and the continuous conduction band in the semi-classical theory changed into a series of subbands [3] as the wave nature of carriers manifests. Carriers are no longer locally distributed. The concentration of carriers in the inversion layer is described by the wave function of density probability. The concept of the conduction band and valence band effective density-of-states is no longer valid. With the development of the VLSI technology, the influences of quantum mechanical effects (QMEs) on MOS devices characteristics are more and more significant [4–6] and have been studied extensively by experimental investigations [7,8], theoretical calculations [9,10] as well as modeling studies [11–14]. Despite the extensive numerical and analytical investigation, no simple, physical, flexible concept like the conduction band and valence band effective density-of-states is proposed to make up the vacancy in the quantum mechanical description of the MOS inversion layer. The situation seems to have changed since the pioneer work of the authors [15]. In Ref. [15] the concept of surface layer effective density-of-states (SLEDOS) is proposed as the base of the threshold voltage model considering the quantization of the inversion layer. The concept of SLEDOS
0026-2692/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(00)00041-0
398
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
Fig. 1. Semi-classical surface layer effective density-of-states (SLEDOS) as a function of inversion layer charge density. Different lines represent different substrate doping levels.
is applicable for both the quantum mechanical description and the semi-classical description of the MOS inversion layer. Our further works demonstrate the suitability and flexibility of the concept in the modeling of the quantized and semi-classical inversion layers in the MOS structure. In this paper, the concept of SLEDOS is presented and thoroughly discussed. And then its applications in the modeling of the MOS inversion layer are outlined to demonstrate its usability.
electron in inversion layer [16]. Ndep in Eq. (3) is the depletion charge. The quantum mechanical SLEDOS Ncqm is introduced as XX 2Eij 4 Di·kB T·exp Ncqm kB T i j then qm Ninv
2. Concept of surface layer effective density-of-states In the quantized inversion layer in the MOS structure, the total electron concentration is given by [15] XX qm Nij Ninv i
2Eij Ef 1 qfs X X Di·kB T·exp kB T kB T i j
1
where Nij is the carrier density in the jth subband in the ith group [3], Di gi mdei =p"2 the states density of the ith group of subbands, Ef the Fermi energy level respect to the conduction band edge in bulk, f s the surface potential with respect to the substrate. The energy level Eij under triangular potential well approximation [3] is !1=3 i2=3 h "2 3 pqEeff j 1 34 2 Eij 2 2mi in which Eeff is the surface electric field and given by Eeff
q h·Ninv 1 Ndep : e0 es
3
where h is a weighting coefficient and adopted as 0.75 for
5
In the semi-classical cases, the carrier sheet density is derived as [2] class Ninv
Zd n z dz 0
j
exp
Ncqm exp
Ef 1 qfs kB T
exp
Ef 1 qfs Zd q fs 2 f z Nc ·exp 2 dz kB T kB T 0 6
The semi-classical SLEDOS is defined as Zd q fs 2 f z Nc ·exp 2 dz Ncclass kB T 0 q fs 2 f df N ·exp 2 Zc s c ÿ kB T Ef 0
7
Thus the inversion layer charge sheet density is derived from Eqs. (6) and (7) Ef 1 qfs class Ncclass exp 8 Ninv kB T E(f ) in Eq. (3) is the transverse electric field in the space
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
399
Fig. 2. Quantum mechanical surface layer effective density-of-states (SLEDOS) as a function of inversion layer charge density. Different lines represent different substrate doping levels.
charge region and is given as [2] p F NA E z es q
ft e2f y=ft 1 f y 2 ft 1 e22fF =ft ft ef y=ft 2 f y 2 ft
9 p in which F 2qes and ft kT=q is the thermal voltage. Thus, the inversion carrier density in the quantum qm and in the semi-classical case mechanical case Ninv class Ninv can be represented as a simple function of surface potential by the aid of SLEDOS (Eqs. (5) and (8)). As will be seen in Section 3, this concise relation between the inversion layer sheet density and the surface potential provides a flexible method to characterize the MOS device behavior. In Figs. 1 and 2, SLEDOS versus the inversion layer carrier density both in the semi-classical case and in the quantum mechanical case are presented, respectively. It is clear that significant distinction exists in the two SLEDOSs. Firstly, SLEDOS including QMEs is systematically lower than that of the semi-classical theory. This corresponds to the carrier density reduction effect and the threshold voltage shift effect due to QMEs. Secondly, in the strong inversion region, the quantum mechanical SLEDOS decreases more rapidly than the semi-classical counterpart, which reveals that the variation of the space distribution of carriers due to QMEs (inversion layer capacitance), would cast more influences on device characteristics than in the semi-classical cases. Ncclass can also be represented as a function of the inversion layer carrier sheet density. It is shown in Fig. 1 that in the lower level of the carrier density, Ncclass is nearly constant with respect to carrier density while in the high level region, Ncclass is determined absolutely by the carrier
sheet density without an interrelation with the substrate doping level. Using the method in Ref. [17], an analytical formula of Ncclass versus Ninv can be reached q 10 Ncclass f 2 f 2 2 Ncclass Nsub ·Ncclass Ninv with f 0:5· Ncclass Nsub 1 1 1 dNcclass Ninv
11
Ncclass Nsub 66:8557 2 1:2776 ln Nsub
12
Ncclass Ninv 75:6511 2 2:28626 ln Ninv
13
where Nsub is the substrate doping concentration; d is a small coefficient to adjust the smooth degree between the two functions. 3. Applications in MOS structure modeling In this section, the applications of SLEDOS in deriving the analytical threshold voltage shift model, in building a new method to calculate the surface potential and the carrier sheet density and in developing a new capacitance model are presented. 3.1. Threshold voltage shift model We define DENc 2kB T·ln Ncqm =Ncclass
14
then Ncqm Ncclass ·exp
2DENc kB T
15
400
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
Fig. 3. Threshold voltage shift due to QMEs. The experimental data are from Ref. [7] and the self-consistent numerical data are from Ref. [5]. The present model gives consistent results with experimental and self-consistent calculation results.
Then from Eqs. (5) and (15) it is straightforward to get Ef 1 qfs 2 DENc qm 16 Ninv Ncclass ·exp kB T Comparing Eqs. (8) and (16), it is seen that in order to reach the threshold point, the surface potential in the quantum mechanical case must be larger than the semi-classical one by the quantity of DENc ; i.e. qm class DENc =q Dfqm s fs 2 fs
17
The analytical model of the Vth shift due to QMEs is derived
in the depletion approximation s! 1 eSi qNsub qm DVT Dfs 1 1 2Cox fb
18
in which kB T Nsub ·ln fb ni q is the body Fermi potential. Results of the present model are calculated and compared
Fig. 4. Inversion layer charge density comparison of model results and numerical results. Dashed lines are semi-classical results by simplified method in the present work. Solid lines are quantum mechanical results by simplified method. Symbols are numerical results. Open symbols are for semi-classical cases by solving Poisson equation and closed symbols are for quantum mechanical case by self-consistent solution of Schrodinger and Poisson equations. Gate oxide thickness is 5 nm in the calculation.
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
401
Fig. 5. Surface potential comparison of model results and numerical results. The meanings of symbols and lines are same as that in Fig. 4.
with experiment data of Ref. [7] and self-consistent calculation results from Ref. [5] as shown in Fig. 3. Both experimental results and numerical calculations give proofs of the accuracy of the model. 3.2. Reversed iteration method Based on the concept of SLEDOS, a new iteration method to calculate the inversion layer carrier sheet density and the surface potential is developed. In the iteration procedure, the gate voltage is given as the starting point. The inversion layer carrier density is initialized by Ninv Cox · Vgate 2 Vth =q
19
where Vth is threshold voltage and Cox the gate oxide capacitance. Then the surface layer effective density of states Ncqm is derived by Eq. (4), and from Eq. (5) the surface potential is given by
fs kB T 2Ef 1 ln Ninv =Ncqm =q
20
and then Ninv Cox Vgate 2 fs =q 2 Ndep
21
Thus one loop of iteration is implemented. Several more iterations are needed to achieve reasonable accuracy. Usually, four or five iterations is sufficient to reach a relative error of less than 0.5% in the carrier sheet density. The feature of the iteration method lies in the derivation of new f s from Ninv by Eq. (20). Since Ninv is an exponential function of surface potential, this procedure is inherent to shrink the iterative error and thus results in a high efficiency. It is straightforward to propose the concept of SLEDOS and the iteration procedure in the semi-classical treatment. Figs. 4 and 5 give the numerical verification of the model in both semi-classical and quantum mechanical cases. The
results show that in a wide range of substrate doping concentration and gate bias the results of the simplified iteration method coincide with the numerical results very well. 3.2.1. Capacitance model Based on the concept of SLEDOS and the formula developed in Section 2, it is feasible to derive a gate capacitance model suitable for semi-classical case and quantum mechanical case. The gate capacitance is given by Cgate
q·d Ninv 1 Ndep dQg dVgate dVgate
22
In strong inversion region in quantum mechanical case, disregarding the dependence of Ndep on gate bias, it is deduced from Eq. (5) that qm q· Cgate
dNinv dVgate
! dNcqm dNinv Ef 1 qfs 1 q·Ncqm q· · ·exp dNinv dVgate kB T ! Ef 1 qfs q df s qm dNcqm · · × exp Cgate · kB T dVgate kB T dNinv ! Ef 1 qfs q df s · 1 q·Ninv · × exp (23) kB T kB T dVgate and from Eq. (21) qm Cgate
dfs Cox · 1 2 dVgate
! 24
402
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
Fig. 6. Gate capacitance in quantum mechanical and in semi-classical cases in the inversion region. Flat band voltage is assumed to be zero in the calculation. QMEs result in a substantially lower gate capacitance in the strong inversion region.
Combining Eqs. (23) and (24) results in
References
2
qm Cgate
q Ninv Cox dNcqm E f 1 qf s q2 Ninv 1 Cox ·kB T 1 2 ·exp dNinv kB T 25
Since Ncqm is an analytical function of Eeff and then of Ninv, Eq. (25) actually gives a straightforward method to calculate gate capacitance. In the semi-classical case, the only difference is the class is readily available by SLEDOS. The expression of Cgate replacing Ncqm in Eq. (25) by Ncclass. The gate capacitance in the whole gate bias range (weak inversion region and strong inversion region) is presented in Fig. 6. The gate oxide thickness is 3 nm and the substrate doping concentration is 10 18 cm 23. The flat band voltage is assumed to be zero in calculation. As can be seen, the gate capacitance for the quantum mechanical case is substantially lower than the semi-classical one in the strong inversion region. 4. Conclusions In this paper, the concept of SLEDOS is presented, based on which the carrier distribution in the inversion layer in the MOS structure is formulated in both semi-classical and quantum mechanical cases. The applications of the concept of SLEDOS in the quantized and unquantized inversion layer modeling are presented demonstrating the feasibility of the concept. These applications include the threshold voltage model, a new method to calculate the charge density and the surface potential, and the gate capacitance model. The model results are compared with the experimental and numerical data to demonstrate the accuracy of the models.
[1] Y. Taur, D.A. Buchanan, W. Chen, D.J. Frank, K.E. Ismail, S.H. Lo, G.A. SaiHalasz, R.G. Viswanathan, H.J.C. Wann, S.J. Wind, H.S. Wong, CMOS scaling into the nanometer regime, Proc. IEEE 85 (4) (1997) 486–504. [2] S.M. Sze, Physics of Semiconductor Devices, 2 ed., Wiley, New York, 1981. [3] F. Stern, Self-consistent results for n-type Si inversion layers, Phys. Rev. B 5 (12) (1972) 4891. [4] T. Janik, B. Majkusiak, Influence of carrier energy quantization on threshold voltage of metal-oxide-semiconductor transistor, J. Appl. Phys. 75 (10) (1994) 5186–5190. [5] S. Jallepalli, J. Bude, W.-K. Shih, M.R. Pinto, C.M. Maziar, A.F. Tasch Jr, Electron and hole quantization and their impact on deep submicron silicon p- and n-MOSFET characteristics, IEEE Trans. Electron Devices 44 (3) (1997) 297. [6] B.K. Ip, J.R. Brews, Quantum effects upon drain current in a biased MOSFET, IEEE Trans. Electron Devices 45 (10) (1998) 2213–2221. [7] M.J. Van Dort, et al., Influence of high substrate doping levels on the threshold voltage and the mobility of deep-submicrometer MOSFET’s, IEEE Trans. Electron Devices 39 (4) (1992) 932. [8] C. Chindalore, S.A. Hareland, S. Jallepalli, F.A. Tasch Jr., C.M. Maziar, V.K.F. Chia, S. Smith, Experimental determination of threshold voltage shifts due to quantum mechanical effects in MOS electron and hole inversion layer, IEEE EDL 18 (5) (1997) 206–208. [9] T. Janik, B. Majkusiak, Analysis of the MOS transistor based on the self-consistent solution to the Schrodinger and Poisson equations and on the local mobility model, IEEE Trans. Electron Devices 45 (6) (1998) 1263. [10] A. Pacelli, Self-consistent solution of the Schrodinger equation in semiconductor devices by implicit iteration, IEEE Trans. Electron Devices 44 (7) (1997) 1169. [11] R. Rios, N.D. Arora, C.-L. Huang et al. A physical compact MOSFET model, including quantum mechanical effects, for statistical circuit design applications, IEDM95, 1995, p. 937. [12] M.J. vanDort, P.H. Woerlee, A.J. Walker, A simple model for quantization effects in heavily-doped silicon MOSFET’s at inversion conditions, Solid-State Electron. 37 (3) (1994) 435.
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403 [13] S.A. Hareland, M. Manassian, W.K. Shin, et al., Computational efficient models for quantization effects in MOS electron and hole accumulation layers, IEEE Trans. Electron Devices 45 (7) (1998) 1487. [14] W. Liu, X. Jin, Y. King, C. Hu, An Efficient and accurate compact model for thin-oxide-MOSFET intrinsic capacitance considering the finite charge layer thickness, IEEE Trans. Electron Devices 46 (5) (1999) 1070. [15] Y. Ma, Z. Li, L. Liu, L. Tian, Z. Yu, effective density-of-states
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approach to QM correction in MOS structures, Solid-State Electronics 44 (3) (2000) 40. [16] Y. Ma, L. Liu, Z. Yu, Z. Li, Validity and applicability of triangular potential well approximation in modeling of MOS structure inversion and accumulation layer, IEEE Trans. Electron Devices (2000), in press. [17] A. Chatterjee, C.F. Machala, P. Yang, A submicron DC MOSFET model for simulation of analog circuits, IEEE Trans. Computer-Aided Design 14 (1995) 1193–1207.
Surface layer effective density-of-states (SLEDOS) and its applications in MOS devices modeling Y. Ma*, L. Liu, Z. Li Institute of Microelectronics, Tsinghua University, Beijing 100084, People’s Republic of China Accepted 2 March 2000
Abstract The concept of surface potential effective density-of-states (SLEDOS) is presented in modeling the quantized inversion layer. Carrier distribution models both in semi-classical and quantum mechanical cases are developed based on SLEDOS. Threshold voltage shift model due to QMEs, a new iteration method to calculate the inversion carrier sheet density and surface potential, as well as a gate capacitance model are built based on the concept of SLEDOS. It is demonstrated that the concept of SLEDOS reveals the physical nature of inversion layer quantization and provides a feasible method to characterize MOS inversion layer. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Surface potential effective density-of-states; Metal-oxide-semiconductor-field-effect-transistor; Maxwell–Boltzmann statistics
1. Introduction VLSI technology, the base of electrical industry is progressing rapidly. The key element in VLSI circuits, MetalOxide-Semiconductor-Field-Effect-Transistor (MOSFET), is scaling down from the sub-micron through the deep submicron to the sub-100 nm regime [1]. Based on the energy band theory [2], the semi-classical theory of semiconductor physics supports the mansion of modern electronics. In semiconductor physics, the conduction band and valence band effective density-of-states ! mdn kB T 3=2 Nc 2 × 2p"2 and
Nv
mdp kB T 2× 2p"2
3=2 !
are key parameters. From Nc and Nv, assisted by Fermi energy level (EF), following the Maxwell–Boltzmann statistics carrier concentration (from now on, we take electron as example, same conclusions are suitable for hole) at an arbitrary point r in a semiconductor is given as: n r Nc ·exp EF 2 Ec =kB T: It is the invention of the concept of the conduction band and valence band effective * Corresponding author. E-mail address: [email protected] (Y. Ma).
density-of-states that make a clear and concise formula for the carrier concentration at any point possible. However, things changed owing to the quantum mechanical nature of the mobile carrier in semiconductors. Carriers in the MOS inversion layer are actually confined in a deep potential well near the Si–SiO2 interface. As a result of the confinement, carrier energy is quantized and the continuous conduction band in the semi-classical theory changed into a series of subbands [3] as the wave nature of carriers manifests. Carriers are no longer locally distributed. The concentration of carriers in the inversion layer is described by the wave function of density probability. The concept of the conduction band and valence band effective density-of-states is no longer valid. With the development of the VLSI technology, the influences of quantum mechanical effects (QMEs) on MOS devices characteristics are more and more significant [4–6] and have been studied extensively by experimental investigations [7,8], theoretical calculations [9,10] as well as modeling studies [11–14]. Despite the extensive numerical and analytical investigation, no simple, physical, flexible concept like the conduction band and valence band effective density-of-states is proposed to make up the vacancy in the quantum mechanical description of the MOS inversion layer. The situation seems to have changed since the pioneer work of the authors [15]. In Ref. [15] the concept of surface layer effective density-of-states (SLEDOS) is proposed as the base of the threshold voltage model considering the quantization of the inversion layer. The concept of SLEDOS
0026-2692/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(00)00041-0
398
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
Fig. 1. Semi-classical surface layer effective density-of-states (SLEDOS) as a function of inversion layer charge density. Different lines represent different substrate doping levels.
is applicable for both the quantum mechanical description and the semi-classical description of the MOS inversion layer. Our further works demonstrate the suitability and flexibility of the concept in the modeling of the quantized and semi-classical inversion layers in the MOS structure. In this paper, the concept of SLEDOS is presented and thoroughly discussed. And then its applications in the modeling of the MOS inversion layer are outlined to demonstrate its usability.
electron in inversion layer [16]. Ndep in Eq. (3) is the depletion charge. The quantum mechanical SLEDOS Ncqm is introduced as XX 2Eij 4 Di·kB T·exp Ncqm kB T i j then qm Ninv
2. Concept of surface layer effective density-of-states In the quantized inversion layer in the MOS structure, the total electron concentration is given by [15] XX qm Nij Ninv i
2Eij Ef 1 qfs X X Di·kB T·exp kB T kB T i j
1
where Nij is the carrier density in the jth subband in the ith group [3], Di gi mdei =p"2 the states density of the ith group of subbands, Ef the Fermi energy level respect to the conduction band edge in bulk, f s the surface potential with respect to the substrate. The energy level Eij under triangular potential well approximation [3] is !1=3 i2=3 h "2 3 pqEeff j 1 34 2 Eij 2 2mi in which Eeff is the surface electric field and given by Eeff
q h·Ninv 1 Ndep : e0 es
3
where h is a weighting coefficient and adopted as 0.75 for
5
In the semi-classical cases, the carrier sheet density is derived as [2] class Ninv
Zd n z dz 0
j
exp
Ncqm exp
Ef 1 qfs kB T
exp
Ef 1 qfs Zd q fs 2 f z Nc ·exp 2 dz kB T kB T 0 6
The semi-classical SLEDOS is defined as Zd q fs 2 f z Nc ·exp 2 dz Ncclass kB T 0 q fs 2 f df N ·exp 2 Zc s c ÿ kB T Ef 0
7
Thus the inversion layer charge sheet density is derived from Eqs. (6) and (7) Ef 1 qfs class Ncclass exp 8 Ninv kB T E(f ) in Eq. (3) is the transverse electric field in the space
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
399
Fig. 2. Quantum mechanical surface layer effective density-of-states (SLEDOS) as a function of inversion layer charge density. Different lines represent different substrate doping levels.
charge region and is given as [2] p F NA E z es q
ft e2f y=ft 1 f y 2 ft 1 e22fF =ft ft ef y=ft 2 f y 2 ft
9 p in which F 2qes and ft kT=q is the thermal voltage. Thus, the inversion carrier density in the quantum qm and in the semi-classical case mechanical case Ninv class Ninv can be represented as a simple function of surface potential by the aid of SLEDOS (Eqs. (5) and (8)). As will be seen in Section 3, this concise relation between the inversion layer sheet density and the surface potential provides a flexible method to characterize the MOS device behavior. In Figs. 1 and 2, SLEDOS versus the inversion layer carrier density both in the semi-classical case and in the quantum mechanical case are presented, respectively. It is clear that significant distinction exists in the two SLEDOSs. Firstly, SLEDOS including QMEs is systematically lower than that of the semi-classical theory. This corresponds to the carrier density reduction effect and the threshold voltage shift effect due to QMEs. Secondly, in the strong inversion region, the quantum mechanical SLEDOS decreases more rapidly than the semi-classical counterpart, which reveals that the variation of the space distribution of carriers due to QMEs (inversion layer capacitance), would cast more influences on device characteristics than in the semi-classical cases. Ncclass can also be represented as a function of the inversion layer carrier sheet density. It is shown in Fig. 1 that in the lower level of the carrier density, Ncclass is nearly constant with respect to carrier density while in the high level region, Ncclass is determined absolutely by the carrier
sheet density without an interrelation with the substrate doping level. Using the method in Ref. [17], an analytical formula of Ncclass versus Ninv can be reached q 10 Ncclass f 2 f 2 2 Ncclass Nsub ·Ncclass Ninv with f 0:5· Ncclass Nsub 1 1 1 dNcclass Ninv
11
Ncclass Nsub 66:8557 2 1:2776 ln Nsub
12
Ncclass Ninv 75:6511 2 2:28626 ln Ninv
13
where Nsub is the substrate doping concentration; d is a small coefficient to adjust the smooth degree between the two functions. 3. Applications in MOS structure modeling In this section, the applications of SLEDOS in deriving the analytical threshold voltage shift model, in building a new method to calculate the surface potential and the carrier sheet density and in developing a new capacitance model are presented. 3.1. Threshold voltage shift model We define DENc 2kB T·ln Ncqm =Ncclass
14
then Ncqm Ncclass ·exp
2DENc kB T
15
400
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
Fig. 3. Threshold voltage shift due to QMEs. The experimental data are from Ref. [7] and the self-consistent numerical data are from Ref. [5]. The present model gives consistent results with experimental and self-consistent calculation results.
Then from Eqs. (5) and (15) it is straightforward to get Ef 1 qfs 2 DENc qm 16 Ninv Ncclass ·exp kB T Comparing Eqs. (8) and (16), it is seen that in order to reach the threshold point, the surface potential in the quantum mechanical case must be larger than the semi-classical one by the quantity of DENc ; i.e. qm class DENc =q Dfqm s fs 2 fs
17
The analytical model of the Vth shift due to QMEs is derived
in the depletion approximation s! 1 eSi qNsub qm DVT Dfs 1 1 2Cox fb
18
in which kB T Nsub ·ln fb ni q is the body Fermi potential. Results of the present model are calculated and compared
Fig. 4. Inversion layer charge density comparison of model results and numerical results. Dashed lines are semi-classical results by simplified method in the present work. Solid lines are quantum mechanical results by simplified method. Symbols are numerical results. Open symbols are for semi-classical cases by solving Poisson equation and closed symbols are for quantum mechanical case by self-consistent solution of Schrodinger and Poisson equations. Gate oxide thickness is 5 nm in the calculation.
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
401
Fig. 5. Surface potential comparison of model results and numerical results. The meanings of symbols and lines are same as that in Fig. 4.
with experiment data of Ref. [7] and self-consistent calculation results from Ref. [5] as shown in Fig. 3. Both experimental results and numerical calculations give proofs of the accuracy of the model. 3.2. Reversed iteration method Based on the concept of SLEDOS, a new iteration method to calculate the inversion layer carrier sheet density and the surface potential is developed. In the iteration procedure, the gate voltage is given as the starting point. The inversion layer carrier density is initialized by Ninv Cox · Vgate 2 Vth =q
19
where Vth is threshold voltage and Cox the gate oxide capacitance. Then the surface layer effective density of states Ncqm is derived by Eq. (4), and from Eq. (5) the surface potential is given by
fs kB T 2Ef 1 ln Ninv =Ncqm =q
20
and then Ninv Cox Vgate 2 fs =q 2 Ndep
21
Thus one loop of iteration is implemented. Several more iterations are needed to achieve reasonable accuracy. Usually, four or five iterations is sufficient to reach a relative error of less than 0.5% in the carrier sheet density. The feature of the iteration method lies in the derivation of new f s from Ninv by Eq. (20). Since Ninv is an exponential function of surface potential, this procedure is inherent to shrink the iterative error and thus results in a high efficiency. It is straightforward to propose the concept of SLEDOS and the iteration procedure in the semi-classical treatment. Figs. 4 and 5 give the numerical verification of the model in both semi-classical and quantum mechanical cases. The
results show that in a wide range of substrate doping concentration and gate bias the results of the simplified iteration method coincide with the numerical results very well. 3.2.1. Capacitance model Based on the concept of SLEDOS and the formula developed in Section 2, it is feasible to derive a gate capacitance model suitable for semi-classical case and quantum mechanical case. The gate capacitance is given by Cgate
q·d Ninv 1 Ndep dQg dVgate dVgate
22
In strong inversion region in quantum mechanical case, disregarding the dependence of Ndep on gate bias, it is deduced from Eq. (5) that qm q· Cgate
dNinv dVgate
! dNcqm dNinv Ef 1 qfs 1 q·Ncqm q· · ·exp dNinv dVgate kB T ! Ef 1 qfs q df s qm dNcqm · · × exp Cgate · kB T dVgate kB T dNinv ! Ef 1 qfs q df s · 1 q·Ninv · × exp (23) kB T kB T dVgate and from Eq. (21) qm Cgate
dfs Cox · 1 2 dVgate
! 24
402
Y. Ma et al. / Microelectronics Journal 31 (2000) 397–403
Fig. 6. Gate capacitance in quantum mechanical and in semi-classical cases in the inversion region. Flat band voltage is assumed to be zero in the calculation. QMEs result in a substantially lower gate capacitance in the strong inversion region.
Combining Eqs. (23) and (24) results in
References
2
qm Cgate
q Ninv Cox dNcqm E f 1 qf s q2 Ninv 1 Cox ·kB T 1 2 ·exp dNinv kB T 25
Since Ncqm is an analytical function of Eeff and then of Ninv, Eq. (25) actually gives a straightforward method to calculate gate capacitance. In the semi-classical case, the only difference is the class is readily available by SLEDOS. The expression of Cgate replacing Ncqm in Eq. (25) by Ncclass. The gate capacitance in the whole gate bias range (weak inversion region and strong inversion region) is presented in Fig. 6. The gate oxide thickness is 3 nm and the substrate doping concentration is 10 18 cm 23. The flat band voltage is assumed to be zero in calculation. As can be seen, the gate capacitance for the quantum mechanical case is substantially lower than the semi-classical one in the strong inversion region. 4. Conclusions In this paper, the concept of SLEDOS is presented, based on which the carrier distribution in the inversion layer in the MOS structure is formulated in both semi-classical and quantum mechanical cases. The applications of the concept of SLEDOS in the quantized and unquantized inversion layer modeling are presented demonstrating the feasibility of the concept. These applications include the threshold voltage model, a new method to calculate the charge density and the surface potential, and the gate capacitance model. The model results are compared with the experimental and numerical data to demonstrate the accuracy of the models.
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