# Impact of Band Nonparabolicity on Threshold Voltage of Nanoscale SOI MOSFET

Authors: Yasuhisa Omura
Journal: Active and Passive Electronic Components
 (2016)

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## Abstract

This paper reconsiders the mathematical formulation of the conventional nonparabolic band model and proposes a model of the effective mass of conduction band electrons including the nonparabolicity of the conduction band. It is demonstrated that this model produces realistic results for a sub-10-nm-thick Si layer surrounded by an SiO2 layer. The major part of the discussion is focused on the low-dimensional electron system confined with insulator barriers. To examine the feasibility of our consideration, the model is applied to the threshold voltage of nanoscale SOI FinFETs and compared to prior experimental results. This paper also addresses a model of the effective mass of valence band holes assuming the nonparabolic condition.


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## Body

## 1. Introduction

In the last 3 decades, silicon-on-insulator (SOI) MOSFETs have been attracting attention because of their high short-channel effect immunity [1] and excellent potential with regard to future nanoscale devices [2]. Therefore, many studies on quantum confinement effects have been performed. The author predicted that the threshold voltage (Vth) of ultrathin-body (UTB) SOI MOSFET rises as SOI body thickness (tS) is reduced due to the quantum-mechanical confinement effect [3]. Several studies have demonstrated experiments [4] that, they believe, demonstrate the effect of quantum confinement. However, the author demonstrated that the apparent rise of Vth of UTB SOI MOSFETs is due to more than the quantum effect; it includes the semiclassical effect [5]. Other experiments strongly suggested that a simple parabolic band model is not appropriate in analyzing Vth of nanoscale SOI MOSFETs [6, 7]; in these experiments, the threshold voltage was lower than the simulation values that assume the parabolic conduction band (X bands for Si) for tS<5 nm. The possibility of determining the limits of the conventional parabolic band approximation is a motivation of this study. Recently, a couple of research articles demonstrated the importance of band nonparabolicity in the analysis of the transport characteristics of nanoscale materials [8, 9], where the first principle calculation and the tight-binding method are used to compute the electronic states. In practical situations, however, demand an analytical closed form for the quantized energy levels and an effective mass tensor is requested for electronic device designs, since they greatly reduce the time costs.This paper reconsiders the mathematical formulation of the conventional nonparabolic band model. This paper examines whether some perturbations can be added to the conventional model for convenience. In the following discussion, this paper focuses on a low-dimensional electron system confined by insulator barriers. We discuss the impact of the nonparabolic conduction band in Si on the effective mass and propose an analytical expression for the effective mass of electrons including the conduction band nonparabolicity. The model is applied to the threshold voltage of nanoscale SOI FinFETs, and its validity is examined. By examining the mathematical basis for the effective mass of electrons that have conduction band nonparabolicity, this paper also illuminates a model for the effective mass of holes having valence band nonparabolicity.


## 2. Modeling Nonparabolic Band Structure

When an isotropic band structure is assumed, it is conventionally known that its form can be expressed generally as [10](1)En-par1+αEn-par=Epar=ħ2k22m∗,where m∗ is the effective mass of electrons in the isotropic band, α is the nonparabolic band factor, En-par is the energy of electrons in the nonparabolic band, and Epar is the electron energy expression in the form of the parabolic band scheme. Here we reconsider how the realistic effective mass for device analysis should be estimated; the above simplified nonparabolic band model is assumed for the conduction band electrons of a three-dimensional quasi-free electron system.When an external field effect is taken account of, we must examine whether the following formulation is theoretically valid or not:(2)En-par1+αEn-par=Epar+βEext,where Eext is the perturbation energy corresponding to the 1st-order perturbation generated by the external electric field and β is the perturbation factor. When (2) is valid, (2) can be rewritten as(3)En-par=1+4αEpar+βEext-12α.In order to examine the availability of (3), the right-hand side of (3) is changed into Tayler’s power series.(4)En-par=12α1+∑n=1∞cnEpar+βEextn-1=12α1+2αEpar+βEext+∑n=2∞cnEpar+βEextn-1, where cn is the expansion coefficient and c1=2α. If (4) is meaningful, En-par should be the eigenvalue of the corresponding Schrödinger’s equation. That is, (4) should be equivalent to the following operator representation:(5)H^n-par=12α1+2αH^par+βH^ext+∑n=2∞cnH^par+βH^extn-1=H^par+H^′,where all suffice and correspond to those in (4) and H^′ is the perturbation term that is expressed as(6)H^′=βH^ext+12α∑n=2∞cnH^par+βH^extn.When we can assume that the amplitude of H^′ is smaller than that of H^par [11], we approximately have(7)En-park≈Epark-EC+βEext+12α∑n=2∞cnEpark-EC+βEextn=12α1+2αEpark-EC+βEext+∑n=2∞cnEpark-EC+βEextn-1=1+4αEpark-EC+βEext-12α.Equation (7) has the same form as (3). However, as the above formulation indicates, the meaning of (7) differs from that of (3) because (7) has been examined with the Hamiltonian operator representation in spite of the approximations used. Provided that prominent nonlinear effects are not significant, it can be concluded that (7) holds important physical meaning in terms of evaluating nonparabolic band effects on transport characteristics.For the two-dimensional electron system, (7) can be rewritten as [11](8)En-par,nk=1+4αEnkn-EC+βEext+Efreek∥-12α,for the nth subband attributed to the specific conduction band, where En is the subband energy level of electrons, kn is the discrete wavenumber of the subband labeled “n” in the quantum well, and Efree(k∥) is the energy component of the transport direction. Equation (8) is valid only for E1-EC≫Eext based on the same assumption used for (7). Actually, it is easily found that Eext is much smaller than E1-EC for electric fields less than 107 V/cm and tS < 10 nm [10]; therefore, (8) is applicable to the analysis of nanoscale MOSFET characteristics.The result described above yields an expression for the effective mass tensor (m∗n-par,ij) for two-dimensional electron systems around the subband bottom as in [14]. Before using (8) for direct calculations, we review the conventional idea of the effective mass. We must recall the fact that electronic states with finite dimensions are inherently discrete; the difference from a very small size material is just the magnitude of Δki between adjacent states. Naive calculation discards the term of the group velocity, but this is erroneous in terms of physics. The correct calculation result is given as(9)1mn-par,ij=1ħ2∂2En-par,nk∂kj∂ki=1+4αEjkj-EC+βEext+Efreek∥-1/21mpar,ij-2α1+4αEjkj-EC+βEext+Efreek∥-3/2vcpar,jvcpar,i,where “j” means confinement direction and vcpar,j is the “effective group velocity” along the direction labeled by “j.” Given that label “i” indicates the transport direction, we have(10)vcpar,i=1ħ∂Efreek∥∂ki.Given that label “j” represents the confinement direction, we have(11)vcpar,j=1ħ∂Ejkj+βEext∂kj=ħkjmjj∗+1ħ∂βEext∂kj,where kj is the discrete wavenumber and mjj∗ is the effective mass along the confinement direction. The 2nd term of the right-hand side is the perturbation term raised by the external field. When the semiconductor layer thickness is of the order of nanometers, the contribution of the 2nd term of (11) is quite small. Thus, the effective mass tensor (m∗n-par,ij) value of low-dimensional electron systems can be calculated around the subband bottom [15]; the important point is the fact that the effective mass of electrons is larger by the factor of (1+4α(Ej(0)-EC+βEext))1/2 than that estimated assuming a parabolic band.


## 3. Calculation Results of Effective Mass of Low-Dimensionality Electrons

Assuming a thin Si layer, we calculated effective mass values (ml/m0 and mt/m0) for the (001) surface, where it is assumed that 2-fold and 4-fold X-band electrons are confined along the 001 axis. Physical parameters assumed in calculations are summarized in Table 1. Figure 1 shows the effective mass of electrons occupying the ground state as a function of Si layer thickness (tS), where the nonparabolicity factor α is assumed to be 0.5 eV−1 and, for comparison, 1st principle calculation results [8, 9] are also shown; the impact of band nonparabolicity on the effective mass of conduction band electrons appears when tS < 5 nm [8, 9]. The present model successfully reproduces the 1st principle calculation results. Figure 2(a) shows the effective mass of electrons occupying the 1st excited state as a function of Si layer thickness. For comparison, 1st principle calculation results [8, 9] are also shown in Figure 2(a). The present model does not reproduce the 1st principle calculation results, so the conventional value of α is not appropriate. We, therefore, varied the value of α until the calculated curves fitted the 1st principle calculation results. Figure 2(b) shows the calculation results of the effective mass value of conduction band electrons occupying the 1st excited state, including the nonparabolicity effect, where it is assumed that α = 0.1 eV−1 for ml/m0 and α = 0.05 eV−1 for mt/m0. It is seen that the present model for effective mass successfully reproduces the theoretical simulation results. It is strongly suggested that the reason why the value of α for the 1st excited state is smaller than expected stems from the approximation that stripped higher-order expansion terms from (7).Table 1
Physical parameters assumed in calculations [12, 13].
Notations Values Comments m l ∗ / m 0 0.92 Longitudinal mass of electrons m t ∗ / m 0 0.19 Transverse mass of electrons m h h ∗ / m 0 0.49 Heavy holes m l h ∗ / m 0 0.16 Light holes m 0 9.1 × 10−31 [kg] Free electrons n i (@300 K) 9.7 × 109 [cm−3] Bulk Si E g (@300 K) 1.1 [eV] Permittivity of Si (εS) 12ε0 [F/cm] Permittivity of SiO2 (εOX) 3.8ε0 [F/cm] Permittivity of vacuum (ε0) 8.9 × 10−14 [F/cm]Figure 1
Calculated effective mass values of electrons occupying the ground state as a function of semiconductor layer thickness [11]. It is assumed that α = 0.5 eV−1 in order to calculate the nonparabolic band effect.Figure 2
Calculated effective mass values of electrons occupying the 1st excited state as a function of semiconductor layer thickness [11]. (a) Effective mass dependence on Si layer thickness. It is assumed that α=0.5 eV−1 in order to calculate the nonparabolic band effect. (b) Effective mass dependence on Si layer thickness. It is assumed that α = 0.1 eV−1 for ml/m0 and α = 0.05 eV−1 for mt/m0 in calculating the nonparabolic band effect.
(a) (b)


## 4. Applying Effective Mass Model to Express Threshold Voltage of Nanoscale MOSFETs

Since the threshold voltage (Vth) of ultrathin-body SOI MOSFETs is effectively ruled by the lowest energy level [3], it is easily anticipated that the impact of band nonparabolicity on Vth is very significant. The author already clarified that the rise in threshold voltage stems from not only the quantum-mechanical mechanism [3] but also the semiclassical mechanism [5]. When it is assumed that the threshold voltage rising stems from the quantum confinement [3, 6, 15], Vth for ultrathin-bodyn-channel SOI MOSFET can be expressed as [3](12a)Vth=VFB+ϕsth+QBCox,(12b)ϕsth=ϕF,QM+EG2q+En1-ECq,where VFB is the flat-band voltage, Cox is the gate oxide capacitance per unit area, QB is the depletion charge density, ϕF,QM is the “effective Fermi potential” including quantum effects (defined later), EG is the bandgap energy, and En1 is the ground-state level energy of confined electrons in the conduction band. Fundamentally, the above expression for the threshold voltage of ultrathin-body SOI MOSFETs should yield the threshold voltage at low temperatures because the Fermi-Dirac function has a step-like function at low temperatures. At room temperature, the Fermi-Dirac function is thermally deformed, and the Fermi level does not simply define the threshold voltage; finite inversion layer charges are needed at the onset of the threshold [5]. From Boltzmann’s approximation, at room temperature we have [16](13)ϕF,QM=Ei-EFq=1qEi-Ep1+kBTln⁡NAtSqDospkBT,where Dosp is the density of states of two-dimensional valence-band holes, Ep1 is the ground-state level energy of confined holes in the valence band, and NA is the doping density; this expression is available for Ea-Ep1≫kBT, where Ea is the acceptor level energy. This expression is approximately derived from the following expression for intrinsic carrier density (ni∗), which should be called “effective intrinsic carrier density”, in the quantum well defined at room temperature [16].(14)ni∗T=DosnDosp1/2kBTtSexp-EG∗2kBT,where Dosn is the density of states of two-dimensional conduction band electrons; EG∗ is the “effective bandgap energy” of the quantum well and can be expressed as(15)EG∗=En1-Ep1.In calculating Vth of ap-channel SOI MOSFET, we can also apply the present nonparabolic band model to holes (see the Appendix section).On the other hand,Vth for a quantum-wire SOI fin field-effect transistor (FinFET) is expressed as [15, 17](16a)ϕsth=ϕF,QM+EG2q+En,ij-ECq,(16b)En,ij-EC=h28i2m∗nz,itS2+j2m∗nx,jwS2, where h is Planck’s constant, tS is fin height, wS is fin width, En,ij is the quantum level energy, and m∗nx,j (or m∗nz,i) is the effective mass of electrons occupying the corresponding subband. In (16b), it is assumed that the Si wire is confined along x-axis and z-axis.CalculatedVth dependence on fin width of SOI FinFET two-dimensionally confined with (001) and (011) surfaces [6] and on SOI layer thickness of thin SOI MOSFET [7] is shown in Figure 3, where the top surface of the Si fin body is (001) and the side surface is (011). Physical parameters assumed in the calculations are summarized in Table 1. In this calculation of threshold voltage of FinFET, it is assumed that electrons of the Si body can be represented as quasi-two-dimensional system because the value of tS is larger than the minimal value of wS. The conventional parabolic band model overestimates the quantum confinement effect in the range defined by sub-5-nm tS. This suggests that the nonparabolicity should be taken into account when estimating the Vth values of nanoscale SOI MOSFETs. On the other hand, experimental data of FinFET for tS > 5 nm take values higher than two curves. It is anticipated, as one possibility, that fabrication-induced local variation of SOI layer thickness influences the threshold voltage of devices [6, 7, 18].Figure 3
V t h dependence on fin width or SOI layer thickness of Si SOI MOSFET device. The devices shown in [6] are two-dimensionally confined with (001) and (011) surfaces. It is assumed that the gate oxide layer is 3 nm thick, the body silicon layer is 7 nm thick for [6], and α = 0.5 eV−1. The devices shown in [7] are two-dimensionally confined with (001) surface. It is assumed that the gate oxide layer is 50 nm thick, the body silicon layer is ranging from 3 nm to 6 nm thick for [7], and α = 0.5 eV−1. It is also assumed that m∗/m0 = 0.916 for (001) Si surface and m∗/m0 = 0.314 for (011) Si surface.Ge-on-insulator (GOI) devices are now attracting attention from the viewpoints of high-speed device applications. However, it is known that Ge demonstrates stronger conduction band nonparabolicity than Si. Therefore, the discussion given here is critical when considering nanoscale GOI device characteristics [19].


## 5. Conclusion

This paper reconsidered the mathematical formulation of the conventional nonparabolic band model. Since the conventional simplified model for band nonparabolicity does not include the perturbation created by the external potential effect, we examined whether such perturbations could be added to the conventional model for convenience. When the perturbation energy is smaller than the unperturbed energy, the insertion of a perturbation term into the conventional expression for the nonparabolic band model is valid; it was confirmed that this approximation is acceptable given a sub-10-nm-thick Si layer surrounded by an SiO2 layer. The major discussion focused on the low-dimensional electron system confined by an insulator barrier. For the purpose of verifying this consideration, we addressed the influence of band nonparabolicity on the threshold voltage of Si-based nanoscale SOI FinFETs; calculation results yielded by the proposed model were compared to experimental results and the validity of the model proposed here was confirmed.



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