Numerical method for a 2D drift diffusion model arising in strained n-type MOSFET device

P

— journal of June 2016

physics pp. 1391–1400

Numerical method for a 2D drift diffusion model arising in strained n-type MOSFET device

RACHIDA BENSEGUENI^∗and SAIDA LATRECHE

Laboratoire Hyperfréquences & Semi-conducteurs (LHS), Département d’Electronique, Faculté des Sciences de la Technologie, Université Constantine1, Constantine 25000, Algérie

∗Corresponding author. E-mail: rachidabensegueni@gmail.com

MS received 18 February 2015; revised 27 April 2015; accepted 1 June 2015 DOI:10.1007/s12043-015-1135-0; ePublication:4 March 2016

Abstract. This paper reports the calculation of electron transport in metal oxide semiconductor field effects transistors (MOSFETs) with biaxially tensile strained silicon channel. The calculation is formulated based on two-dimensional drift diffusion model (DDM) including strain effects. The carrier mobility dependence on the lateral and vertical electric field model is especially considered in the formulation. By using the model presented here, numerical method based on finite difference approach is performed. The obtained results show that the presence of biaxially tensile strain enhances the current in the devices.

Keywords.Simulation; model; finite difference method; electron transport; enhancement mobility.

PACS Nos 85.30.De; 73.63.−b; 02.60.Cb

1. Introduction

The Si-based semiconductor industry made considerable progress in developing transistors with feature size well below 100 nm. However, as the transistor gate lengthL_gis scaled below 45 nm and the gate oxide thickness drops to 1 nm [1], physical limitations, such as gate leakage current [2], and the degradation in transport start to limit the enhance- ment in transistor performances. It is believed that conventional MOSFET is reaching its scaling limits, and the incorporation of new materials become inevitable.

Recently, newly engineered substrate materials like strained silicon have become increasingly integrated in the semiconductor industry. Indeed, the considered devices present higher electrical performances than the silicon one: Theoretical calculations [3–7]

predict electron and hole mobility enhancements in strained Si. For an n MOSFET sur- face channel, the strained Si can be engineered by depositing a thin Si film on a relaxed Si₁₋xGexsubstrate [8].

In this work, we develop numerical simulation of the 2D drift diffusion model (DDM) using our own simulator SIBIDIF (simulation bidimensional by finite difference), to

evaluate the influence of biaxial tensile strain effects on the derived current and elec- tron mobility improvement in n MOSFET structures. SIBIDIF [9] is mainly based on Gummel’s decoupling method, a finite difference (FD) meshing scheme and a monotone iterative (MI) method. First of all, the partial differential equations (PDEs) in the set of drift diffusion model are discretized using an adaptive finite difference (FD) mesh. Accor- ding to Gummel’s procedure, all the discretized PDEs are decoupled and then solved by means of the Gauss Seidel’s method. Simulation results obtained in this study are then discussed.

2. Mathematical model

Figure 1 shows the schematic cross-sectional view of the strained Si n MOSFET structure.

A layer of Si is grown pseudomorphically on the relaxed Si₁₋xGex layer, wherex is the mole fraction of Ge which causes strain in the Si layer, due to lattice mismatching with Si1−xGex [10]. Figure 2 displays the change in silicon energy band structure because of strain in the silicon channel [11,12].

The effect of strain on the electron affinity, the band-gap energy and the permittivity of strained Si/SiGe film can be modelled as follows [13,14]:

E_gs-Si+E_Cs-Si=E_g(Si1

−xGex)+E_Vs-Si, (1)

E_Vs-Si(x)=0.57x, (2)

E_Cs-Si(x)=0.6x, (3)

ε(SiGe)(x)=11.8+4.2x, (4)

Figure 1. Cross-section of a strained Si n MOSFET.

Figure 2. Effect of strain on silicon band structure.

whereE_gs−Si(x)is the decrease in band-gap energy of silicon due to strain;E_Vs-Siis the valence band discontinuity;E_Cs-Si(x)is the decrease in electron affinity of silicon due to a strain;ε_(SiGe)(x)is the permittivity of the SiGe layer.

The 2D drift diffusion model of the software SIBIDIF, thus, has been modified accord- ing to the effects of strain on the Si band structure. It is well known that the drift diffusion model (Poisson’s and carrier continuity equations), governs the distribution of electro- static potentialϕand carrier concentrations (nandp) [15–18]. By considering the effects of strain on the band structure of s-Si/SiGe film, a set of 2D drift diffusion equations (DDM) is derived as follows:

(1) 2D Poisson equation:

div(grad(ϕ))= −q

p−n+N_D⁺−N_A⁻

ε_Si,SiGe . (5)

(2) Continuity equations for electrons and holes:

∂n

∂t =g_n+1 q

dJn

dx , (6)

∂p

∂t =g_p−1 q

dJp

dx . (7)

(3) Current equations for electrons and holes:

Jn= −qnμndϕ_n

dx , (8)

Jp =qpμpdϕ_p

dx . (9)

In Poisson equation,ε_Siandε_SiGeare the permittivity of Si and relaxed Si1−xGex;N_D⁺and N_A⁻ are the ionized impurity concentrations;q indicates the absolute value of the elec- tronic charge.g_nandg_p are the net recombination rate of electron–hole pairs;ϕ_nandϕ_p are the quasi-Fermi levels for electron and holes, respectively.

ϕ_n= −1

qE_FN, (10)

ϕp = −1

qE_FP, (11)

E_FN =E_C+KT ln n

N_C

+KT lnγ_n, (12)

E_FP=E_V−KTln p

N_V

+KTlnγp. (13) E_FN andE_FP are the quasi-Fermi energies; E_C andE_V are the conduction and valence band energies,N_C andN_V are the effective density of states in conduction and valence bands;γnandγpare parameters specified to the statistics of Fermi–Dirac

E_C= −qϕ+E_g

2 +E_Cs-Si(x), (14)

E_V= −qϕ+E_g

2 −E_Vs-Si(x), (15)

μ_nandμ_pin eqs (8) and (9) are the electron and hole mobility models. To calculate the effects of electric field on mobility, we use a simplified mobility model [19], see eq. (16).

The effects of the lateral and vertical electric fields are separately considered. Equation (17) presents the correlation existing between vertical electric field and carrier mobility [19].

μ(N, E_⊥, E)=μ(N, E_⊥)∗

1+

μ(N, E_⊥)E V_sat

β₋1/β

, (16)

μ(N, E_⊥)=μ (N)∗

1+ E_⊥ E_crit

₋α

, (17)

whereμ(N)andV_satare low-field mobility and saturation velocity, respectively;E_⊥is the vertical electric field created by the gate voltage andE_critis a reference value.

Moreover, the effect of strain on the electron mobility is considered for both Si₁₋xGex

and tensile-strained Si layers [20] as follows:

μ_n=

μ(N)×(1+7.969x−10.90x²), 0≤x≤0.15,

μ(N)×(1.789+1.708x−2.663x²), 0.15≤x ≤0.4. (18)

3. Discretization and iterative solution method

The numerical scheme used in these calculations is based on the finite difference method.

Using this approach, the continuous functions of the equations are represented by vectors of function values at a series of grid points, and the differential operators turn into finite difference operators. Each equation is integrated over a small volume enclosing each node, yielding 3(n×m) nonlinear algebraic equations, where (n×m) is the number of grid points for the unknown potential and free-carrier concentration. For numerical implantation eqs (5)–(7) can be abstractly represented by

G_kϕ_k−1 +B_kϕ_k−n+H_kϕ_k+n+D_kϕ_k+1−C_kϕ_k

−exp(ϕk)· kⁿ+exp(−ϕk)· k^p−DOP=0, (19)

Gⁿ_kN_kⁿ₋₁+B_kⁿN_kⁿ_−n+H_kⁿN_kⁿ_+n+D_kⁿN_kⁿ₊₁−C_kⁿN_kⁿ+G(k)=0, (20) G^p_kP_kⁿ₋₁+B_k^pP_k^p_−n+H_k^pP_k^p_+n+D^p_kP_k^p₊₁−C_k^pP_k^p+G(k)=0, (21)

where

n=exp(−ϕ_n), (22)

p =exp(ϕ_p). (23)

In the above equations, B_k, H_k, D_k, G_k, C_k are constants depending only on the considered grid dimensions.

The next step is to linearize the discretized PDE equations. We may write the resulting system of algebraic equations as

G_kXk−1+B_kXk−n+D_kXk+1+H_kXk+n−C_kXk=Sk, (24) where X is a vector consisting of discrete values ofϕ, N andP, and S is the vector resulting from the source terms.

In the numerical solution, the linearized equations are then decoupled with the Gum- mel’s method. In this method, the equations are solved sequentially. First the Poisson equation is solved assuming fixed quasi-Fermi potentials. Then the new potential is sub- stituted into the continuity equations, which are linear and can be solved directly. The new free carrier concentrations are substituted back into the charge term of the Poisson equation and another iteration begins.

To solve the linear system (24), we use the Gauss–Seidel iterative method. The aim is to build a sequence of approximationsX⁽⁰⁾_k , X⁽¹⁾_k , ... , X_k⁽ⁿ⁾, that converges to the true solution X_k. For any initial approximationX_k⁰, the sequence{X_kⁱ}ⁿ_i=0of approximations defined by

X⁽ⁱ_k⁺¹⁾= 1

Ck S_k−

G_kX⁽ⁱ⁾_k−1+B_kX⁽ⁱ⁾_k−n+D_kX_k⁽ⁱ⁾₊₁+H_kX⁽ⁱ⁾_k+n

. (25)

This method assures a very fast convergence. We stop the calculation when two successive values ofX_kare nearly equal. We use an absolute convergence in the case of potential.

X^(m+1)_k −X_k^(m)≤ε (26)

and we use a relative convergence in the case of electrons and holes density:

X^(m_k ⁺¹⁾−X^(m)_k X_k^(m+1)

≤ε. (27) The stopping criterion for the iteration is fixed around 1e-9, for all computed physical quantities.

4. Numerical results

We applied our modelling methodology to simulate a 130 nm strained Si n LDD MOSFET (lightly doped drain n-type MOSFET) device with a thin gate oxide thickness of 3 nm and uniform channel doping of 1e 17 cm⁻³. The thickness of the strained Si layer is 13 nm and the Ge contentxin the pseudosubstrate varies from 0 to 30%. Note that the main parameters for device simulation are selected from those reported in an experimental work [21] (listed in table 1).

In figures 3a–3d, we show the results of macroscopic quantities for the strained MOS- FET with 4018 nodes established in the considered mesh. We assume applied biases of V_source =0 V, V_gate =1 V and V_drain =0.5 V. The surface concentrations in the source/drain region is about 1e 20 cm⁻³. The deep strained channel implantation exhibits a local doping maximum at about the same depth as the LDD regions. Figure 3b shows the electron density distribution calculated by SIBIDIF software. The surface concentra- tion of electron in the channel is relatively high because of the small threshold voltage.

Table 1. The parameters considered for the simulation of strained Si n LDD MOSFETs.

Parameter Value

Gate oxide thickness (T_ox)(nm) 3

Gate lengthL_g(nm) 130

Strained silicon film thickness (nm) 13

Channel doping (cm⁻³) 1 e 17

Source/drain doping (cm⁻³) 1 e 20

LDD source/drain doping (cm⁻³) 5 e 19

Si substrate doping (cm⁻³) 2 e 16

SiGe layer doping (cm⁻³) 1 e 18

Drain biasV_DS(V) 0.01–2

Gate bias,V_GS(V) 0.01–2

Ge mole fraction of SiGe layer,x 10–30%

Temperature (K) 300

Figure 3. Strained n MOSFET device. DDM results atV_DS=0.5 V andV_GS = 1 V. (a) Electrostatic potential distribution; (b) electron density; (c) longitudinal electric field distributions; (d) transverse electric field distributions.

Figure 3a presents the resulting electrostatic potential along the structure. As can be seen, in the highly doped source/drain and the LDD regions the potential appears constant while in the space zone of the reverse biased drain/substrate diode it falls monotonically.

We can observe only a very small barrier between the LDD-source and the channel. Also shown in figure 3a is the linear variation of the simulated potential, along the interface Si−SiO2. In figures 3c and 3d, we show the calculated longitudinal and transverse electric fields in the vertical direction of the strained n MOSFET channel. As can be observed, there is a significant difference between the two electric field distributions. Along the channel, obviously, the simulated longitudinal electric fieldE_xis constant and the conduc- tion system is purely resistive, then near the drain regionE_x reaches high values. Finally, noticeable transverse electric field Ey can be observed all along the channel, although mostly near the interface Si−SiO2. Figure 4 shows the transverse electric field (E_⊥) dependence of the electron mobilityμn for various Ge mole fractions (x =10, 20 and 30%) in the relaxed SiGe layers. The drain voltage (V_DS)is fixed at 0.1 V. We note that as the transverse electric field (E_⊥) increases, the electron mobilityμnin the inversion

Figure 4. Transverse electric field (E_⊥) dependence of electron mobility μ_n for various Ge mole fractions in the relaxed SiGe layer of strained n MOSFETs at V_DS=0.1 V.

Figure 5. The drain currentI_Dis plotted againtsV_Dfor conventional n MOSFTT and strained n MOSFETs for different Ge mole fractions.

layers decreases. This behaviour can be observed at strained and unstrained MOSFETs. It is clear that the strain of the Si layer which increases in proportion to the Ge mole fraction (x=10%, 20%) causes an increase in the electron mobility (up to a factor of 1.5 and 1.9, respectively), relative to the unstrained devices, over the entire range of transverse electric field(E_⊥). The drain current is plotted againstV_DS in figure 5. Clearly, the strained n MOSFET has a higher drain current and hence, higher speed. These results are in good agreement with those presented in [22,23].

5. Conclusion

In this work, we have developed an accurate finite difference solver ‘SIBIDIF’ for numerical simulation of two-dimensional drift diffusion equation (DDE) describing two- dimensional semiconductor devices, including the biaxially tensile strained n channel LDDMOSFET device. The implemented solution methodology mainly relies on finite difference mesh Gummel’s procedure and a Gauss–Seidel iterative method. We have proved that this approach assures a very fast convergence for each finite difference dis- cretized device equation. The simulation results demonstrate the superior capability of the SIBIDIF solver, which can be used to obtain accurate results with a given mesh. We have used the solver to calculate the current–voltage characteristics of the strained MOSFETs, as well as internal variables (electron concentration, electrostatic potential, electric fields, etc.). Our simulator predicted that the drive current of strained MOSFET is higher than that of conventional MOSFET. Likewise, in the inversion region, the strained MOSFET showed a maximum mobility enhancement up to a factor of 1.9 (for Ge mole fractionx = 20%) compared to the conventional devices.

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