A quantum energy transport model for semiconductor device simulation

Journal of Computational Physics 235 (2013) 486–496

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A quantum energy transport model for semiconductor device simulation Shohiro Sho a,⇑, Shinji Odanaka b a b

Graduate School of Information Science and Technology, Osaka University, Osaka, Japan Computer Assisted Science Division, Cybermedia Center, Osaka University, Osaka, Japan

a r t i c l e

i n f o

Article history: Received 1 February 2012 Received in revised form 7 September 2012 Accepted 30 October 2012 Available online 19 November 2012 Keywords: Simulation Numerical method Quantum energy transport model Semiconductor

a b s t r a c t This paper describes numerical methods for a quantum energy transport (QET) model in semiconductors, which is derived by using a diffusion scaling in the quantum hydrodynamic (QHD) model. We newly drive a four-moments QET model similar with a classical ET model. Space discretization is performed by a new set of unknown variables. Numerical stability and convergence are obtained by developing numerical schemes and an iterative solution method with a relaxation method. Numerical simulations of electron transport in a scaled MOSFET device are discussed. The QET model allows simulations of quantum confinement transport, and nonlocal and hot-carrier effects in scaled MOSFETs. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The semiconductor devices are scaled down into the nanoscale regime to achieve high circuit performance in the future integrated system. The performance of nanoscale semiconductor devices primarily relies on carrier transport properties in the short channels. Quantum energy transport (QET) models have been developed to understand such physical phenomena in scaled semiconductor devices. A full QET model has been derived from the collisional Wigner–Boltzmann equations using the entropy minimization principle [1]. Numerical simulations using this model, however, have not been performed [2]. Simplified QET models have been proposed as the energy transport extension of the quantum drift diffusion (QDD) model with Fourier law closure and numerically investigated [3,4]. In Ref. [4], the carrier temperature in the current density is further approximated by the lattice temperature to bring the model into a self-adjoint form. In this paper, we develop numerical methods for a QET model derived from a quantum hydrodynamic (QHD) model. To overcome the difficulties associated with the Fourier law closure, we newly derive a four-moments QET model similar with a classical energy transport (ET) model [5]. The numerical stability is achieved by developing numerical schemes and an iterative solution method in terms of a new set of variables. Numerical results in a scaled MOSFET are demonstrated. The paper is organized as follows: In Section 2, a four-moments QET model is derived from the QHD model. In Section 3, we present nonlinear discretization schemes and an iterative solution method to solve the QET system. In Section 4, numerical simulations of electron transport in a scaled MOSFET are discussed. Some conclusions are addressed in Section 5.

2. 4 Moments quantum energy transport model The QET models are obtained by using a diffusion scaling in the quantum hydrodynamic equations, similar as in the classical hydrodynamic model [5]. The QHD model has been derived from the collisional Wigner-Boltzmann equations, assuming ⇑ Corresponding author. E-mail address: [email protected] (S. Sho). 0021-9991/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcp.2012.10.051

S. Sho, S. Odanaka / Journal of Computational Physics 235 (2013) 486–496

487

Fourier law closure [6]. For classical hydrodynamic simulations, the closure relation based on the four-moments of the Boltzmann equation has been discussed [7–9], and the four-moments ET models are developed for simulations of thin body MOSFETs [5,10]. In this work, we derive a four-moments QET model from four moments equations derived from the collisional Wigner–Boltzmann equation. For simplicity, we consider only the case of electrons. The four moment equations have the same form as the classical hydrodynamic equations [7],

@ t n þ r ðnv Þ ¼ nC n ;

ð1Þ

@ t ðnpÞ þ r ðnUÞ nFE ¼ nC p ;

ð2Þ

@ t ðnwÞ þ r ðnSÞ nv FE ¼ nC ;

ð3Þ

r ðnRÞ nðwI þ UÞ FE ¼ nC p ;

ð4Þ

where n; p, and w are the electron density, momentum, and kinetic energy, respectively. v ; U; S and R are the velocity, second moment tensor, energy flow, and fourth moment tensor, respectively. I is the identity tensor. F E ¼ qE, where E is the electric field. C n ; C p ; C , and C p are the electron generation rate, the production of crystal momentum, the energy production, and the production of the energy flux, respectively. (1), (2), (3), and (4) represent conservation of particles, momentum, energy, and energy flux, respectively. By assuming parabolic bands, we give the following closure relations for p and U as

p ¼ mv ;

ð5Þ

Pij U ij ¼ mv i v j ; n

ð6Þ

where m is an effective mass. The quantum correction to the stress tensor Pij was proposed by Ancona and Iafrate [11], and the quantum correction to the energy density W ¼ nw was first derived by Wigner [12], which are given by 2

Pij ¼ nkT n dij þ

h @2 4 log n þ Oð h Þ; n 12m @xi @xj

ð7Þ

2

W¼

1 3 h @2 4 mnv 2 þ nkT n n 2 log n þ Oðh Þ; 2 2 24m @xk

ð8Þ

where T n and h are the electron temperature and Plank’s constant, respectively. For the collision terms, we employ a macroscopic relaxation time approximation to drive a QET model as follows:

C n ¼ 0; Cp ¼ C ¼

ð9Þ p

sp

ð10Þ

;

w w0

s

ð11Þ

;

where sp and s are the momentum and energy relaxation times, respectively. Substituting (5)–(7) into (1) and (2), we obtain moment equations for conservation of electron number and momentum

@n @ þ ðnv i Þ ¼ 0; @t @xi

ð12Þ 2

2

!

@ @ h @ ðmnv i Þ þ mnv i v j þ knT n log n n @t @xj 12m @xi @xj

¼ n

@V mnv i : @xi sp

ð13Þ

We further get the following relation:

@ @2 @ 1 @ 2 pffiffiffi pffiffiffi n: n log n ¼ 2n @xi @xi @xj @xj n @x2i

ð14Þ

With the relation (14), the quantum correction term in (13) is written as

! 2 2 h @ @2 h n @ 1 @ 2 pffiffiffi @ pffiffiffi 2 n ¼ qn n log n ¼ c; 6m @xi @xi n 12m @xi @xi @xj n @xj

ð15Þ

where the term 2

cn ¼

h 1 @ 2 pffiffiffi pffiffiffi 2 n 6mq n @xj

ð16Þ

488


is the quantum potential. Then, the conservation of momentum is given by

@ @ @ @V mnv i ðmnv i Þ þ ðmnv i v j þ knT n Þ qn c ¼ n : @t @xj @xi n @xi sp

ð17Þ

We can define the current density and the electric charge as J j ¼ qnv j and q is the positive electric charge. Using a diffusion scaling in (17), we obtain

sp where

@ @ @ @V J kln ðnT n Þ þ qnln c ¼ ln n Ji ; @t i @xi @xi n @xi

ð18Þ

ln ¼ þ qmsp is the electron mobility. The potential energy is given by V ¼ qu:

ð19Þ

From (12), (18) and (19), we obtain the current continuity equation as follows:

1 div Jn ¼ 0; q kT n nrðu þ cn Þ : J n ¼ qln r n q

ð20Þ ð21Þ

The energy balance equation is derived from (3) and (4) [7]. The collision term in (2) is rewritten as

Cp ¼

qv

ln

ð22Þ

:

In analogy to (22), the collision term in (4) is modeled as

C p ¼ where

qS

ls

ð23Þ

;

ls is the energy flow mobility. Neglecting the time derivative term in (2), we get nFE ¼ r ðnUÞ þ n

qv

ln

ð24Þ

:

Substituting (24) into (4), the expression of energy flux S is given as

S¼

ls l ðwI þ UÞ v þ s ððwI þ UÞ r ðnUÞ r ðnRÞÞ: ln qn

ð25Þ

Assuming a heated Maxwellian distribution, the fourth moment tensor R is specified by the classical form as

R¼

5 2 2 k T n I: 2

ð26Þ

Using closure (26), an expression for the energy flux density Sn ¼ nS is obtained as

Sn ¼

ls l 5 2 2 nk T n I ðWI þ nUÞ v þ s ðwI þ UÞ r ðnUÞ r 2 ln q

:

ð27Þ

The second term of (27) is the diffusive contributions to the energy flux density which includes the classical form of R. In this work, we develop a QET model, neglecting quantum corrections in the diffusive contributions to the energy flux density. Substituting (6)–(8) into (27), the quantum corrections to the energy density W and stress tensor P ij are included in the drift contributions to the energy flux density Sn and neglected in the diffusive contributions. As a result, we obtain a quantum corrected expression for the energy flux density as

!

2 2 l 5 kT n h h @2 l 5 k 2 Sn ¼ s D log n log n J n s qln nT n rT n : ln 2 q 24mq 12mq @xi @xj ln 2 q

ð28Þ

From (3), we get

3 2

r Sn ¼ Jn ru kn Assuming that the velocity

Tn TL

s

:

ð29Þ

v is slowly varying in the device region, the following term in (29) is approximated as

! ! ! 2 2 h @ @2 h @ @2 h @ v j @2 @ nv j log n ¼ vj n log n þ n log n J n c: @xj n 12m @xi @xi @xj 12m @xi @xi @xj 12m @xi @xi @xj 2

ð30Þ


489

Then, we obtain a four-moments QET model as follows:

Du ¼ qðn p CÞ;

ð31Þ

1 div J n ¼ 0; q kT n J n ¼ q ln r n nrðu þ cn Þ ; q bn r ðqn run Þ

ð33Þ

kT n q q un ¼ n ðu un Þ; q n 2 3 2

r Sn ¼ Jn ru kn Sn ¼

ð32Þ

Tn TL

s

ð34Þ ð35Þ

; !

2 ls 5 kT n h l 5 k 2 D log n cn Jn s qln nT n rT n ; ln 2 q 24mq ln 2 q

ð36Þ

where v n ¼ ðuþc2n un Þ and un ¼ kTqn v n . u; un , and p are the electrostatic potential, chemical potential, and hole density, respectively. qn is the the root-density of electrons. ; q, and k are the permittivity of semiconductor, electronic charge, and Boltzmann’s constant. C and T L are the ionized impurity density and the lattice temperature, respectively. The value of effective mass is given by a single parameter m ¼ 0:26m0 in the silicon devices, where m0 is the mass of a stationary electron. The quantum parameter for electrons becomes 2

bn ¼

h : 12qm

ð37Þ

For a temperature dependent mobility model, we apply the simplified Hänsch’s mobility model [5],

lðT n Þ 3 l0 k ¼ 1þ ðT n T L Þ 2 qs v 2s l0

1 ð38Þ

;

where l0 and v s are the low-field mobility and saturation velocity, respectively. From (16), the quantum potential equation is derived as

2bn r2 qn cn qn ¼ 0:

ð39Þ

In our model, (39) is replaced by (34) with respect to the variable un by employing an exponential transformation of variable pffiffiffi pffiffiffiffi qn ¼ n ¼ ni expðkTqn v n Þ [13]. If the variable un is uniformly bounded, the electron density is maintained to be positive. As mentioned below, this approach provides a numerical advantage for developing the iterative solution method of the QET model as well as the QDD model [13]. The system (31)–(36) are solved in the bounded domain X. The boundary @ X of the domain X splits into two disjoint part CD and CN . The contacts of semiconductor devices are modeled by the boundary conditions on CD , which fulfill charge neutrality and thermal equilibrium. We further assume that no quantum effects occur at the contacts. Here, the boundary conditions are given as follows:

u ¼ ub þ uappl ; n ¼ nD ; un ¼ uD ; T n ¼ T L on CD ; ru m ¼ rJn m ¼ run m ¼ rSn m ¼ 0 on CN ; where ub is a built-in potential and uappl is an applied bias voltage. uD ¼ positive constant at the silicon dioxide interface.

ð40Þ ð41Þ q ub kT L 2

on the contacts and un ¼ u0 , where u0 is a small

3. Discretization and iterative solution method 3.1. Discretization Space discretization of the four-moments QET model is performed by a new set of unknown variables (u; un ; n; T n ). For the current density, we have

kT n q kT n J n ¼ q ln r n n rðu þ cn Þ : kT n q q

ð42Þ

As pointed out in discretization of classical hydrodynamic models [17,18], the total energy flow H ¼ Sn þ uJ n , which consists of both the thermal energy flow Sn and the electrical flow uJ n , is used to solve the energy balance equation. The total energy flow can be rewritten as

490


H ¼ Sn þ u J n ¼ e Sn þ u þ 5 e Sn ¼ 2

ls h2 D log n þ cn ln 24mq

!! Jn ;

ð43Þ

2 ls kT n 5 ls k qln nT n rT n : Jn 2 ln q ln q

ð44Þ

Substituting (33) into (44), for the energy flow, we have

5 e Sn ¼ 2

ls kT n kT kT kT kT ql rn n n nrðu þ cn Þ þ n nr n ln n q q q q q

! 2 2 5 kT n q kT n ¼ qls rn n rðu þ cn Þ : 2 kT n q q

ð45Þ

When the variable n is defined as n ¼ n kTqn ¼ ng in the current density J n and n ¼ nðkTqn Þ2 ¼ ng2 in the energy flow e S n ; J n and e Sn can be written in the same form, similar as in the classical ET models [10,14],

r F ¼ r C rn

q nrðu þ cn Þ ; kT n

ð46Þ

where F is the flux. The constant C is defined as C ¼ qln in J n and C ¼ 52 qls in e S n . By projecting (46) onto a grid line and Rx using the variable g ¼ xi kTqn rðu þ cn Þ, a one-dimensional self-adjoint form is obtained as

d d d F ¼ ðCeg ðeg nÞÞ: dx dx dx

ð47Þ

For space discretization, the simulation region is divided into computational cells Xij centered at ðxi ; yj Þ. In a staggered Cartesian grid, each computational cell is rectangular, and the variables u; un ; n; T n are defined at cell centers and the flux is defined at cell interfaces. For space discretization of (47), we construct high-accuracy nonlinear schemes, applying the finite-volume method to construct multidimensional schemes. For the flux F ¼ Ceg rðeg nÞ, we integrate (47) over the computational cells Xij . Using Green’s theorem, we obtain a discrete form as

Z

Xij

r Fdx ¼ aj F iþ12 F i12 þ ai F jþ12 F j12 ;

ð48Þ

where ai and aj are the cell sizes of the computational cell Xij . In order to find F iþ1 at cell interfaces, integrating the flux F over 2 the interval ½xi ; xiþ1 , an approximation F iþ1 yields 2

F iþ1 2

Cðwiþ1;j wi;j Þ ; ¼ R xiþ1 g e dx xi

ð49Þ

where w ¼ eg n. A similar expression is obtained for F i1 ; F jþ1 , and F j1 . The accuracy of the numerical flux depends on the Rx 2 2 2 explicit integration xiiþ1 eg dx in (49). In order to construct a higher accuracy nonlinear scheme, an explicit integration R xiþ1 g e dx is obtained by the piecewise linear approximation of u and T n on the interval ½xi ; xiþ1 [15,16]. Then we have xi

F iþ1 2

! nxiþ1;j nxi;j C x x ; ¼ x x BðDiþ1 Þ BðDiþ1 Þ giþ1;j gi;j hiþ1 hiþ1 x

ð50Þ x

where BðÞ is the Bernoulli function. hiþ1 is defined as hiþ1 ¼ ðaxiþ1 þ axi Þ=2. The variables hxiþ1 ; Dxiþ1 are calculated as follows:

!

hxiþ1 ¼ log

giþ1;j =ðgiþ1;j gi;j Þ; gi;j

Dxiþ1 ¼ hxiþ1 ððuiþ1;j ui;j Þ þ ðcniþ1;j cni;j Þ ðgiþ1;j gi;j ÞÞ:

ð51Þ ð52Þ

Such schemes to J n and e S n result in a consistent generalization of the Scharfetter–Gummel type schemes to the QET equations. The energy balance equation is further discretized using (49). To conserve the total energy flow H ¼ Sn þ uJ n (43), discretization of the carrier heating term is another key issue [17,18]. Integrating (35) over the computational cell yields

Z

r eS n dx

Xij

¼

Z l bn 3 Tn TL kn J n r u þ s cn þ D log n dx dx: l 2 s Xij Xij 2 n

Z

ð53Þ

Here, quantum corrections are included in the carrier heating term. From Gauss’s theorem, the first term on the right hand side of (53) can be calculated as


Z l bn l bn J n r u þ s cn þ D log n dx ¼ J n u þ s cn þ D log n mdx: ln 2 ln 2 Xij @ Xij

Z

491

ð54Þ

Assuming the Boltzmann statics, the electron density is expressed as

n ¼ ni exp

qðu þ c un Þ ¼ ni expð2un Þ; kT n

ð55Þ

where ni is the intrinsic density. Then, the discretization for D log n ¼ 2Dun in (54) is obtained by a standard five-point approximation:

Dh uhn

! y y hjþ1 þ hj hxiþ1 þ hxi 1 1 1 1 ¼ y y ui;jþ1 þ y y ui;j1 þ x x uiþ1;j þ x x ui1;j ui;j : y y þ x x x aj hjþ1 a j hj ai hiþ1 ai hi ayj hjþ1 hj ai hiþ1 hi

ð56Þ

The discrete form of the carrier heating term in (53) yields

Z

ls l ðc þ b Dh uh Þ dx axi Jn 1 ujþ1 þ s cn 1 þ bn Dh uhn 2 jþ ln n n n ln jþ2 2 Xij l l l Jn 1 uj1 þ s cn 1 þ bn Dh uhn ayj Jn 1 uiþ1 þ s cn 1 þ bn Dh uhn Jn 1 ui1 þ s cn 1 þ bn Dh uhn : 2 2 2 j iþ i ln j2 ln iþ2 ln i2 2 2 2 Jn r

uþ

ð57Þ

Space discretization of (34) is performed following our previous works [13,19] to achieve a Scharfetter–Gummel type scheme, i.e.,

ayj x hiþ1

bn e

uniþ1;j

uni;j Þ

Bðuniþ1;j uni;j Þðuniþ1;j uni;j Þ ayj y

hj

uni;j

bn e

ayj x hi

uni;j

bn e

Bðuni;j uni1;j Þðuni;j uni1;j Þ þ

axi uni;jþ1 Bðuni;jþ1 uni;j Þðuni;jþ1 y bn e hjþ1

Bðuni;j uni;j1 Þðuni;j uni;j1 Þ gij unij Kij

1 ¼ ðuij unij ÞKij ; 2 where Kij ¼

R Xij

ð58Þ

qn dx, which is approximated as

! x y x y x y x y hi hj hiþ1 hj hi hjþ1 hiþ1 hj 1 unij Kij ¼ e ui1;j ui;j ui;j1 ui;j þ uiþ1;j ui;j ui;j1 ui;j þ ui1;j ui;j ui;jþ1 ui;j þ uiþ1;j ui;j ui;jþ1 ui;j : 4 B B B B B B B B 2 2 2 2 2 2 2 2

ð59Þ

3.2. Iterative solution method We develop an iterative solution method of the QET model by constructing a Gummel map [20] with a new set of unknown variables (u; un ; n; T n ) as follows: mþ1 (P1) Let um ; nm ; pm ; T m , n are given, solve the nonlinear Poisson equation with respect to the electrostatic potential u where m is the number of iteration. Eq. (31) is linearized using a Newton method. Then the linearized equation becomes

Dumþ1

q2 n m p q2 nm p umþ1 ¼ qðnm pm CÞ um : m þ m þ k Tn Tp k Tn Tp

ð60Þ

m mþ1 (P2) Let umþ1 ; Sm ; um . n ; T n are given, solve the potential un

mþ1 mþ1 bn r ðqm Þ gm qm ¼ n run n un

qmn 2

ðumþ1 um n Þ:

ð61Þ

Then, using umþ1 the quantum potential is further calculated as n mþ1 cmþ1 ¼ 2gm umþ1 þ um : n n n u mþ1 ; Tm n n

(P3) Let umþ1 ; c

ð62Þ

are given, solve the electron density nmþ1 .

1 div J n ¼ 0; q

ð63Þ

J n ¼ qln eg rðeg nmþ1 gm Þ:

ð64Þ

We set the generalized chemical potential by

umn ¼ gm log

nmþ1 þ umþ1 þ cmþ1 : n ni

ð65Þ

492

S. Sho, S. Odanaka / Journal of Computational Physics 235 (2013) 486–496 mþ1 (P4) Let umþ1 ; cmþ1 ; nmþ1 ; T m . n n are given, solve the electron temperature T n

3 nmþ1 T mþ1 n 2 s l 3 nmþ1 T L mþ1 ¼ J n r umþ1 þ s ðcmþ1 þ b D u Þ þ k : n n n 2 ln s

r eS n þ k

ð66Þ

An iterative solution method, which consists of the inner and outer iteration loops, is developed, as shown in Fig. 1. The algorithm using the variable un in (34) ensures the positivity of the root-density of electrons without introducing damping parameters [13]. In fact, it is a critical issue to solve for the root-density qn the quantum potential equation

2bn r2 qn þ cn qn ¼ 0:

ð67Þ

In this case, the iterative solution method requires an additional iteration loop to maintain positive solutions for the rootdensity of electrons in the inner iteration loop as pointed out in Ref. [21]. Hence, in the inner iteration loop, (67) is replaced by (34). Therefore, we can enhance the robustness of the iterative solution method by introducing an under relaxation method with a parameter a; 0 < a < 1, in the outer iteration loop:

T mþ1 ¼ T m þ aðT mþ1 T m Þ:

ð68Þ

The convergence behavior of electron temperature is shown in Fig.2 as a function of the relaxation parameter. It is clear that the numerical stability is obtained by the relaxation method. 4. Numerical results The numerical results are obtained for a 35 nm MOSFET having thin gate oxide thickness of 1.5 nm, uniform substrate concentration of 2:0 1018 cm3 , and n-type doping concentration of 1:0 1020 cm3 . The energy relaxation time s of 0:1 1012 ps and a ratio ls =ln of 0.8 are chosen. The MOSFET structure is shown in Fig. 3. The QET model includes a

Fig. 1. An iterative solution method with a relaxation algorithm.

493


100 alpha=0.2 alpha=0.5 alpha=0.8

RELATIVE ERROR

10 1 0.1 0.01 0.001 0.0001

0

5

10

15 20 25 30 35 NUBMER OF ITERATIONS

40

45

50

Fig. 2. Relative error of electron temperature vs. number of iterations at different relaxation parameters.

two-dimensional calculation of the electrostatic potential in the region with boundary A-G-L-F, and a two-dimensional calculation of the variables n; un , and T n in the silicon region with boundary A-B-E-F. The mixed boundary conditions for the QET system are assigned as follows: For the electrostatic potential u

u ¼ uappl þ ub ;

ð69Þ

at source and drain regions, and back gate, where uappl is the applied bias voltage, and ub is the built-in potential, respectively. The gate region is also treated as a Dirichlet boundary condition with an approximated work function of the material. At the sides A–B, H–I, J–K, E–F, we have the homogeneous Neumann condition

@u ¼ 0: @m

ð70Þ

For the variables n; un , and T n , we have the constant Dirichlet conditions

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < ðC þ C 2 þ 4n2i Þ=2 at sides B—C and D—E; n¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 2n2 =ðC þ C 2 þ 4n2 Þ at the back gate; i i

T n ¼ T L at sides B—C; D—E; and A—F; ðqub Þ=ð2kT n Þ at sides B—C; D—E; and A—F; un ¼ u0 at the silicon-oxide interface C—D;

Fig. 3. Two-dimensional cross section of a 35 nm MOSFET.

ð71Þ

494


where u0 is the small positive constant. At the sides A–B and E–F, the homogeneous Neumann conditions read:

@n @T n @un ¼ ¼ ¼ 0; @m @m @m

ð72Þ

at the side C–D,

@n @T n ¼ ¼ 0: @m @m

ð73Þ

In Fig. 4 and Fig. 5, we compare the electron density distributions calculated by QDD, QET and classical ET models. The device was biased with Vg = 0.8 V and Vd = 0.8 V. The simulated density distributions are plotted at different positions of the channel. Fig. 4 shows the electron density distributions perpendicular to the interface at the source end of the channel. The electron density distributions calculated from the QET and QDD models are almost identical in the inversion layers. Carrier heating due to the short channel effects results in the spread of electrons towards the bulk in simulations using the QET and ET models. As a result, the profiles between two models are almost identical at the bulk. The electron density distributions

ELECTRON DENSITY(cm-3)

1e+020 QET ET QDD

1e+018

1e+016

1e+014

1e+012

1e+010

0

0.005

0.01 DEAPTH(µm)

0.015

0.02

Fig. 4. Electron density distributions perpendicular to the interface at the source end of the channel for a 35 nm MOSFET.

ELECTRON DENSITY(cm-3)

1e+020 QET ET QDD

1e+018

1e+016

1e+014

1e+012

1e+010

0

0.005

0.01 DEAPTH(µm)

0.015

0.02

Fig. 5. Electron density distributions perpendicular to the interface at the drain end of the channel for a 35 nm MOSFET.

495


ELECTRON TEMPERATURE(k)

3000

ET vg=1.2V QET vg=1.2V QCET vg=1.2V

2500 2000 1500 1000 500 0 0

0.02

0.04 0.06 0.08 0.1 LATERAL POSITION(µm)

0.12

0.14

Fig. 6. Lateral profiles of electron temperature distributions calculated by ET (solid line), QET, and QCET models at the same drain bias of Vd = 0.8 V and the same gate bias of Vg = 1.2 V.

ELECTRON TEMPERATURE(k)

1400 ET vg=1.2V QET vg=1.6V

1200 1000 800 600 400 200

0

0.02

0.04 0.06 0.08 0.1 LATERAL POSITION(µm)

0.12

0.14

Fig. 7. Lateral profiles of electron temperature distributions calculated by ET (solid line) and QET (dotted line) models at the same drain bias of Vd = 0.8 V. ET model at Vg = 1.2 V, QET model at Vg = 1.6 V.

perpendicular to the interface at the drain end of the channel are shown in Fig. 5. The results clearly indicate that the quantum confinement effect is reduced by the enhanced diffusion towards the bulk due to the high electron temperature near the drain. The QET model allows simulations of quantum confinement transport with hot-carrier effects in MOSFETs. Fig. 6 shows lateral profiles of electron temperature calculated by the QET, QCET, and ET models at the same gate voltage of 1.2 V. In Fig. 7, we compare the results calculated by the ET model at Vg = 1.2 V and the QET model at Vg = 1.6 V. The simulations are done at the same drain voltage of 0.8 V. The quantum corrected ET (QCET) model is a simplified QET model based on [4] with a temperature dependent mobility model (38). In the QCET model, the quantum correction to the energy density is neglected, and the carrier temperature in the current density is approximated by the lattice temperature [4]. As shown in Fig. 6, the QET model exhibits a sharper distribution of electron temperature at the lateral direction, when compared to that calculated by the classical ET model. The electron temperature calculated by the QCET model is further increased. This difference is caused by the threshold voltage shift due to the quantum confinement transport in the channel. Therefore, as shown in Fig. 7, the shape of electron temperature distributions calculated by the QET model at Vg = 1.6 V is close to that obtained by the ET model at Vg = 1.2 V. In Fig. 8, we present the x-component of the current densities calculated by the QET and ET models. The results verify that the magnitude of the current density calculated by the QET model at Vg = 1.6 V corresponds to that calculated by the ET model at Vg = 1.2 V.

496


-1.2e+008

2

CURRENT DENSITY Jx(A/cm )

ET vg=1.2V -1e+008

QET vg=1.6V QET vg=1.2V

-8e+007 -6e+007 -4e+007 -2e+007

0

0

0.005

0.01 DEAPTH( µm)

0.015

0.02

Fig. 8. x-Component of current densities perpendicular to the interface for a 35 nm MOSFET. ET model at Vg = 1.2 V, QET model at Vg = 1.2 V and Vg = 1.6 V.

5. Conclusion A four-moments QET model has been derived by using a diffusion scaling in the quantum hydrodynamic model. Space discretization of the four-moments QET model has been performed by a new set of unknown variables. Numerical schemes result in a consistent generalization of the Scharfetter-Gummel type scheme to the QET equations. We can enhance the robustness of the iterative solution method by introducing a relaxation method. The QET model allows simulations of quantum confinement transport with hot-carrier effects in scaled MOSFETs. The simulation results reveal the difference of electron temperature distributions between the QET and ET models due to the quantum confinement effects. Acknowledgement The authors thank Dr. Shimada for numerical simulations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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