CN104076266A - Method for extracting subthreshold swing of MOSFET of double-material double-gate structure - Google Patents

Abstract

The invention belongs to the technical field of semiconductors, and particularly discloses a method for extracting the subthreshold swing of an MOSFET of a double-material double-gate structure. Potential distribution of the MOSFET of the double-material double-gate structure is acquired, and then an analysis model of the subthershold swing is acquired through acquired potentials according to the definition of the subthreshold swing. The analysis model of the subthreshold swing is simple in form and clear in physical concept, and a rapid tool is provided for circuit stimulation software on research on a novel device of the double-material double-gate structure.

Description

A kind of method of extracting two material double-gate structure MOSFET subthreshold swings

Technical field

The invention belongs to technical field of semiconductors, be specifically related to a kind of method of extracting two material double-gate structure metal-oxide semiconductor fieldeffect transistor (MOSFET) subthreshold swings.

Background technology

Along with integrated circuit (IC) chip integrated level improves constantly, device geometries is constantly dwindled, and MOSFET device progressively develops from planar structure to on-plane surface spatial structure.And in all kinds of non-traditional planer device structures, the grid control ability of double-gate structure MOSFET is strong, can better suppress short-channel effect, reduce the power consumption that waits for quietly of device.Two material gate MOS FET are combined with double grids MOSFET, just can be in conjunction with both advantages, make device have better short-channel properties and performance.Owing to using the less material of work function to do grid near drain terminal, can reduce flat-band voltage, increase effective grid voltage, can reduce drain terminal along channel direction electric field, thereby reduce hot carrier's effect.Due to above advantage, this double grids MOSFET Structure Creating analytic model is become to particularly important, and its subthreshold swing extraction model receives industry member concern day by day.Tradition bulk silicon MOSFETs subthreshold swing model is no longer applicable, and this modeling and simulation for novel multiple-grid nano-device has brought new challenge.

Subthreshold swing is MOSFET one of important parameter the most, and it is defined as: subthreshold value region, ten times of the every variations of electric current, the required variable quantity of grid bias.In order to use correctly mimic channel characteristic of circuit simulation software, it is very important setting up accurate subthreshold swing model.

Summary of the invention

The object of the invention is to provide a kind of convenience, the correct method of extracting two material double grids MOSFET subthreshold swings.

The method of the two material double grids MOSFET subthreshold swings of extraction provided by the invention, key is to set up that form is succinct, clear physics conception, and the high two material double grids MOSFET subthreshold swing analytic models of precision.

Two material double-gate structure MOSFET subthreshold swing analytic models that the present invention sets up, for circuit simulation software provides a kind of accurate Analysis double-gate structure model fast.

Concrete steps are as follows:

(1) set up two material double grids MOSFETs

Two material double grids MOSFETs and double grids MOSFET are similar, and centre is silicon.Raceway groove adopts p-type doping, and source is leaked and adopted N-shaped heavy doping.Grid adopts asymmetric structure, and one of them grid adopts two kinds of material preparations that work function is different.For the groove potential that obtains notch cuttype distributes, material M2 adopts the less N-shaped heavily doped polysilicon (work function is 4.17eV) of work function, and material M1 adopts the p-type polysilicon that work function is higher (work function is 5.25eV).The external same bias voltage of two ends grid.

Channel length is that the grid of L one end is divided into two parts, respectively corresponding two kinds of grid materials that work function is different.The length that material M1 is corresponding is L ₁, the length that material M2 is corresponding is L ₂, L=L ₁+ L ₂.The work function of material M1 is 5.25eV, and the work function of material M2 is 4.17eV.T _ox1, t _ox2for front grid and back of the body gate oxide thickness, t _sifor channel thickness.

(2) solve the Poisson equation of groove potential, obtain groove potential

The Poisson equation of groove potential can be expressed as:

$\frac{{\∂}^2\mathrm{\φ}(x,y)}{{\∂x}^2}+\frac{{\∂}^2\mathrm{\φ}(x,y)}{\∂y^2}=\frac{q{N}_A}{{\mathrm{\ϵ}}_{\mathrm{si}}}---\left(1\right)$

Wherein q is electron charge, N _afor the doping content of raceway groove, ε _sifor the specific inductive capacity of silicon, φ (x, y) is groove potential.

According to electric field at raceway groove, oxide layer interface continuously and the voltage at source, leakage two ends, boundary condition can be expressed as:

φ(x＝0,y)＝V _S (2)

φ(x＝L,y)＝V _S+V _DS (3)

$\mathrm{\φ}(x,y=0)=V_{\mathrm{GFF}}+\frac{{\mathrm{\ϵ}}_{\mathrm{si}}}{C_{\mathrm{ox}1}}\frac{\∂\mathrm{\φ}}{\∂y}{|}_{y=0}---\left(4\right)$

$\mathrm{\φ}(x,y=t_{\mathrm{si}})=V_{\mathrm{GFB}}+\frac{{\mathrm{\ϵ}}_{\mathrm{si}}}{C_{\mathrm{ox}2}}\frac{\∂\mathrm{\φ}}{\∂y}{|}_{y=t_{\mathrm{si}}}---\left(5\right)$

Wherein, VS is Built-in potential, and VDS is drain-source voltage, and COX1, COX2 are respectively grid and lower gate oxide unit-area capacitance, V _gFF, V _gFBbe respectively grid and the effective grid voltage of lower grid.

The grid of double-gate materials adopts two kinds of different materials of work function to form.Because effective grid voltage expression formula is V _gFF=V _gS-V _fBF, V _fBFfor front grid flat-band voltage, and V _fBFthe different value of correspondence in bi-material, the therefore V in (4) formula _gFFshould be a piecewise function.If definition V _fBF1and V _fBF2be respectively the flat-band voltage of bi-material, so V _gFFjust can be expressed as V _gFF=V _gS-V _fBF1(0<x<L ₁), V _gFF=V _gS-V _fBF2(L ₁<x<L ₁+ L ₂).V _fBF1and V _fBF2can calculate by the work function of material.Definition r ₁=V _gFB/ V _gFF, r ₂=C _ox2/ C _ox1, and substitution formula (5), boundary condition can be expressed as again:

$\mathrm{\&phi;}(x,y=t_{\mathrm{si}})=r_1{V}_{\mathrm{GFF}}-\frac{{\mathrm{\&epsiv;}}_{\mathrm{si}}}{r_2{C}_{\mathrm{ox}1}}\frac{\&PartialD;\mathrm{\&phi;}}{\&PartialD;y}{|}_{y=t_{\mathrm{si}}---\left(6\right)}$

The form of separating can be expressed as:

$\mathrm{\&phi;}(x,y)=V_S+\frac{V_{\mathrm{DS}}}{L}x+\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{A}_n\left(y\right)\sin \frac{\mathrm{n\&pi;x}}{L}---\left(7\right)$

Wherein n is integer, and An is undetermined coefficient.

By formula (7) substitution Poisson equation (1), can obtain:

$\frac{d^2{A}_n\left(y\right)}{{\mathrm{dy}}^2}-k_n^2{A}_n\left(y\right)=f_n---\left(8\right)$

Wherein k _n=n π/L, f _n=(2qN _a/ n π ε _si) [1-(1) ⁿ].Formula (8) is an ordinary differential equation, Xie Wei:

$A_n\left(y\right)=C_n{e}^{k_{n}y}+D_n{e}^{-k_{n}y}-f_n/k_n^2---\left(9\right)$

By formula (9) substitution boundary condition (4) and (6), then make Fourier expansion, can further obtain coefficient C _nwith D _nbe respectively:

$C_n=\frac{C_{\mathrm{ox}1}[({\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n-r_2{C}_{\mathrm{ox}1})(f_n-G_n{k}_n^2)+r_2(C_{\mathrm{ox}1}+{\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n)(f_n-H_n{k}_n^2)e^{k_n{t}_{\mathrm{si}}}]}{k_n^2[C_{\mathrm{ox}1}{\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n(1+r_2)(1+e^{{2k}_n{t}_{\mathrm{Si}}})-(r_2{C}_{\mathrm{ox}1}^2+{\mathrm{\&epsiv;}}_{\mathrm{si}}^2{k}_n^2)(1-e^{{2k}_n{t}_{\mathrm{Si}}})]}---\left(10\right)$

And

$C_n=C_{\mathrm{ox}1}{e}^{k_n{t}_{\mathrm{si}}}\frac{[({\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n+r_2{C}_{\mathrm{ox}1})(f_n-G_n{k}_n^2)e^{k_n{t}_{\mathrm{si}}}-r_2(C_{\mathrm{ox}1}-{\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n)(f_n-H_n{k}_n^2)]}{k_n^2[C_{\mathrm{ox}1}{\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n(1+r_2)(1+e^{{2k}_n{t}_{\mathrm{Si}}})-(r_2{C}_{\mathrm{ox}1}^2+{\mathrm{\&epsiv;}}_{\mathrm{si}}^2{k}_n^2)(1-e^{{2k}_n{t}_{\mathrm{Si}}})]}---\left(11\right)$

Wherein G _n, H _nbe respectively

$G_n=\frac{2}{\mathrm{n\&pi;}}[V_S(1-{(-1)}^n)-V_{\mathrm{GFF}1}(1-\cos \frac{\mathrm{n\&pi;}{L}_1}{L})-V_{\mathrm{GFF}2}(\cos \frac{\mathrm{n\&pi;}{L}_1}{L}-{(-1)}^n)-{(-1)}^n{V}_{\mathrm{DS}}]---\left(12\right)$

$H_n=\frac{2}{\mathrm{n\&pi;}}[(V_S-r_1{V}_{\mathrm{GFF}})(1-{(-1)}^n)-{(-1)}^n{V}_{\mathrm{DS}}]---\left(13\right)$

So just can obtain the groove potential expression formula of two material double grids.If introduce the material of more different work functions, can solve by similar method so.

(3) the sub-threshold values amplitude of oscillation analytical expression of the two material double grids MOSFETs of foundation

Subthreshold swing is defined as:

$S=\frac{\&PartialD;V_{\mathrm{GS}}}{\&PartialD;{\log }_{10}{I}_{\mathrm{DS}}}---\left(14\right)$

In subthreshold value region, two material double grids MOSFETs are under weak transoid condition, and source-drain current is pressed and can be approximated by:

I _ds∝n _min(y) (15)

N _min(y) can be expressed as $n_{\mathrm{miin}}\left(y\right)=(n_i^2/N_A)e^{q{\mathrm{\&phi;}}_{\min }\left(y\right)/\mathrm{kT}}.$ φ _minfor the electromotive force minimum value of x direction.Substitution formula (15) afterwards, can be rewritten as the expression formula of subthreshold swing again:

$S=2.3V_t{\left[\frac{\&PartialD;\mathrm{\&phi;}(x_0,y)}{\&PartialD;V_{\mathrm{GS}}}\right]}^{-1}---\left(16\right)$

Wherein V _t=k _bt/q, k _bfor Boltzmann constant, T is temperature.

By result substitution (16) formula of front portion groove potential, the analytical expression that can obtain subthreshold swing is:

$\begin{array}{c}S=2.3V_t\{\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{2}{\mathrm{n\&pi;}}\sin \left(\frac{\mathrm{n\&pi;}{x}_0}{L}\right)(1-{(-1)}^n)\\ \frac{C_{\mathrm{ox}1}[e^{k_{n}y}({\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n-r_2{C}_{\mathrm{ox}1})+e^{k_n(2t_{\mathrm{si}}-y)}({\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n+r_2{C}_{\mathrm{ox}1})}{C_{\mathrm{ox}1}{\mathrm{\&epsiv;}}_{\mathrm{si}}{k}_n(1+r_2)(1+e^{2k_n{t}_{\mathrm{Si}}})-(r_2{C}_{\mathrm{ox}1}^2+{\mathrm{\&epsiv;}}_{\mathrm{si}}^2{k}_n^2)(1-e^{{2k}_n{t}_{\mathrm{Si}}})}{\}}^{-1}\end{array}---\left(17\right)$

When calculating, y can use effective conductive path y of raceway groove _effsubstitute, the namely equivalent barycenter of electric charge,

$y_{\mathrm{eff}}={\&Integral;}_0^{t_{\mathrm{si}}}\mathrm{yn}(x_0,y)\mathrm{dy}/{\&Integral;}_0^{t_{\mathrm{si}}}n(x_0,y)\mathrm{dy}---\left(18\right)$

Wherein, n _ifor intrinsic carrier concentration.Y _effexpression formula can be rewritten as

$y_{\mathrm{eff}}={\&Integral;}_0^{t_{\mathrm{si}}}{\mathrm{ye}}^{\mathrm{\&phi;}(x_0,y)/V_t}\mathrm{dy}/{\&Integral;}_0^{t_{\mathrm{si}}}{e}^{\mathrm{\&phi;}(x_0,y)/V_t}\mathrm{dy}---\left(19\right)$

Like this, just obtain the subthreshold swing analytic model of two material double grids MOSFETs.

Brief description of the drawings

The two material double grids MOSFET two-dimensional structure schematic diagram of Fig. 1.

Fig. 2 electromotive force is in the distribution situation of channel surface.

Fig. 3 in the time of different body silicon thicknesses, the variation relation of two material double grids MOSFET subthreshold swings and channel length.

Fig. 4 in the time of different oxidated layer thickness, the variation relation of two material double grids MOSFET subthreshold swings and channel length.

Fig. 5 subthreshold swing modeling schematic flow sheet.

Embodiment

Analytic model by us calculates, and as shown in Figure 3, the present invention is to two material double grids MOSFET subthreshold swings analysis result with changes in channel length under consubstantiality silicon thickness not.Fig. 4 is two material double grids MOSFET subthreshold swings analysis results with changes in channel length under different gate oxide thickness.The same with common double grids MOSFET, the subthreshold swing of two material double grids MOSFETs reduces with channel length equally.Meanwhile, the increase of gate oxide and the increase of channel thickness, all can make the Sub-Threshold Characteristic variation of raceway groove, and subthreshold swing increases.

According to our analytic model, can the two material double grids MOSFET subthreshold swings of very convenient, accurate extraction.

Claims (1)

1. a method of extracting two material double-gate structure MOSFET subthreshold swings, is characterized in that concrete steps are as follows:

(1) set up two material double grids MOSFETs

Two material double grids MOSFETs, centre is silicon, and raceway groove adopts p-type doping, and source is leaked and is adopted N-shaped heavy doping; Grid adopts asymmetric structure, and one of them grid adopts two kinds of material preparations that work function is different; The second material M2 adopts the less N-shaped heavily doped polysilicon of work function, and the first material M1 adopts the p-type polysilicon that work function is higher; The external same bias voltage of two ends grid;

Channel length is lthe grid of one end is divided into two parts, respectively corresponding two kinds of grid materials that work function is different; The length that the first material M1 is corresponding is l ₁ , the length that the second material M2 is corresponding is l ₂ , L=L ₁ + L ₂ ; The work function of the first material M1 is 5.25eV, and the work function of the second material M2 is 4.17eV; t _ox1 , t _ox2 for front grid and back of the body gate oxide thickness, t _si for channel thickness;

(2) solve the Poisson equation of groove potential, obtain groove potential

The Poisson equation of groove potential is expressed as:

（1）

Wherein, for electron charge, for the doping content of raceway groove, for the specific inductive capacity of silicon, for groove potential;

According to electric field at raceway groove, oxide layer interface continuously and the voltage at source, leakage two ends, boundary condition is expressed as:

（2）

（3）

（4）

（5）

Wherein, vSfor Built-in potential, vDSfor drain-source voltage, cOX1, cOX2be respectively grid and lower gate oxide unit-area capacitance, v _gFF , v _gFB be respectively grid and the effective grid voltage of lower grid; The grid of double-gate materials adopts two kinds of different materials of work function to form, in (4) formula v _gFF it is a piecewise function; Definition , , and substitution formula (5), boundary condition is expressed as again:

（6）

The form of separating is expressed as:

（7）

Wherein nfor integer, anfor undetermined coefficient;

By formula (7) substitution Poisson equation (1), obtain:

（8）

Wherein, , ;

The solution of formula (8) is:

（9）

By formula (9) substitution boundary condition (4) and (6), then make Fourier expansion, obtain coefficient with for:

（10）

And

（11）

Wherein, , be respectively

（12）

（13）

Like this, obtain the groove potential expression formula of two material double grids;

(3) the sub-threshold values amplitude of oscillation analytical expression of the two material double grids MOSFETs of foundation

Subthreshold swing is defined as:

（14）

In subthreshold value region, two material double grids MOSFETs are under weak transoid condition, and source-drain current is pressed and is approximated by:

（15）

be expressed as , for xthe electromotive force minimum value of direction; Substitution formula (15), is rewritten as the expression formula of subthreshold swing:

（16）

Wherein, , for Boltzmann constant, for temperature;

By result substitution (16) formula of groove potential, the analytical expression that obtains subthreshold swing is:

（17）

When calculating, effective conductive path of available raceway groove substitute the namely equivalent barycenter of electric charge:

（18）

Wherein, , for intrinsic carrier concentration, expression formula be rewritten as:

（19）。