Analytical model for quasi-ballistic transport in MOSFET including carrier backscattering

Abstract

In this paper, a charge-based analytical model is proposed for double-gate MOSFETs working in the quasi-ballistic regime. The model includes both Lundstrom backscattering theory and conventional drift–diffusion theory. Both the theories are used to model the charge density along channel length, which are used to solve Poisson's equation to get the variation of the channel potential. To compute the ballistic segments and diffusive segments of the current, the calculated charge density and surface potential are used. The model is validated with reported numerical data of different channel length and found to be accurate.

Appendices

Appendix 1

A charge-based approach has been used in our work for calculating the charge density, surface potential and current. The model is based on a simple scattering picture as shown in Fig.

Fig. 10

A simple representation of the proposed model for evaluating the charge density along the channel. At each λ interval (mean free path length), a specific fraction (S), of approaching carrier gets scattered and a fraction r of scattered carrier gets backscattered and the rest move toward drain

10. where it is assumed that the scattering will take place at every λ (mean free path length) interval and a fraction S (scattering parameter) of total carrier will be scattered; also a fraction r (backscattering coefficient) of scattered carrier will be backscattered in move toward source. The parameters λ and S depend on channel material properties and doping concentration [11].

First consider the charge density is injecting from source end only, where N_SOURCE is the total charge density that is injecting from the source end. It is expected that carrier will scatter at every λ interval, after scattering a fraction part S of the total approaching carrier will get scattered [11]; also a fraction r of scattered carrier get backscattered [2]. This event happened at each point (point × 1, × 2, × 3…) separated by λ interval.

1.1 1st scattering (λ distance from source)

Charge density injected from source end (N_SOURCE) approach at point x₁ and scattered, after 1st scattering (at point x₁) a fraction S of the total carrier will be scattered and rest move without scattering.

• At point x₁ total approaching carrier = N_SOURCE

• At point x₁ total scattered carrier = S × N_SOURCE

• At point x₁ total back scattered carrier = r × (S × N_SOURCE)

• At point x₁ total unscattered carrier = N_SOURCE—S × N_SOURCE = (1 − S) N_SOURCE

1.2 2nd scattering (2λ distance from source)

Total carrier which remains unscattered at point x₁ will reach at point x₂ where they will be scattered again.

• At point x₂ total approaching carrier = (1 − S) N_SOURCE

• At point x₂ total scattered carrier = S × (1 − S) N_SOURCE

• At point x₂ total back scattered carrier = r × (S × (1 − S) N_SOURCE)

• At point x₂ total unscattered carrier = (1 − S) N_SOURCE—S × (1 − S) N_SOURCE = (1 − S)² N_SOURCE

1.3 3rd scattering (3λ distance from source)

Similarly the total carrier which remain unscattered at point x₂ will reach at point x₃ where they will be scattered again.

• At point x₃ total approaching carrier = (1 − S)² N_SOURCE

• At point x₃ total scattered carrier = S × (1 − S)² N_SOURCE

• At point x₃ total back scattered carrier = r × (S × (1 − S)² N_SOURCE)

• At point x₃ total unscattered carrier = (1 − S)² N_SOURCE—S × (1 − S)² N_SOURCE = (1 − S)³ N_SOURCE

1.4 mth scattering (at x = mλ distance from source)

Similarly the charge density at x_x (or x distance from source) can be obtained as.

• At point x_x total approaching carrier = (1 − S)^m−1 N_SOURCE.

• At point x_x total scattered carrier = S × (1 − S)^m−1 N_SOURCE.

• At point x_x total back scattered carrier = r × (S × (1 − S)^m−1 N_SOURCE).

• At point x_x total unscattered carrier = (1 − S)^m−1 N_SOURCE − S × (1 − S)^m−1 N_SOURCE = (1 − S)^m N_SOURCE.

From the above approach, the charge density of unscattered (ballistic) carrier injected from source at distance x is obtained as

$$N_{B - {\rm SOURCE}} (x) = (1 - S)^{m} N_{\rm SOURCE} = (1 - S)^{x/\lambda } N_{\rm SOURCE}$$

(25)

This Eq. (25) is similar to Eq. (1).

By putting x = L into (25), total unscattered carrier reached at drain end (which contribute in ballistic current) can be obtained as:

$$N_{B - {\rm SOURCE}} = (1 - S)^{L/\lambda } N_{\rm SOURCE}$$

(26)

Similarly scattered charge density at distance x is obtained as

$$N_{DD - {\rm SOURCE}} (x) = S(1 - S)^{m - 1} N_{\rm SOURCE} = (1 - S)^{{\frac{x}{\lambda } - 1}} N_{\rm SOURCE}$$

(27)

Device ballisticity Device ballisticity (bal) can be defined as the ratio of total charge which reached at drain side (N_B-SOURCE) without being scattered to the total charge which injects into channel (N_SOURCE) and can be obtained as

$$bal = \frac{{N_{B - {\rm SOURCE}} }}{{N_{\rm SOURCE} }} = \frac{{(1 - S)^{L/\lambda } N_{\rm SOURCE} }}{{N_{\rm SOURCE} }}$$

(28)

From (4), scattering parameter (S) can be obtained in term of device ballisticity (bal) as

$$S = 1 - bal^{\lambda /L}$$

(29)

From (5), for a known device (known ballisticity), the scattering parameter (S) can be obtained and used in model.

For the low drain-to-source bias, there will be carrier injection from the drain side to source side [3], so the same approach can be applied to the drain side injected carrier.

Appendix 2

Figure 2 shows a schematic of the n channel double-gate MOSFET. The Poisson’s equation governing the potential in channel (φ(x)) can be expressed as

$$\frac{{\partial^{2} \phi \left( {x,y} \right)}}{{\partial x^{2} }} + \frac{{\partial^{2} \phi \left( {x,y} \right)}}{{\partial y^{2} }} = \frac{q}{{\varepsilon_{\rm sub} }}\left( {N_{a} + n(x)} \right)$$

(30)

where x is the distance along the channel length, y is the distance along the channel thickness, ε_sub is the permittivity of silicon, N_a is the channel doping, n(x) is the mobile charge density, and q is intrinsic charge.

The solution of (30) is realized by invoking the parabolic potential approximation [17] as

$$\phi \left( {x,y} \right) = a_{11} \left( x \right) + a_{21} \left( x \right)\,y + a_{31} \left( x \right)\,y^{2}$$

(31)

The boundary conditions for evaluating the coefficients a₁₁(x), a₂₁(x) and a₃₁(x) are as follows.

• (i) Same values of surface potential (φ_S(x)) at front (y = T_sub/2) and back (y = -T_sub/2) interfaces due to symmetric operation

  $$\phi \left( {x,y = T_{\rm sub} /2} \right) = \phi \left( {x,y = - T_{\rm sub} /2} \right) = \phi_{S} \left( x \right)$$

  (32)

• (ii) Boundary condition at center of the channel can be expressed as

  $$\left. {\frac{d\phi (x,y)}{{dy}}} \right|_{y = 0} = 0$$

  (33)
  $$\phi \left( {x,y = 0} \right) = \phi_{C} \left( x \right)$$

  (34)

• (iii) Electric fields at front and back interfaces can be expressed as

  $$\left. {\frac{{\partial \phi \left( {x,y} \right)}}{\partial y}} \right|_{{y = - T_{\rm sub} /2}} = \frac{{\varepsilon_{ox} \,}}{{\varepsilon_{\rm sub} T_{ox} }}\left( {\phi_{S} \left( x \right) - (V_{g} - V_{fb} )} \right)$$

  (35)
  $$\left. {\frac{{\partial \phi \left( {x,y} \right)}}{\partial y}} \right|_{{y = T_{\rm sub} /2}} = \frac{{\varepsilon_{ox} \,}}{{\varepsilon_{\rm sub} T_{ox} }}\left( {(V_{g} - V_{fb} ) - \phi_{S} \left( x \right)} \right)$$

  (36)

  where φ_S(x) is the surface potential, V_g is the gate voltage, V_fb is the flat-band voltage, ε_sub is the permittivity of silicon, T_sub is the channel thickness, ε_ox is the permittivity of the oxide layer, and T_ox is the thickness of oxide layer. Using (32)–(36), the coefficients a₁₁(x), a₂₁(x) and a₃₁(x) can determined and (31) can be written as

  $$\phi \left( {x,y} \right) = \phi_{C} \left( x \right) - \frac{{\varepsilon_{ox\,} \,\left( {\phi_{S} \left( x \right) - (V_{g} - V_{fb} )} \right)}}{{\varepsilon_{\rm sub} T_{\rm sub} T_{ox} }}{\kern 1pt} y^{2}$$

  (37)

Therefore, putting the (37) into (30) the Poisson’s equation is simplified in terms of the center potential as

$$\frac{{\partial^{2} \phi_{C} \left( x \right)}}{{\partial x^{2} }} + \frac{1}{\varepsilon }\left( {V_{g} - V_{fb} - \phi_{C} \left( x \right)} \right) = \frac{{q\,(N_{ch} + n(x))}}{{\,\varepsilon_{\rm sub} }}$$

(38)

where ε is the square of natural/characteristic length of DG transistor, and is given as

$$\varepsilon = \frac{{\varepsilon_{\rm sub} T_{ox} T_{\rm sub} }}{{2\,\varepsilon_{ox} }}\left( {1 + \frac{{\varepsilon_{ox} \,T_{\rm sub\,} }}{{4\,\varepsilon_{\rm sub} T_{ox} }}} \right)$$