Strain-induced, off-diagonal, same-atom parameters in empirical tight-binding theory suitable for [110] uniaxial strain applied to a silicon parametrization

State-of-the-art transistors achieve their improved performance through strain engineering. The somewhat unusual uniaxial [110] strain is of particular importance as it provides a significant mobility increase for electrons. Empirical tight binding has shown tremendous benefits in modeling realistically large structures including standard strain conditions, but often fails to predict the correct uniaxial [110] strain behavior because most treatments neglect the same-atom different-orbital matrix elements induced by this strain. Two separate mechanisms are responsible for these conditions: Löwdin orbital changes and displacement of nearest-neighbor potentials. We present a model which separately includes both mechanisms via parameters whose range of validity can be independently determined. Using this method we optimize a set of strain parameters for Si. The combination of both effects is able to reproduce the $\text{Si}{X}_z$-valley transverse mass splitting under uniaxial [110] strain. We then use this model to calculate the drain current of a strained double-gate, ultrathin-body metal-oxide-semiconductor field-effect transistor, finding experimentally plausible results.

Figure 1

Conduction- and valence-band edges of Si under uniaxial [001] strain as reproduced by this model (lines) and as predicted by the model solid theory of Ref. 20 (symbols).

Figure 2

Conduction-band masses of Si under uniaxial [001] strain as reproduced by this model ($X_{x,y}$: solid lines and $X_z$: dotted lines) and as predicted by VASP (Ref. 21) ($X_{x,y}$: solid symbols and $X_z$: open symbols), in units of the free-electron mass, $m_0$. Although the longitudinal masses in the two calculations follow opposite trends, the maximum difference between the tight-binding and VASP (Ref. 16) results is less than 4.5%.

Figure 3

Conduction- and valence-band edges of Si under uniaxial [110] strain as reproduced by this model (lines) and as predicted by the model solid theory of Ref. 20 (symbols). The experimentally relevant region, $\left|{\epsilon }_{110}\right|\lesssim 0.015$, is indicated by vertical dashed lines.

Figure 4

Conduction-band masses of Si under uniaxial [110] strain as reproduced by this model (lines) and as predicted by VASP (Ref. 21) (symbols), in units of the free-electron mass, $m_0$. The transverse masses are those of the $X_z$ valleys.

Figure 5

Conduction- and valence-band edges of Si under hydrostatic strain as reproduced by this model (lines) and as predicted by the model solid theory of Ref. 20 (symbols).

Figure 6

Conduction-band masses of Si under hydrostatic strain as reproduced by this model (lines) and as predicted by VASP (Ref. 21) (symbols) in units of the free-electron mass, $m_0$.

Figure 7

Drain ON current, $I_d$ (open symbols) as a function of strain and percent change relative to the drain ON current at zero strain (closed symbols) for a Si, double-gate, ultrathin-body MOSFET (see text); lines are guides to the eyes. The operating conditions are: $V_{gs}=V_{ds}=V_{DD}=1\text{V}$. The curve labeled “Old Model” is a calculation using the strain model of Ref. 14.

Figure 8

Schematic of the same-atom matrix element $⟨{\Phi }_{i,x}|{\widehat{V}}_j^{\left(i\right)}|{\Phi }_{i,xy}⟩$ for $l_j=n_j=0,m_j=1$; $\mathbf{R}$ is the location of the nearest-neighbor potential. Because the potential is attractive and nearest to a region of positive orbital product ($+/+,-/-$ orbital overlap), this matrix element is expected to be negative.