When a metal is joined by a semiconductor, and thermodynamic equilibrium is reached, chemical bonding has to take place. At an ordinary, polycrystalline MS interface, the bonding geometry likely changes from place to place, leading to a locally varying interface dipole. The measured SBH then reflects some weighted average of this interface dipole. Because of the randomness of the interface structure, there is the expectation that the interface dipole can perhaps be estimated using bulk-derived properties. It therefore makes some sense to analyze the electric dipole at a MS interface using established techniques in chemical physics, developed largely for molecular systems. The total energy of a multi-atomic molecule can be written, neglecting higher order terms, as

   E_tot( Q_A, ... Q_N) = S [ E_A^o + U_AQ_A + (1/2) Y_AQ_A² ] + S (1/2) Q_AQ_BJ_AB},

where E_A^o is the energy of atom A in the uncharged state, -eQ_A is the net charge on atom A, U_A (= c_A /2 + I_A /2 ) is the Mulliken potential, Y_A (= I_A - c_A ) is the idempotential, and J_AB (= e² / e_o d_AB ) is the Coulombic interaction between two charges which are situated at atomic positions A and B. In the above, c is the electron affinity, I is the ionization potential, and d_AB is the distance between atoms A and B. In thermal equilibrium, one can apply the condition that the chemical potential is a constant for all the atoms in the same molecule, and be able to estimate the charge transfer between atoms. Because this procedure predicts finite charge transfers even for atoms placed far apart, it is only physically meaningful when the interatomic distances are near their equilibrium values. Potential faults notwithstanding, this approach has yielded dipoles for small heteronuclear molecules in good quantitative agreement with experiment.

To apply the above method to an MS interface, the conglomerate of the entire metal-semiconductor region can be regarded as a giant ``molecule''. The truncated lattices of a metal and a semiconductor are assumed to form bonds on an atomically flat interface plane. A density of chemical bonds, N_B, is assumed to form across the MS interface. In general, N_B needs not equal, and is likely less than, the total number of semiconductor (or metal) atoms per unit area on each plane parallel to the interface. Lattice mismatch, structure incompatibility, interface mixing, the formation of tilted bonds, etc. all tend to reduce the number of effective bonds formed across an MS interface. Without loss of generality, we assume chemical bonds to form in a square array with a lateral dimension of d_B ( = N_B^-1/2 ). We can write a total energy equation according to (4) for the entire metal-semiconductor system. For simplicity, we assume Q's to be non-zero only for those metal and semiconductor atoms on the immediate interface planes which are involved in the bonding. Furthermore, we keep only the interactions between charged atoms which are nearest neighbors.

   E_MS = S (E_M^o + U_M Q_{M_i} + (1/2) Y_MQ_{M_i} ² ) + S( E_S^o + U_S Q_{S_i} + (1/2) Y_S Q_{S_i}² )

   E_MS = S Q_{S_i} J_MS + S [ J_MM Q_{M_i} Q_{M_j} + J_SS Q_{S_i} Q_{S_j} ] .       (6)

Now we require that the chemical potential of a metal atom at the interface, m_{M_i} = dE_MS/ dQ_{M_i}, to equal that of a semiconductor atom, m_{S_i} = dE_MS/ dQ_{S_i}. Because of symmetry, every bonded metal atom at the interface has the same net charge -eQ_M and every semiconductor atom has a net charge of +eQ_M. One thus obtains,

   Q_M = [U_S - U_M] / [Y_M + Y_S + 4J_MM -2J_MS +4J_SS ] ,       (7)

where we have used the fact that there are four in-plane nearest neighbors. In the spirit of analyzing charge transfer between two crystals, rather than between atoms, we can let the atoms acquire bulk characteristics. For a bulk metal, the ionization potential and the electron affinity are both identified as the work function of the metal, f_M, and Y_M =0. For a semiconductor, the ionization potential and the electron affinity, c_S, differ by its band gap, E_g. Therefore, U_S = c_S + E_g/2 and Y_S = E_g. To account for the fact that the Coulombic interactions take place inside a solid, screening by the respective dielectric medium is also assumed. Eq. (7) becomes

    Q_M = [ f_M - c_S - E_g /2 ] / [ E_g + k ] ,       (8)

where k is the sum of all the hopping interactions, i.e. k = 4e² / ( e_S d_B ) - 2e² / ( e_it d_MS ), and d_MS is the distance between metal and semiconductor atoms at the interface. The voltage drop across this interfacial dipole layer can now be calculated to be

   V_"int" = [ -ed_MS N_BQ_M / e_it ] = ( -ed_MS N_B / e_it ) x [ f_M - c_S - E_g /2 ] / ( E_g + k ) .      (9)

Combining Eqs. (1) and (9), one gets

   F_B,n^o = [ 1 - e² d_MSN_B e_it^-1 ( E_g + k )^-1 ] ( f_M - c_S ) + e²d_MSN_B E_g e_it^-1 (E_g + k )^-1 /2 ,

which simplifies to

    F_B,n^o = g_B (f_M - c_S ) + (1 - g_B ) E_g/2 ,        (10)

where

   g_B = 1 - e² d_MS N_B e_it^-1 ( E_g + k )^-1 .       (11)

Obviously, Eq. (10) predicts a dependence of the SBH on the metal work function which is similar to that predicted by gap state models, Eq. (3).

In Eq. (11), the dielectric constant of the interface region, e_it, should have a value somewhere between the dielectric constant of the semiconductor, e_inf, and that of the metal, e_M ( = infinity ) . A simple average e_it^-1 = e_M^-1 /2 + e_inf^-1 /2 ) leads to an estimate of 2e_inf. Here the optical dielectric constant is used because the charge transfer occurs between the semiconductor and the metal, so there should be little ionic contribution to the screening. To test the validity of Eqs. (10) and (11), we plot in Fig. 2 the experimentally observed S_F in the form of [ e_inf ( 1 - S_F ) ]^-1 against the band gap of the semiconductor. According to Eq. (11), the quantity plotted in Fig. 2 is 2 e_o ( E_g + k ) / ( e d_MS N_B ), which should display a linear behavior if d_MS, N_B, and k do not vary appreciably with semiconductors. Indeed, a roughly linear relationship is observed, which approximately extrapolates through the origin, suggesting that, due to screening, k is small compared with typical band gaps. A slope of \(ap 0.13 eV^-1 is deduced from a linear fit, which yields a ( d_MS N_B ) of ~ 9 x 10⁶ cm^-1. Taking d_MS to be 0.25 nm, one gets an N_B of ~ 4 x 10¹⁴ cm^-2 which is a very reasonable estimate of the number of available bonds on a typical semiconductor surface. Overall, Fig. 2 provides strong support for the chemical bonding picture of FL pinning.

The polarization of the chemical bonds at MS interfaces leads to a weak dependence of the SBH on the work function and a natural tendency for the SBHs to converge toward one half of the band gap, both of which are in agreement with experimental results and have long been thought to be attributable only to the presence of interface gap states. The excellent agreement of experimental data with the present theory suggest that interface bonding is a primary mechanism of SBH formation. Even though the present theory seems to have captured the essence of the SBH formation mechanism, it is only an approximation and should not be regarded as numerically accurate. For numerical comparison with SBH experiments, one should always rely on first principles calculations for a more accurate treatment of the interface dipole.

The chemical bonding scheme described here is most applicable for abrupt interfaces and for semiconductors with large band gaps. In the present approach, charge is allowed to transfer only between atoms situated at the interface plane. In reality, this charge transfer likely takes place over several atomic planes and the sign of the charge transfer may even oscillate with the number of planes. A more sophisticated model of the interface can perhaps be attempted, but doing so is not advisable given the simplicity of the present molecular approach and because precise knowledge on the atomic structure would then be required. In deriving Eq. 8, we have used quantities characteristic of isolated crystals to represent the ability of individual atoms to attract electrons. Since the metal work function and the semiconductor electron affinity both contain a surface term which depends on the structure of the free surface and does not appear relevant to the interface charge transfer, it may be argued that only the bulk contributions to the work function and electron affinity should be used. This modification shifts the apparent ``pinning'' position slightly from midgap of the semiconductor. Since reliable data for the surface dipoles are generally lacking, the impact of this possible modification is not clear. One may comment in passing that the work function and electron affinity in Eq. 1 presumably should not be modified in the same way, because without surface terms the discrepancy between the charge densities on metal and semiconductor Wigner-Seitz cell boundaries would remain.

There are some similarities between Eqs. (4) and (11), especially if N_B / E_g can be identified with D_gs. Since gap states undoubtedly accommodate some of the charge transfer in chemical bonding, the question may arise as to whether the two approaches are in fact equivalent. They are not. The charge in the gap state models is assumed to be controlled by the FL position. So, the gap state models artificially build in a negative-feedback mechanism, with the charge neutrality level as the back drop. If one imagines slowly "turning on" (or off) the MIGS mechanism by increasing (or decreasing) D_gs, the interface FL cannot cross over the neutrality level ( g_gs in Eq. (4) cannot become negative). In the present theory, both the pinning and the pinning position (around midgap) come as a result of the minimization of the total energy. The FL is not explicitly involved in the evaluation of the interface dipole, and therefore the contradiction associated with the bias-dependence of the SBH does not arise. The distribution of gap states occurs as a result of bonding and will vary with the metal and the interface structure, which is in good agreement with the structure dependence of the SBH seen at epitaxial interfaces, and with the observation of SBH inhomogeneity at polycrystalline interfaces.

When the experimentally measured interface behavior parameters, S_F, are plotted in a fashion suggested by the division-by-epsilon model of Schottky barrier height formation, the agreement is poor. The agreement is also poor between experiment and the prediction of the metal-induced gap-state model of Schottky barrier height formation.

To read more about applying chemical concept at metal-semiconductor interfaces, consult Physical Review B64, 205310 (2001).