Source code for simudo.example.heterojunction.silicongermanium

from math import pi
from simudo.physics import Material

from .silicon import SiliconMaterial
from .germanium import GermaniumMaterial


class SiliconGermaniumAlloy(Material):
    """Silicon-Germanium material data based on Palankovski.

       This is for Si(1-x)Ge(x) strained to the Silicon lattice.
       mole fraction should be specified by the MoleFractionX spatial rule.

       V. Palankovski and R. Quay, "Analysis and Simulation of Heterostructure Devices",
       Springer-Verlag (2004).
    """

    name = "SiliconGermanium"

    def get_dict(self):
        d = super().get_dict()
        U = self.unit_registry
        X = self.problem_data.pdd.spatial.value_rules["MoleFractionX"][0].value
        Silicon = SiliconMaterial(problem_data=self.problem_data).get_dict()
        Germanium = GermaniumMaterial(problem_data=self.problem_data).get_dict()
        # print(X)

        def vegard(param, C):
            """ use vegard's law to interpolate parameter param with bowing parameter C."""
            return (
                Silicon[param] * (U("1") - X)
                + Germanium[param] * X
                + C * (U("1") - X) * X
            )

        def linear_interp(param):
            return Silicon[param] * (U("1") - X) + Germanium[param] * X

        d.update(
            {
                "poisson/permittivity": vegard("poisson/permittivity", U("0")),
                # Table 3.9
                "CB/EgX_0K": vegard("CB/EgX_0K", U("-0.4 eV")),
                "CB/ChiX_0K": vegard("CB/ChiX_0K", U("0.4 eV")),
                "CB/varshni_alphaX": Silicon["CB/varshni_alphaX"],
                "CB/varshni_betaX": Silicon["CB/varshni_betaX"],
                # Table 3.21
                "CB/MCX": U("6"),
                # recombination velocity at interfaces with other materials
                "CB/SRV": U("500 cm/s"),
                "VB/SRV": U("500 cm/s"),
                # SRH recombination parameters
                "SRH/CB/tau": U("1e-9 s"),
                "SRH/VB/tau": U("1e-6 s"),
                # Properties for band-to-band tunneling
                "CB/eff_mass_tunnel": U("0.05"),
                "VB/eff_mass_tunnel": U("0.1"),
                "CB/tunnel_offset_energy": U("0 eV"),
                "VB/tunnel_offset_energy": U("0 eV"),
            }
        )

        T = self.temperature
        Eg = d["CB/EgX_0K"] - d["CB/varshni_alphaX"] * T ** 2 / (
            d["CB/varshni_betaX"] + T
        )
        Chi = d["CB/ChiX_0K"] + d["CB/varshni_alphaX"] * T ** 2 / (
            d["CB/varshni_betaX"] + T
        )

        mn = vegard("CB/mDOS", U("-0.183"))
        mp = vegard("VB/mDOS", U("-0.096"))
        m_e = U.electron_mass
        k_B = U.boltzmann_constant
        h = U.planck_constant
        NC = 2 * d["CB/MCX"] * (2 * pi * mn * m_e * k_B * T / h ** 2) ** (3 / 2)
        NV = 2 * (2 * pi * mp * m_e * k_B * T / h ** 2) ** (3 / 2)

        ###########################
        # TODO: Update these for SiGe. Below is just the calculation for silicon
        # Effective masses and thermal velocity from Green (1990)
        # mtc actually has a small temperature dependence acording to Green
        mtc = U("0.28")
        vth_c = (
            8 * U.boltzmann_constant * T / (pi * mtc * U.electron_mass)
        ) ** 0.5

        mtv = (
            0.3676
            + U("1.98738e-5 /K^2") * T ** 2
            - U("2.588144e-7 /K^3") * T ** 3
            + U("1.415372e-9 /K^4") * T ** 4
            - U("3.919169e-12 /K^5") * T ** 5
            + U("5.410849e-15 /K^6") * T ** 6
            - U("2.959797e-18 /K^7") * T ** 7
        )
        vth_v = (
            8 * U.boltzmann_constant * T / (pi * mtv * U.electron_mass)
        ) ** 0.5
        #################################

        d.update(
            {
                "CB/mDOS": mn,
                "VB/mDOS": mp,
                "CB/energy_level": -Chi,
                "VB/energy_level": -Chi - Eg,
                "CB/effective_density_of_states": NC,
                "VB/effective_density_of_states": NV,
                # TODO - update these from Palankovski
                "CB/mobility": U("1400 cm^2/V/s"),
                "VB/mobility": U(" 450 cm^2/V/s"),
                "CB/vth": vth_c,
                "VB/vth": vth_v,        
                # SRH energy level
                "SRH/energy_level": -Chi - Eg / 2,
            }
        )

        return d