Schottky barrier

Input Files:

• 1DSchottky_barrier_GaAs_nn3.in

• 2DSchottky_barrier_GaAs_nn3.in

• 1DSchottky_barrier_GaAs_ohmic_nn3.in

• 1DSchottky_barrier_GaAs_SchottkyBarrier0V_nn3.in

• 1DSchottky_barrier_GaAs_surface_density_nn3.in

• 1DSchottky_barrier_GaAs_surface_states_acceptor_nn3.in

Scope:

In this tutorial we simulate Schottky contacts.

Introduction

When a metal is in contact with a semiconductor, a potential barrier is formed at the metal-semiconductor interface. In 1938, Walter Schottky suggested that this potential barrier arises due to stable space charges in the semiconductor. At thermal equilibrium, the Fermi levels of the metal and the semiconductor must coincide. There are two limiting cases:

1. Ideal Schottky barrier:

   • metal/n-type semiconductor: The barrier height \(\phi_B\) is the difference of the metal work function \(\phi_M\) and the electron affinity (\(\chi\)) in the semiconductor.

   \[\mathrm{e} \phi_B = \mathrm{e} ( \phi_M - \chi_s )\]

   • metal/p-type semiconductor: The barrier height \(\phi_{B,p}\) is given by:

   \[\mathrm{e} \phi_{B,p} = \mathrm{e} ( \phi_M - \chi_s ) - E_\mathrm{gap}\]

2. Fermi level pinning:

   If surface states on the semiconductor surface are present: The barrier height is determined by the property of the semiconductor surface and is independent of the metal work function

Consequence: The Schottky barrier sets a (Dirichlet) boundary condition for the electrostatic potential, i.e. the solution of the Poisson equation in the semiconductor, because the conduction and valence band edge energies are in a definite energy relationship with the Fermi level of the metal.

The Schottky barrier model implemented in nextnano++ is basically a Fermi level pinning and does not take into account the work function of the metal: The barrier height is independent of the metal work function and is entirely determined by the surface states and the doping.

contacts{
    schottky{                   # Schottky barrier (Fermi level pinning)
        name = contact
        bias = 0.0              # [V] apply voltage
        barrier = 0.53          # [V] GaAs, S.M. Sze, "Physics of Semiconductor Devices", p. 275 (2nd ed.)
    }
}

The n-type donor concentration in \(GaAs\) has been taken to be 1 \(\cdot\) 10¹⁸ cm^-3 (fully ionized). The temperature is set to 300 K.

Figure 3.4.13 shows the conduction band edge profile for n-type \(GaAs\) in equilibrium with

• a Schottky barrier of 0.53 V, i.e. the conduction band edge is pinned 0.53 eV above the Fermi level (which is at 0 eV)

• a Schottky barrier of 0 V

• an ohmic contact at 10 nm.

(The contact region is from 0 nm to 10 nm but no equations are solved inside the contact region.)

Note that in equilibrium the Femi level is constant and equal to 0 eV in the whole device. If the semiconductor is doped, the conduction and valence band edges are shifted with respect to this Fermi level, i.e. relative to 0 eV and are thus dependent on doping. This is a bulk property and independent of surface effects, like ohmic contacts or Schottky barrier height (see right part of the figure). At the left boundary, however, the band profile is affected by the type of contact.

Figure 3.4.13 Calculated conduction band profile

Note

A Schottky barrier of 0 V is not equivalent to an ohmic contact!

An ohmic contact corresponds to a Neumann boundary condition for the Poisson equation, i.e. \(d\phi/dx\) = 0 (constant electrostatic potential) or flat band condition which is equivalent to \(E\) = 0. A Schottky barrier \(\phi_B\) is a Dirichlet boundary condition for the Poisson equation, i.e. the value of the conduction band edge at the boundary is fixed with respect to the Fermi level:

\[E_c - E_F = \mathrm{e} \phi_B\]

In this particular example, an artificial Schottky barrier of -0.04184 V would be equivalent to an ohmic contact, (i.e. flat band condition), but only for the same temperature and the same doping concentration.

Interface charges (surface states)

Input file: 1DSchottky_barrier_GaAs_surface_density.in

Instead of specifying a Schottky barrier, the user can alternatively specify a fixed surface charge density.

structure{
    ...
    region{ # charge sheet
        line{ x = [10E0, 10E0 + $Width]}            # between contact/GaAs at 10 nm
        doping{
            constant{
                name = "negative-interface-charge"   # name of impurity
                conc = 2.7675e20                     # doping concentration [cm-3]
            }
        }
    }
}

impurities{
    ...
    charge{
        name = "negative-interface-charge"          # refer to region with name negative-interface-charge
        type = negative
    }
}

Figure 3.4.14 shows that the red curve ( = “ohmic” contact with interface charge density \(\sigma\) (surface states) of -2.7675 \(\cdot\) 10¹² \(|e|\) /cm² = -4.4340 \(\cdot\) 10^-3 C/m²) is equivalent to the black curve (Schottky barrier of 0.53 eV).

A sheet charge density of -2.7675 \(\cdot\) 10¹² cm^-2 corresponds to a volume charge of -2.7675 \(\cdot\) 10²⁰ cm^-3 if one assumes this charge to be distributed over a grid spacing of 0.1 nm. In this case, the interface charge density corresponds to a Neumann boundary condition for the derivative of the electrostatic potential \(\phi\):

\[\frac{d\phi}{dx} = - E_x = \mathrm{const},\]

where \(E_x\) is the electric field component along the x direction. \(E_x\) is related to the interface charge as follows:

\[E_x = \frac{\sigma}{\epsilon_r\epsilon_0}\]

where \(\epsilon_0\) is the permittivity of vacuum and \(\epsilon_r\) is the dielectric constant of the semiconductor. In this example:

• \(\epsilon_r\) = 12.93 for GaAs

• \(E_x\) = 387.3 kV/cm

The output for the electric field (in units of [kV/cm]) can be found in this file: electric_field.dat

Figure 3.4.14 Calculated conduction band profile

The output for the interface densities can be found in this file: material\density_fixed_charge.dat.

Surface states - Acceptors

Input file: 1DSchottky_barrier_GaAs_surface_states_acceptor_nnp.in

Instead of specifying a Schottky barrier, the user can alternatively specify a density of acceptor surface states (p-type doping). Essentially, this can be done by specifying a p-type doping region that is very thin, i.e. the doping is specified only on one grid point.

In this example, we use a doping area of 0.1 nm at the surface that we dope p-type with a volume density of 276.75 \(\cdot\) 10¹⁸ cm^-3. This corresponds to a sheet charge density of 2.7675 \(\cdot\) 10¹² cm^-2 where we assume the states to be fully ionized.

impurities{
    ...
    acceptor{ # p-type
        name = "impurity"   # refer to region with name impurity
        energy = -1000E0    # all ionized
        degeneracy = 4      # degeneracy of energy levels, 2 for n-type, 4 for p-type
    }
}

The results are the same as shown in Figure 3.4.14 for the interface charges.