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nextnano3 - Tutorial

next generation 3D nano device simulator

1D Tutorial

Schottky barrier

Note: This tutorial's copyright is owned by Stefan Birner, www.nextnano.com.

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DSchottky_barrier_GaAs.in                         - input file for the nextnano3 and nextnano++ software (1D simulation)
-> 2DSchottky_barrier_GaAs.in                         -
input file for the nextnano3 and nextnano++ software (2D simulation)
-> 1DSchottky_barrier_GaAs_ohmic.in                   -
input file for the nextnano3 and nextnano++ software
-> 1DSchottky_barrier_GaAs_SchottkyBarrier0V.in       -
input file for the nextnano3 and nextnano++ software
-> 1DSchottky_barrier_GaAs_surface_density.in         -
input file for the nextnano3 and nextnano++ software
-> 1DSchottky_barrier_GaAs_surface_states_acceptor.in -
input file for the nextnano3 and nextnano++ software


Schottky barrier

-> 1DSchottky_barrier_GaAs.in
-> 2DSchottky_barrier_GaAs.in
-> 1DSchottky_barrier_GaAs_ohmic.in
-> 1DSchottky_barrier_GaAs_SchottkyBarrier0V.in

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:

  • a) Ideal Schottky barrier:

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

        e phiB = e ( phiM - chis )

    - metal/p-type semiconductor: The barrier height phiB,p is given by:

      - e phiB,p =  e ( phiM - chis ) - Egap
  • b) 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.

 

  $poisson-boundary-conditions
   poisson-cluster-number   = 1
   region-cluster-number    = 1
 ! boundary-condition-type  = ohmic      !
ohmic contact
   boundary-condition-type  = schottky   !
Schottky barrier (Fermi level pinning)
   contact-control          = voltage
   applied-voltage          = 0.0d0      ! [V]
   schottky-barrier         = 0.53d0     ! [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 x 1018 cm-3 (fully ionized).
The temperature is set to 300 K.

The following figure 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.

Note that 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. derivate of electrostatic potential = 0 (constant electrostatic potential), i.e. flat band condition which is equivalent to electric field = 0).
A Schottky barrier phiB 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: Ec EFermi = e phiB
(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)

-> 1DSchottky_barrier_GaAs_surface_density.in

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

 $material-interfaces

  interface-number               = 1             !
sigma
  apply-between-material-numbers = 1 2           !
between contact/GaAs at 10 nm
  state-numbers                  = 1             ! 1 =
fixed charge

 $end_material-interfaces

 $interface-states
  state-number                   = 1             !
between contact/GaAs at 10 nm
  state-type                     = fixed-charge  !
sigma
  interface-density              = -2.7675d12    ! -2.7675 * 1012 [|e|/cm2]
 !interface-density              = 0d0           !       0 * 1012 [|e|/cm2]
 $end_interface-states

The following figure shows that the red curve ("ohmic" contact with interface charge density sigma (surface states) of -2.7675 * 1012 |e|/cm2 = -4.4340 * 10-3 C/m2 is equivalent to the black curve (Schottky barrier of 0.53 eV).
A sheet charge density of -2.7675 * 1012 cm-2 corresponds to a volume charge of -2.7675 * 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:
dphi / dx = - Fx = constant /= 0, where Fx is the electric field component along the x direction.
Fx is related to the interface charge as follows: Fx = sigma / (epsilon0 epsilon) where epsilon0 is the permittivity of vacuum and epsilon is the dielectric constant of the semiconductor.
In this example (epsilon = 12.93 for GaAs), Fx = 387.3 kV/cm.
 static-dielectric-constants = 12.93d0 12.93d0 12.93d0

The output for the electric field (in units of [kV/cm]) can be found in this file:
band_structure/electric_field1D.dat

The output for the interface densities can be found in this file: densities/interface_densitiesD.txt

   -----------------
   INTERFACE CHARGES
   -----------------

   Interface number 1 at position 10 nm
   interface charge: -4.434023546775000E-003 C / m^2
   interface charge: -2.76750000000000       1E12 e/cm^2

 

Surface states - Acceptors

-> 1DSchottky_barrier_GaAs_surface_states_acceptor.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 * 1018 cm-3.
This corresponds to a sheet charge density of 2.7675 * 1012 cm-2 where we assume the states to be fully ionized.

  $doping-function
   ...
   doping-function-number      = 2               !
   impurity-number             = 2               !
properties of this impurity type have to be specified below
   doping-concentration        = 276.75d0        ! 276.75 * 10^18 cm^-3
   only-region                 = 10.0d0   10.1d0 ! [nm]  xmin  xmax

  $end-doping-function

  $impurity-parameters
   ...
   !---------------------------------------------------------------------------------
   ! 'p-type' means a negative background charge density of ionized acceptors NA-.
   !---------------------------------------------------------------------------------
   impurity-number             = 2               !
   impurity-type               = p-type          !
   number-of-energy-levels     = 1               !
   energy-levels-relative      = -1000d0         ! [eV]
a negative value means fully ionized
   degeneracy-of-energy-levels = 4               !
degeneracy of energy levels, 2 for n-type, 4 for p-type

  $end_impurity-parameters

The results are the same as shown in the figure above for the interface charges.

 

  • Please help us to improve our tutorial. Send comments to support [at] nextnano.com.