Calculating the Schottky barrier

The Schottky barrier is the energy barrier that forms at a metal–semiconductor junction. The barrier height is an important characteristic for charge transport across the junction and a critical parameter in the design of semiconductor contacts, transistors, and rectifying diodes. The p-type Schottky barrier height [math]\displaystyle{ \varphi_{\mathrm{p}} }[/math] describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height [math]\displaystyle{ \varphi_{\mathrm{n}} }[/math] describes the barrier seen by electrons in the conduction band. In VASP, the potential alignment methodcitation needed can be used to estimate the Schottky barrier.

Three quantities from bulk calculations, the Fermi energy [math]\displaystyle{ E_{\mathrm{F}} }[/math] of the metal, and the valence-band maximum [math]\displaystyle{ E_{\mathrm{VBM}} }[/math] and band gap [math]\displaystyle{ E_{\mathrm{g}} }[/math] of the semiconductor, are needed for the calculation. As well as the macroscopic electrostatic potential difference [math]\displaystyle{ \Delta\bar{V} }[/math] across the interface, which is extracted from a separate interface slab calculation.

Step-by-step instructions

Important: For the calculation of the p- and n-type Schottky barriers, energies, and potentials from different VASP calculations are compared. It is crucial to set a single value for ENCUT for all calculations.

Step 1: Compute the Fermi energy of the metal.

After ensuring that the bulk geometry is optimized, a separate static calculation should be performed with a dense, Γ-centered k-point mesh and ISMEAR = -5 (thedrahedron method with Blöchl corrections), to get an accurate Fermi energy. It can be easily accessed by py4vasp:

import py4vasp as pv
calc = pv.Calculation.from_path("Al_bulk_static")
E_F = calc.dos.read()["fermi_energy"]
print(f"E_F = {E_F:.4f} eV")

Step 2: Compute the band gap and valence-band maximum of the semiconductor.

Band gaps are notoriously underestimated by standard DFT, so for accurate values, it is usually necessary to use DFT+U, hybrid functionals, or GW calculations. The level of theory to choose depends on the semiconductor of interest. Generally, it is advisable to select BANDGAP = KPOINT for more verbose output, and ideally to make a full bandstructure calculation, since the extrema of the bands will be located at high symmetry lines.

Step 2a: Geometry optimization of bulk Si

Use the PBEsol functional for an accurate equilibrium lattice constant:

GGA = PS
IBRION = 1 ; ISIF = 7 ; NSW = 50
ISMEAR = -5 ; SIGMA = 0.05
ENCUT = 300 ; EDIFF = 1E-6 ; EDIFFG = -0.01

Step 2b: Band-structure calculation of bulk Si with HSE06

The band gap and valence-band maximum are computed with the HSE06 hybrid functional, which gives accurate band gaps for semiconductors. This requires a hybrid band-structure calculation; follow Band-structure calculation using hybrid functionals to set up the KPOINTS and KPOINTS_OPT files. The key INCAR settings are:

LHFCALC = .TRUE. ; HFSCREEN = 0.2
GGA = PE ; HFRCUT = -1
ALGO = Normal ; ISMEAR = -5
ENCUT = 300 ; PREC = Accurate ; EDIFF = 1E-6

GGA = PE (PBE) is the correct base functional for HSE06, not PBEsol. HFRCUT = -1 avoids discontinuities in the band structure for gapped systems. Extract [math]\displaystyle{ E_{\mathrm{g}} }[/math] and [math]\displaystyle{ E_{\mathrm{VBM}} }[/math] with py4vasp:

import py4vasp as pv
import numpy as np

calc  = pv.Calculation.from_path("Si_bulk_HSE_bands")
bg    = calc.bandgap.read()
band  = calc.band.read()

E_g   = bg["fundamental"]
E_F   = bg["fermi_energy"]
E_VBM = (band["bands"][0] + E_F)[band["occupations"][0] > 0.5].max()

print(f"E_g   = {E_g:.4f} eV")
print(f"E_VBM = {E_VBM:.4f} eV")

For bulk Si this gives [math]\displaystyle{ E_{\mathrm{g}} = 1.160 }[/math] eV and [math]\displaystyle{ E_{\mathrm{VBM}} = 5.449 }[/math] eV.

Step 3: Build the Al(111)/Si(111) interface structure.

Building a lattice-matched metal–semiconductor interface slab is outside the scope of this page. Tools such as the CoherentInterfaceBuilder in pymatgencitation needed and the interface construction utilities in ASEcitation needed can be used to construct the initial geometry. The structure must satisfy the following requirements before it is passed to Step 4:

• Low lattice mismatch. A coincidence-site lattice supercell should be chosen such that the in-plane periodicities of the two surfaces match to within ~2%, minimising artificial biaxial strain. For Al(111) and Si(111), a 4×4 Al supercell matched to a 3×3 Si supercell achieves a mismatch of 0.9%.

• Sufficient slab thickness. Each material should contain enough layers that a bulk-like region, free of interface perturbations, exists in the centre. A minimum of 8–10 monolayers per material is a practical starting point; convergence with slab thickness should be verified. The slab used here contains 13 Al(111) monolayers and 12 Si(111) monolayers (6 bilayers).

• Selective dynamics. Only the layers immediately adjacent to the interface (2–3 layers on each side) should be free to relax in Step 4; the remaining inner layers are frozen. This substantially reduces the cost of the relaxation.

• Reasonable initial interface geometry. The initial separation between the last metal layer and the first semiconductor layer should be set to a physically plausible value (2–3 Å, close to the equilibrium Al–Si bond length) to ensure smooth convergence.

A pre-optimized POSCAR for this interface will be provided here; readers starting from that file may proceed directly to Step 5.

Step 4: Relax the interface.

The relaxation is performed in two passes. In the first pass, only the ionic positions are updated with the cell held fixed (ISIF = 2). After convergence, copy CONTCAR to POSCAR and run a second pass with full cell relaxation (ISIF = 3), which relieves any residual stress from the lattice mismatch. LREAL = A is recommended throughout for this large supercell.

Step 4a: Ionic relaxation (fixed cell)

ALGO = Fast ; LREAL = A
ISMEAR = 0 ; SIGMA = 0.1
ENCUT = 300 ; PREC = Normal
EDIFF = 1E-6 ; EDIFFG = -0.01
IBRION = 2 ; ISIF = 2 ; NSW = 200

ISMEAR = 0 (Gaussian smearing) is the safe choice for the interface supercell, which contains both metallic and semiconducting regions. LREAL = A uses real-space projection for the non-local operations, which is significantly faster for cells with more than ~100 atoms.

Step 4b: Full cell relaxation

Copy CONTCAR to POSCAR and re-run with cell degrees of freedom enabled:

IBRION = 1 ; ISIF = 3

All other tags are unchanged from Step 4a.

Step 5: Compute the electrostatic potential of the interface.

To extract the potential alignment, a static calculation is performed on the relaxed interface with WRT_POTENTIAL = hartree ionic, which writes the ionic and Hartree contributions to the electrostatic potential into vaspout.h5 for direct access. The electrostatic (ionic + Hartree) part of the potential is the correct choice for potential alignment; the exchange-correlation contribution present in the total potential (LVTOT) must be excluded.

ALGO = Fast ; LREAL = A
ISMEAR = -5
ENCUT = 300 ; PREC = Accurate ; EDIFF = 1E-8
WRT_POTENTIAL = hartree ionic

Step 6: Extract [math]\displaystyle{ \Delta\bar{V} }[/math] from the electrostatic potential.

The planar average [math]\displaystyle{ \bar{V}(z) }[/math] is obtained by averaging the 3D potential over the in-plane (xy) grid at each [math]\displaystyle{ z }[/math] point. A macroscopic average is then computed by applying a uniform box-car convolution of window length [math]\displaystyle{ L }[/math] equal to the bulk layer period of the materials involved; this removes short-range oscillations within each atomic layer while preserving the long-range potential step across the interface. For Al(111) and Si(111), [math]\displaystyle{ L = 9.35 }[/math] Å covers approximately three Al monolayers or three Si bilayers. Read the potential from vaspout.h5, compute both averages, and plot:

import h5py
import numpy as np
from scipy.ndimage import uniform_filter1d
import matplotlib.pyplot as plt

VASPOUT       = "Step5/vaspout.h5"
WINDOW_LENGTH = 9.35              # macroscopic averaging window (Å)

with h5py.File(VASPOUT, "r") as f:
    hartree = f["/results/potential/hartree"][0]   # shape (NGZ, NGY, NGX)
    ionic   = f["/results/potential/ionic"][0]     # shape (NGZ, NGY, NGX)
    lat     = f["/results/positions/lattice_vectors"][:]

c   = np.linalg.norm(lat[2])         # cell length along interface normal (Å)
NGZ = hartree.shape[0]
dz  = c / NGZ

# planar average: mean over the xy-plane at each z
V_planar = (hartree + ionic).mean(axis=(1, 2))    # shape (NGZ,)  [eV]

# macroscopic average: uniform_filter1d is a box-car convolution of `size` points;
# mode='wrap' enforces periodic boundary conditions along z.
n_window = int(round(WINDOW_LENGTH / dz))
V_macro  = uniform_filter1d(V_planar, size=n_window, mode="wrap")

z = np.arange(NGZ) * dz             # z-coordinates in Å

fig, ax = plt.subplots(figsize=(10, 4))
ax.plot(z, V_planar, lw=0.7, color="steelblue", alpha=0.5, label="Planar average")
ax.plot(z, V_macro,  lw=2.0, color="tomato",
        label=f"Macroscopic average (L\u202f=\u202f{WINDOW_LENGTH}\u202fÅ)")
ax.set_xlabel("z (Å)")
ax.set_ylabel("Electrostatic potential (eV)")
ax.set_xlim(0, c)
ax.legend()
plt.tight_layout()
plt.savefig("potential_alignment.png", dpi=150)
plt.show()

Inspect the plot to identify the flat bulk-like plateaus on each side of the interface. The plateau values give [math]\displaystyle{ \bar{V}_{\mathrm{m}} }[/math] and [math]\displaystyle{ \bar{V}_{\mathrm{sc}} }[/math]; adjust the z-range masks below to lie within the flat regions, away from the interface transition zones:

z_frac  = z / c
Al_mask = (z_frac > 0.05) & (z_frac < 0.20)   # bulk-like Al region  (adjust)
Si_mask = (z_frac > 0.42) & (z_frac < 0.58)   # bulk-like Si region  (adjust)

V_Al    = V_macro[Al_mask].mean()
V_Si    = V_macro[Si_mask].mean()
delta_V = V_Al - V_Si

print(f"V̄_Al  = {V_Al:+.4f} eV")
print(f"V̄_Si  = {V_Si:+.4f} eV")
print(f"\u0394V̄     = V̄_Al \u2212 V̄_Si = {delta_V:+.4f} eV")

For the Al(111)/Si(111) interface studied here this gives [math]\displaystyle{ \bar{V}_{\mathrm{m}} = -0.69 }[/math] eV, [math]\displaystyle{ \bar{V}_{\mathrm{sc}} = +1.20 }[/math] eV, and [math]\displaystyle{ \Delta\bar{V} = -1.89 }[/math] eV.

Step 7: Compute the Schottky barrier heights.

With [math]\displaystyle{ E_{\mathrm{F}} }[/math], [math]\displaystyle{ E_{\mathrm{VBM}} }[/math], [math]\displaystyle{ E_{\mathrm{g}} }[/math], and [math]\displaystyle{ \Delta\bar{V} }[/math] in hand, the potential alignment difference [math]\displaystyle{ \Delta\bar{V} = \bar{V}_{\mathrm{m}} - \bar{V}_{\mathrm{sc}} }[/math] (positive if the metal side has the higher average potential) connects the bulk reference frames. The p-type and n-type Schottky barrier heights are then

[math]\displaystyle{ \varphi_{\mathrm{p}} = \Delta\bar{V} + E_{\mathrm{F}} - E_{\mathrm{VBM}}, }[/math]

[math]\displaystyle{ \varphi_{\mathrm{n}} = E_{\mathrm{g}} - \varphi_{\mathrm{p}}. }[/math]

# Values from Steps 1, 2, and 6
E_F     =  8.085   # eV  (bulk Al, Step 1b)
E_VBM   =  5.449   # eV  (bulk Si HSE06, Step 2b)
E_g     =  1.160   # eV  (bulk Si HSE06, Step 2b)
delta_V = -1.891   # eV  (interface potential alignment, Step 6)

phi_p = delta_V + E_F - E_VBM
phi_n = E_g - phi_p

print(f"phi_p = {phi_p:.3f} eV   (p-type Schottky barrier)")
print(f"phi_n = {phi_n:.3f} eV   (n-type Schottky barrier)")

For the Al(111)/Si(111) interface this gives [math]\displaystyle{ \varphi_{\mathrm{p}} = 0.745 }[/math] eV and [math]\displaystyle{ \varphi_{\mathrm{n}} = 0.415 }[/math] eV.

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