Calculating the Schottky barrier - Revision history

Huebsch at 12:22, 27 March 2026

2026-03-27T12:22:38Z

← Older revision		Revision as of 12:22, 27 March 2026
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	'''Step 2:''' Compute the band gap and valence-band maximum of the semiconductor.		'''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 [[:Category:DFT+U\|DFT+U]], [[:Category:Hybrid functionals\|hybrid functionals]], or [[:Category:GW\|GW]] calculations. The level of theory to choose depends on the semiconductor of interest, and HSE06 is oftern a good default for most covalent and ionic semiconductors. Generally, it is advisable to select {{TAG\|BANDGAP\|KPOINT}} to get more verbose output, and ideally to make a full [[:Category:Band_structure\|~~bandstructure~~ calculation]], since the extrema of the bands ~~will be~~ located at high symmetry lines. The fundamental bandgap <math>E_{\mathrm{g}}</math> and the valence band maximum <math>E_{\mathrm{VBM}}</math>, can be read from the {{FILE\|OUTCAR}} file, or with {{py4vasp}}:		: Band gaps are notoriously underestimated by standard DFT, so for accurate values, it is usually necessary to use [[:Category:DFT+U\|DFT+U]], [[:Category:Hybrid functionals\|hybrid functionals]], or [[:Category:GW\|GW]] calculations. The level of theory to choose depends on the semiconductor of interest, and HSE06 is oftern a good default for most covalent and ionic semiconductors.

			: Generally, it is advisable to select {{TAG\|BANDGAP\|KPOINT}} to get more verbose output, and ideally to make a full [[:Category:Band_structure\|band-structure calculation]], since the extrema of the bands are often located at high-symmetry lines. The fundamental bandgap <math>E_{\mathrm{g}}</math> and the valence-band maximum (VBM) <math>E_{\mathrm{VBM}}</math>, can be read from the {{FILE\|OUTCAR}} file, or with {{py4vasp}}:
	:<syntaxhighlight lang="python">		:<syntaxhighlight lang="python">
	import py4vasp as pv		import py4vasp as pv
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	:* ''No vacuum region.'' A cell with a vacuum will result in lateral strain, compressing the cell to reduce the surface energy. It is better to have two equivalent interfaces in the cell without any vacuum at all.		:* ''No vacuum region.'' A cell with a vacuum will result in lateral strain, compressing the cell to reduce the surface energy. It is better to have two equivalent interfaces in the cell without any vacuum at all.

	: For the relaxation, it is usually prudent to fix the layers in the bulk like regions using [[POSCAR#poscar6\|selective dynamics]] in the {{FILE\|POSCAR}} file. Only a couple of layers at the interfaces should be relaxed. Depending on the material pairing and the strain in the interface, it might be beneficial to allow the cell to relax only in the direction normal to the interface plane using the {{TAG\|LATTICE_CONSTRAINTS}} tag.		: For the relaxation, it is usually prudent to fix the layers in the bulk like regions using [[POSCAR#poscar6\|selective dynamics]] in the {{FILE\|POSCAR}} file. Only a couple of layers at the interfaces should be relaxed. Depending on the material pairing and the strain in the interface, it might be beneficial to allow the cell to relax only in the direction normal to the interface plane using the {{TAG\|LATTICE_CONSTRAINTS}} tag or the {{FILE\|ICONST}} file to apply the contraint.

	'''Step 4:''' Compute the electrostatic potential of the interface and calculate the Schottky barrier heights.		'''Step 4:''' Compute the electrostatic potential of the interface and calculate the Schottky barrier heights.

Huebsch: Huebsch moved page Construction:Calculating the Schottky barrier to Calculating the Schottky barrier without leaving a redirect

2026-03-27T12:08:49Z

Huebsch moved page Construction:Calculating the Schottky barrier to Calculating the Schottky barrier without leaving a redirect

← Older revision	Revision as of 12:08, 27 March 2026
(No difference)

Wolloch: /* Example */

2026-03-24T12:46:31Z

Example

← Older revision		Revision as of 12:46, 24 March 2026
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	<!--[[File:Si111_Al111_interface.png\|thumbnail\|120px\|right\|Al(111) Si(111) interface]]-->		<!--[[File:Si111_Al111_interface.png\|thumbnail\|120px\|right\|Al(111) Si(111) interface]]-->

	Here we will consider the Al(111)/Si(111) interface. The bulk Al calculation to determine the Fermi energy is straightforward and yields <math>E_{\mathrm{F}}=8.085</math> eV for the PBE functional ~~and~~ {{TAG\|ENCUT\|300}}, and {{TAG\|ISMEAR\|-5}}.		Here we will consider the Al(111)/Si(111) interface. The bulk Al calculation to determine the Fermi energy is straightforward and yields <math>E_{\mathrm{F}}=8.085</math> eV for the PBE functional with {{TAG\|ENCUT\|300}}, {{TAG\|KSPACING\|0.2}}, and {{TAG\|ISMEAR\|-5}}.

	For the Si bandgap and valence band maximum, a little more care ~~needs to be taken~~. After relaxing with PBEsol to get the correct lattice parameter, a band structure calculation with the HSE06 functional yields a bandgap of <math>E_{\mathrm{g}}=1.160</math> eV and a valence band maximum of <math>E_{\mathrm{VBM}}=5.449</math> eV.		For the Si bandgap and valence band maximum, a little more care is needed. After relaxing with PBEsol to get the correct lattice parameter, a band structure calculation with the HSE06 functional yields a bandgap of <math>E_{\mathrm{g}}=1.160</math> eV and a valence band maximum of <math>E_{\mathrm{VBM}}=5.449</math> eV.

	For Al(111) and Si(111), a 4×4 Al supercell matched to a 3×3 Si supercell achieves a mismatch of 0.9%. For this example, we chose 13 Al layers, and 6 Si bilayers. The starting interface distance before relaxation was set to 2.3 Å We have fully relaxed the cell, but fixed all but the 4 layers closest to the interfaces (2 Al layers, and 1 Si bi-layer for each interface) with the [[POSCAR#poscar6\|selective dynamics]] {{FILE\|POSCAR}} option. The relaxed structure is ~~printed~~ here:		For Al(111) and Si(111), a 4×4 Al supercell matched to a 3×3 Si supercell achieves a mismatch of 0.9%. For this example, we chose 13 Al layers, and 6 Si bilayers. The starting interface distance before relaxation was set to 2.3 Å We have fully relaxed the cell, but fixed all but the 4 layers closest to the interfaces (2 Al layers, and 1 Si bi-layer for each interface) with the [[POSCAR#poscar6\|selective dynamics]] {{FILE\|POSCAR}} option. The relaxed structure with its 316 atoms is given here:
	<div class="toccolours mw-customtoggle-poscar">'''Click to show POSCAR'''</div>		<div class="toccolours mw-customtoggle-poscar">'''Click to show POSCAR'''</div>
	<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-poscar">		<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-poscar">
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	</pre>		</pre>
	</div>		</div>
	After computing the electrostatic potential by setting {{TAG\|WRT_POTENTIAL\|hartree ionic}}, {{TAG\|PREC\|Accurate}}, and {{TAG\|KSPACING\|0.2}}, we can run the ~~python~~ script referenced above in Step 4.		After computing the electrostatic potential by setting {{TAG\|WRT_POTENTIAL\|hartree ionic}}, {{TAG\|PREC\|Accurate}}, and {{TAG\|KSPACING\|0.2}}, we can run the Python script referenced above in Step 4.
	For the Al(111)/Si(111) interface studied, we get the following results: <math>\bar{V}_{\mathrm{m}} = -0.694</math> eV, <math>\bar{V}_{\mathrm{sc}} = +1.192</math> eV, and <math>\Delta\bar{V} = -1.886</math> eV.		For the Al(111)/Si(111) interface studied, we get the following results: <math>\bar{V}_{\mathrm{m}} = -0.694</math> eV, <math>\bar{V}_{\mathrm{sc}} = +1.192</math> eV, and <math>\Delta\bar{V} = -1.886</math> eV.
	[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]		[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]

Wolloch at 12:41, 24 March 2026

2026-03-24T12:41:26Z

← Older revision		Revision as of 12:41, 24 March 2026
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	== Required quantities ==		== Required quantities ==
	Three quantities from bulk calculations, the Fermi energy <math>E_{\mathrm{F}}</math> of the metal, and the valence-band maximum <math>E_{\mathrm{VBM}}</math> and band gap <math>E_{\mathrm{g}}</math> of the semiconductor, are needed for the calculation. ~~As well as~~ the macroscopic electrostatic potential difference <math>\Delta\bar{V}</math> across the interface~~, which~~ is extracted from a separate interface slab calculation. <ref>In {{VASP}}, the <math>\mathbf{G}=0</math> Fourier component of the Hartree potential is set to zero by convention, so all single-particle eigenvalues are already referenced to the average electrostatic potential of their respective unit cell. As a consequence, <math>E_{\mathrm{F}}</math> and <math>E_{\mathrm{VBM}}</math> can be read directly from the bulk {{FILE\|OUTCAR}} files without any further potential referencing, and only the interface calculation requires the electrostatic potential written to {{FILE\|LOCPOT}}.</ref>		Three quantities from bulk calculations, the Fermi energy <math>E_{\mathrm{F}}</math> of the metal, and the valence-band maximum <math>E_{\mathrm{VBM}}</math> and band gap <math>E_{\mathrm{g}}</math> of the semiconductor, are needed for the calculation. In addition, the macroscopic electrostatic potential difference <math>\Delta\bar{V}</math> across the interface is extracted from a separate interface slab calculation<ref>In {{VASP}}, the <math>\mathbf{G}=0</math> Fourier component of the Hartree potential is set to zero by convention, so all single-particle eigenvalues are already referenced to the average electrostatic potential of their respective unit cell. As a consequence, <math>E_{\mathrm{F}}</math> and <math>E_{\mathrm{VBM}}</math> can be read directly from the bulk {{FILE\|OUTCAR}} files without any further potential referencing, and only the interface calculation requires the electrostatic potential to be written out — to {{FILE\|vaspout.h5}} via {{TAG\|WRT_POTENTIAL}} (recommended) or to {{FILE\|LOCPOT}} via {{TAG\|LVHAR}} (legacy).</ref>.

	== Step-by-step instructions ==		== Step-by-step instructions ==
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	'''Step 1:''' Compute the Fermi energy of the metal.		'''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 [[KPOINTS\|'''k'''-point mesh]] and {{TAG\|ISMEAR\|-5}} (~~thedrahedron~~ method with Blöchl corrections), to get an accurate Fermi energy. It can be taken from the {{FILE\|OUTCAR}} and is also easily accessible via {{py4vasp}}:		: After ensuring that the [[Structure optimization\|bulk geometry is optimized]], a separate static calculation should be performed with a dense, Γ-centered [[KPOINTS\|'''k'''-point mesh]] and {{TAG\|ISMEAR\|-5}} (tetrahedron method with Blöchl corrections), to get an accurate Fermi energy. It can be taken from the {{FILE\|OUTCAR}} and is also easily accessible via {{py4vasp}}:
	: <syntaxhighlight lang="python">		: <syntaxhighlight lang="python">
	import py4vasp as pv		import py4vasp as pv
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	'''Step 2:''' Compute the band gap and valence-band maximum of the semiconductor.		'''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 [[:Category:DFT+U\|DFT+U]], [[:Category:Hybrid functionals\|hybrid functionals]], or [[:Category:GW\|GW]] calculations. The level of theory to choose depends on the semiconductor of interest. Generally, it is advisable to select {{TAG\|BANDGAP\|KPOINT}} ~~for~~ more verbose output, and ideally to make a full [[:Category:Band_structure\|bandstructure calculation]], since the extrema of the bands will be located at high symmetry lines. The fundamental bandgap <math>E_{\mathrm{g}}</math> and the valence band maximum <math>E_{\mathrm{VBM}}</math>, can be read from the {{FILE\|OUTCAR}} file, or with {{py4vasp}}:		: Band gaps are notoriously underestimated by standard DFT, so for accurate values, it is usually necessary to use [[:Category:DFT+U\|DFT+U]], [[:Category:Hybrid functionals\|hybrid functionals]], or [[:Category:GW\|GW]] calculations. The level of theory to choose depends on the semiconductor of interest, and HSE06 is oftern a good default for most covalent and ionic semiconductors. Generally, it is advisable to select {{TAG\|BANDGAP\|KPOINT}} to get more verbose output, and ideally to make a full [[:Category:Band_structure\|bandstructure calculation]], since the extrema of the bands will be located at high symmetry lines. The fundamental bandgap <math>E_{\mathrm{g}}</math> and the valence band maximum <math>E_{\mathrm{VBM}}</math>, can be read from the {{FILE\|OUTCAR}} file, or with {{py4vasp}}:
	:<syntaxhighlight lang="python">		:<syntaxhighlight lang="python">
	import py4vasp as pv		import py4vasp as pv
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	:* ''Sufficient slab thickness.'' Each material should contain enough layers that a bulk-like region exists in the centre.		:* ''Sufficient slab thickness.'' Each material should contain enough layers that a bulk-like region exists in the centre.

	:* ''No vacuum region.'' A cell with a vacuum will result in lateral strain, compressing the cell to reduce the surface energy. It is better to have two equivalent interfaces in the cell.		:* ''No vacuum region.'' A cell with a vacuum will result in lateral strain, compressing the cell to reduce the surface energy. It is better to have two equivalent interfaces in the cell without any vacuum at all.

	: For the relaxation, it is usually prudent to fix the layers in the bulk like regions using [[POSCAR#poscar6\|selective dynamics]] in the {{FILE\|POSCAR}} file. Only a couple of layers at the interfaces should be relaxed. Depending on the material pairing and the strain in the interface, it might be beneficial to allow the cell to relax only in the direction normal to the interface plane using the {{TAG\|LATTICE_CONSTRAINTS~~}} {{FILE\|INCAR~~}} tag.		: For the relaxation, it is usually prudent to fix the layers in the bulk like regions using [[POSCAR#poscar6\|selective dynamics]] in the {{FILE\|POSCAR}} file. Only a couple of layers at the interfaces should be relaxed. Depending on the material pairing and the strain in the interface, it might be beneficial to allow the cell to relax only in the direction normal to the interface plane using the {{TAG\|LATTICE_CONSTRAINTS}} tag.

	'''Step 4:''' Compute the electrostatic potential of the interface and calculate the Schottky barrier heights.		'''Step 4:''' Compute the electrostatic potential of the interface and calculate the Schottky barrier heights.

	:Perform a static calculation with ~~thetrahedron~~ smearing ({{TAG\|ISMEAR\|-5}}, and {{TAG\|PREC\|Accurate}} ~~as the final step~~. We need to print out the electrostatic part of the potential (disregarding the XC part) by setting {{TAG\|WRT_POTENTIAL\|hartree ~~inoic~~}}. It is also possible to use {{TAG\|LVHAR\|.TRUE.}} to get the same data, but be aware that {{TAG\|LVTOT}} would include the (semi-)local exchange-correlation potential <math>V_{\text{xc}}(\mathbf{r})</math>, which we want to exclude here.		:Perform a static calculation with tetrahedron smearing ({{TAG\|ISMEAR\|-5}}) and {{TAG\|PREC\|Accurate}}. We need to print out the electrostatic part of the potential (disregarding the XC part) by setting {{TAG\|WRT_POTENTIAL\|hartree ionic}}. It is also possible to use {{TAG\|LVHAR\|.TRUE.}} to get the same data, but be aware that {{TAG\|LVTOT}} would include the (semi-)local exchange-correlation potential <math>V_{\text{xc}}(\mathbf{r})</math>, which we want to exclude here.

	:The planar average <math>\bar{V}(z)</math> of the electrostatic potential is obtained by averaging the 3D potential over the in-plane (xy) grid at each <math>z</math> point (assuming the interface is located in the xy-plane). Then, a macroscopic average ~~can be~~ computed by applying a uniform box-car convolution of window length <math>L</math>~~. Where~~ <math>L</math> ~~is a bit less than~~ the ~~thickness of the thinner of the two materials~~. ~~This~~ removes short-range oscillations within each atomic layer while preserving the long-range potential step across the interface. ~~Read~~ the potential from {{FILE\|vaspout.h5}}, ~~compute~~ both averages, and ~~plot~~:		:The planar average <math>\bar{V}(z)</math> of the electrostatic potential is obtained by averaging the 3D potential over the in-plane (xy) grid at each <math>z</math> point (assuming the interface is located in the xy-plane). Then, a macroscopic average is computed by applying a uniform box-car convolution of window length <math>L</math>, where <math>L</math> equals one bulk primitive-cell repeat period along the interface normal. Averaging over exactly one such period removes the short-range oscillations within each atomic layer while preserving the long-range potential step across the interface. The Python script below attempts to find the optimal <math>L</math> automatically by sweeping a range of values and selecting the one that minimizes the RMS gradient of the macroscopic average in the bulk-like plateau regions of both materials simultaneously. It reads the potential from {{FILE\|vaspout.h5}}, computes both averages, plots the data and reports the macroscopic electrostatic potential difference <math>\Delta\bar{V}</math>:

	:<div class="toccolours mw-customtoggle-~~script~~">'''Click to show python script'''</div>		:<div class="toccolours mw-customtoggle-potentialscript">'''Click to show python script'''</div>
	<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-~~script~~">		<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-potentialscript">

	:<syntaxhighlight lang="python">		:<syntaxhighlight lang="python">
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	</syntaxhighlight>		</syntaxhighlight>
	~~</div>~~
	</div>		</div>

	'''Step 5:''' Compute the Schottky barrier heights.		'''Step 5:''' Compute the Schottky barrier heights.

	:With <math>E_{\mathrm{F}}</math>, <math>E_{\mathrm{VBM}}</math>, <math>E_{\mathrm{g}}</math>, and <math>\Delta\bar{V}</math> in hand, the potential alignment difference <math>\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		:With <math>E_{\mathrm{F}}</math>, <math>E_{\mathrm{VBM}}</math>, <math>E_{\mathrm{g}}</math>, and <math>\Delta\bar{V}</math> in hand, the potential alignment difference <math>\Delta\bar{V} = \bar{V}_{\mathrm{m}} - \bar{V}_{\mathrm{sc}}</math> connects the bulk reference frames. The p-type and n-type Schottky barrier heights are then

	::<math>		::<math>
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	For Al(111) and Si(111), a 4×4 Al supercell matched to a 3×3 Si supercell achieves a mismatch of 0.9%. For this example, we chose 13 Al layers, and 6 Si bilayers. The starting interface distance before relaxation was set to 2.3 Å We have fully relaxed the cell, but fixed all but the 4 layers closest to the interfaces (2 Al layers, and 1 Si bi-layer for each interface) with the [[POSCAR#poscar6\|selective dynamics]] {{FILE\|POSCAR}} option. The relaxed structure is printed here:		For Al(111) and Si(111), a 4×4 Al supercell matched to a 3×3 Si supercell achieves a mismatch of 0.9%. For this example, we chose 13 Al layers, and 6 Si bilayers. The starting interface distance before relaxation was set to 2.3 Å We have fully relaxed the cell, but fixed all but the 4 layers closest to the interfaces (2 Al layers, and 1 Si bi-layer for each interface) with the [[POSCAR#poscar6\|selective dynamics]] {{FILE\|POSCAR}} option. The relaxed structure is printed here:
	<div class="toccolours mw-customtoggle-~~script~~">'''Click to show POSCAR'''</div>		<div class="toccolours mw-customtoggle-poscar">'''Click to show POSCAR'''</div>
	<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-~~script~~">		<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-poscar">
	<pre>		<pre>
	Al208 Si108		Al208 Si108
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	</pre>		</pre>
	</div>		</div>
	~~</div>~~		After computing the electrostatic potential by setting {{TAG\|WRT_POTENTIAL\|hartree ionic}}, {{TAG\|PREC\|Accurate}}, and {{TAG\|KSPACING\|0.2}}, we can run the python script referenced above in Step 4.
	After computing the electrostatic potential by setting {{TAG\|WRT_POTENTIAL\|hartree ionic}}, {{TAG\|PREC\|Accurate}}, and {{TAG\|KSPACING\|0.2}}, we can run the python script referenced above in ~~Step4~~.
	For the Al(111)/Si(111) interface studied, we get the following results: <math>\bar{V}_{\mathrm{m}} = -0.694</math> eV, <math>\bar{V}_{\mathrm{sc}} = +1.192</math> eV, and <math>\Delta\bar{V} = -1.886</math> eV.		For the Al(111)/Si(111) interface studied, we get the following results: <math>\bar{V}_{\mathrm{m}} = -0.694</math> eV, <math>\bar{V}_{\mathrm{sc}} = +1.192</math> eV, and <math>\Delta\bar{V} = -1.886</math> eV.
	[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]		[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]

Wolloch at 09:12, 24 March 2026

2026-03-24T09:12:20Z

← Older revision		Revision as of 09:12, 24 March 2026
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	The [https://en.wikipedia.org/wiki/Schottky_barrier\| 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>\varphi_{\mathrm{p}}</math> describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height <math>\varphi_{\mathrm{n}}</math> describes the barrier seen by electrons in the conduction band. In {{VASP}}, the ~~'''~~potential alignment method~~'''~~ can be used to estimate the Schottky barrier{{cite\|baldereschi:prl:1988}}{{cite\|peressi:1998}}.		The [https://en.wikipedia.org/wiki/Schottky_barrier\| 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>\varphi_{\mathrm{p}}</math> describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height <math>\varphi_{\mathrm{n}}</math> describes the barrier seen by electrons in the conduction band. In {{VASP}}, the potential alignment method can be used to estimate the Schottky barrier{{cite\|baldereschi:prl:1988}}{{cite\|peressi:1998}}.

	== Required quantities ==		== Required quantities ==

Wolloch: Wolloch moved page Construction:Schottky barrier and computing tunneling current to Construction:Calculating the Schottky barrier

2026-03-23T16:24:50Z

Wolloch moved page Construction:Schottky barrier and computing tunneling current to Construction:Calculating the Schottky barrier

← Older revision	Revision as of 16:24, 23 March 2026
(No difference)

Wolloch at 16:24, 23 March 2026

2026-03-23T16:24:19Z

← Older revision		Revision as of 16:24, 23 March 2026
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	The [https://en.wikipedia.org/wiki/Schottky_barrier\| 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>\varphi_{\mathrm{p}}</math> describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height <math>\varphi_{\mathrm{n}}</math> describes the barrier seen by electrons in the conduction band. In {{VASP}}, the ''potential alignment method can be used to estimate the Schottky barrier{{cite~~:balderescchi~~:prl:1988}}{{cite:peressi:1998}}.		The [https://en.wikipedia.org/wiki/Schottky_barrier\| 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>\varphi_{\mathrm{p}}</math> describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height <math>\varphi_{\mathrm{n}}</math> describes the barrier seen by electrons in the conduction band. In {{VASP}}, the '''potential alignment method''' can be used to estimate the Schottky barrier{{cite\|baldereschi:prl:1988}}{{cite\|peressi:1998}}.


	== Required quantities ==		== Required quantities ==

Wolloch at 16:14, 23 March 2026

2026-03-23T16:14:46Z

← Older revision		Revision as of 16:14, 23 March 2026
Line 1:		Line 1:
	The [https://en.wikipedia.org/wiki/Schottky_barrier\| 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>\varphi_{\mathrm{p}}</math> describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height <math>\varphi_{\mathrm{n}}</math> describes the barrier seen by electrons in the conduction band. In {{VASP}}, the ''potential alignment method~~'''''citation needed'''~~ can be used to estimate the Schottky barrier.		The [https://en.wikipedia.org/wiki/Schottky_barrier\| 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>\varphi_{\mathrm{p}}</math> describes the barrier seen by holes in the semiconductor valence band, and the n-type Schottky barrier height <math>\varphi_{\mathrm{n}}</math> describes the barrier seen by electrons in the conduction band. In {{VASP}}, the ''potential alignment method can be used to estimate the Schottky barrier{{cite:balderescchi:prl:1988}}{{cite:peressi:1998}}.

Wolloch: /* Example */

2026-03-23T14:25:29Z

Example

← Older revision		Revision as of 14:25, 23 March 2026
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	[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]		[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]

	Finally we arrive at the Schottky barriers:		Finally, we arrive at the Schottky barriers:

	:<math>\varphi_{\mathrm{p}} = \Delta\bar{V} + E_{\mathrm{F}} - E_{\mathrm{VBM}} = -1.886 + 8.085 - 5.449 = 0.75</math> eV		:<math>\varphi_{\mathrm{p}} = \Delta\bar{V} + E_{\mathrm{F}} - E_{\mathrm{VBM}} = -1.886 + 8.085 - 5.449 = 0.75</math> eV,

	and		which fits well to the experimental values that vary between 0.68 and 0.81 eV, depending on sample and experimental conditions{{cite\|altindal:2007}}, and

	:<math>\varphi_{\mathrm{n}} = E_{\mathrm{g}} - \varphi_{\mathrm{p}} = 1.160 - 0.75 = 0.41</math> eV.		:<math>\varphi_{\mathrm{n}} = E_{\mathrm{g}} - \varphi_{\mathrm{p}} = 1.160 - 0.75 = 0.41</math> eV.

Wolloch: /* Example */

2026-03-23T14:09:43Z

Example

← Older revision		Revision as of 14:09, 23 March 2026
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	[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]		[[File:Potential alignment Si111 Al111.png\|thumbnail\|800px\|center\|Potential alignment plot for the Al(111) Si(111) interface.]]

	Finally we arrive at the Schottky barriers: <math>\varphi_{\mathrm{p}} = \Delta\bar{V} + E_{\mathrm{F}} - E_{\mathrm{VBM}} = -1.886 + 8.085 - 5.449 = 0.75</math> eV and <math>\varphi_{\mathrm{n}} = E_{\mathrm{g}} - \varphi_{\mathrm{p}} = 1.160 - 0.75 = 0.41</math> eV.		Finally we arrive at the Schottky barriers:

			:<math>\varphi_{\mathrm{p}} = \Delta\bar{V} + E_{\mathrm{F}} - E_{\mathrm{VBM}} = -1.886 + 8.085 - 5.449 = 0.75</math> eV

			and

			:<math>\varphi_{\mathrm{n}} = E_{\mathrm{g}} - \varphi_{\mathrm{p}} = 1.160 - 0.75 = 0.41</math> eV.

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