Supercapacitor

The Supercapacitor application module provides two electronic-capacitance workflows:

quantum capacitance from the electronic density of states;
total interfacial capacitance from quantum and double-layer capacitances.

It is available from:

VQkit
 -> Post-processing
 -> [7] Application-oriented analysis
 -> Supercapacitor analysis

The current menu contains:

[1] Quantum capacitance from DOS
    Finite-temperature C_Q(V) using VASP/QE electronic DOS.

[2] Total interfacial capacitance
    Combine C_Q and double-layer capacitance C_EDL in series.

[3] Show equations / conventions

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[99] Exit VQkit

[1] Quantum capacitance from DOS

Choose:

[1] Quantum capacitance from DOS

VQkit evaluates the areal quantum capacitance from the electronic density of states.

The implemented expression is:

\[\frac{C_Q}{A} = \frac{e}{A} \int D(E) \left[ -\frac{\partial f}{\partial E} \right] dE\]

where:

\(e\) is the elementary charge;
\(A\) is the electrode area;
\(D(E)\) is the electronic DOS in states/eV per simulation cell;
\(f(E)\) is the Fermi-Dirac distribution.

The current implementation uses:

\[e = 1.602176634\times10^{-19}\ \mathrm{C}\]

and:

\[k_B = 8.617333262145\times10^{-5}\ \mathrm{eV/K}\]

DOS source

VQkit reads the total DOS automatically using the common DOS post-processing backend.

Supported sources include:

VASP

DOSCAR

Quantum ESPRESSO

Typical QE DOS files produced by dos.x.

The DOS energy axis is referenced to:

E_F = 0 eV

before the quantum-capacitance integral is evaluated.

Spin-polarized DOS

For spin-polarized calculations, VQkit uses the physical total DOS:

\[D(E) = D_\uparrow(E) + D_\downarrow(E)\]

The negative spin-down convention used in some DOS plots is not used in the capacitance integral.

Fermi-Dirac derivative

The thermal broadening kernel is:

\[-\frac{\partial f}{\partial E} = \frac{1}{ 4k_B T \cosh^2\left[ \frac{E-\mu}{2k_B T} \right] }\]

where:

\(T\) is temperature;
\(\mu\) is the electrochemical potential.

In the current VQkit convention, a positive applied potential shifts the electronic chemical potential by:

\[\mu = +V\]

on the DOS energy scale expressed in eV.

Effective DOS

VQkit first evaluates the thermally broadened effective DOS:

\[D_{\mathrm{eff}}(V,T) = \int D(E) \left[ -\frac{\partial f}{\partial E} \right] dE\]

The areal quantum capacitance is then:

\[\frac{C_Q}{A} = \frac{ eD_{\mathrm{eff}} }{ A }\]

Area conversion

The electrode area is entered or obtained in:

Angstrom^2

and VQkit converts it to:

cm^2

using:

\[1\ \mathrm{\AA^2} = 10^{-16}\ \mathrm{cm^2}\]

The final quantum capacitance is reported in:

F/cm^2
microF/cm^2

Electrode-area input

VQkit provides:

[1] Read in-plane area automatically from structure
[2] Enter area manually

Automatic structure detection

VQkit searches for structures such as:

CONTCAR
POSCAR
scf.in
relax.in
vc-relax.in

and other readable QE *.in files.

The in-plane area is calculated from the first two lattice vectors:

\[A = |\mathbf{a}\times\mathbf{b}|\]

This assumes that a and b span the electrode plane.

Manual area

Alternatively, enter the electrode area directly in:

Angstrom^2

Temperature

VQkit asks:

Temperature [300 K]:

The default is:

300 K

The temperature must be greater than zero because the implemented numerical integration uses the finite-temperature Fermi derivative.

Potential modes

VQkit offers:

[1] Single potential
[2] Potential range and C_Q(V) plot

Single potential

VQkit asks:

Potential relative to E_F [0.0 V]:

The result includes:

potential;
effective DOS;
quantum capacitance.

Potential range

VQkit asks:

Start potential [-1.0 V]:
End potential [1.0 V]:
Potential step [0.05 V]:

The potential grid is generated inclusively from start to end.

If start and end are entered in reverse order, VQkit automatically swaps them.

Potential-dependent output

For a potential range, VQkit calculates:

\[C_Q(V)\]

at every selected potential.

The main data file is:

VQkit_QUANTUM_CAPACITANCE.dat

The columns are:

Potential_V
Effective_DOS_states_per_eV
Cq_uF_cm2

Quantum-capacitance summary

VQkit also writes:

VQkit_QUANTUM_CAPACITANCE_summary.txt

The summary records:

DOS source;
electrode area;
temperature;
Fermi-level energy reference;
the equation used;
the location of the potential-dependent data file.

Plots

For a potential range, VQkit generates:

VQkit_QUANTUM_CAPACITANCE.png
VQkit_QUANTUM_CAPACITANCE.pdf

The plot shows:

Potential relative to E_F (V)

versus:

Quantum capacitance (microF/cm^2)

Low-temperature limit

As:

\[T\rightarrow0\]

the Fermi derivative approaches a delta function around the Fermi level.

Therefore the implemented expression approaches:

\[\frac{C_Q}{A} \approx \frac{ eD(E_F) }{ A }\]

when the DOS is expressed in states/eV per simulation cell.

This relation is also displayed in the VQkit equations/conventions menu.

Example quantum-capacitance workflow

Assume a completed electronic-structure calculation contains:

DOSCAR
POSCAR

for VASP, or the corresponding QE DOS and structure files.

Step 1

Open:

Post-processing
Application-oriented analysis
Supercapacitor analysis

Step 2

Choose:

[1] Quantum capacitance from DOS

Step 3

Choose automatic area detection:

[1] Read in-plane area automatically from structure

Step 4

Confirm the detected structure.

Step 5

Accept or change:

Temperature [300 K]

Step 6

Choose:

[2] Potential range and C_Q(V) plot

Step 7

For example:

Start potential : -1.0 V
End potential   :  1.0 V
Potential step  :  0.05 V

Step 8

Inspect:

VQkit_QUANTUM_CAPACITANCE.dat
VQkit_QUANTUM_CAPACITANCE.png
VQkit_QUANTUM_CAPACITANCE.pdf

[2] Total interfacial capacitance

Choose:

[2] Total interfacial capacitance

VQkit combines the quantum capacitance and electric-double-layer capacitance in series.

The implemented relation is:

\[\frac{1}{C_{\mathrm{total}}} = \frac{1}{C_Q} + \frac{1}{C_{\mathrm{EDL}}}\]

Therefore:

\[C_{\mathrm{total}} = \left( \frac{1}{C_Q} + \frac{1}{C_{\mathrm{EDL}}} \right)^{-1}\]

Input

VQkit asks:

Quantum capacitance C_Q (uF/cm^2):
Double-layer capacitance C_EDL (uF/cm^2):

Both values must be positive.

The result is reported as:

Total capacitance : ... uF/cm^2

Physical interpretation

The smaller capacitance tends to dominate the total series capacitance.

For example, if:

C_Q << C_EDL

then:

\[C_{\mathrm{total}} \approx C_Q\]

Conversely, if:

C_{\mathrm{EDL}} << C_Q

then:

\[C_{\mathrm{total}} \approx C_{\mathrm{EDL}}\]

[3] Show equations / conventions

Choose:

[3] Show equations / conventions

VQkit displays the main relations:

\[\frac{C_Q}{A} = \frac{e}{A} \int D(E) \left[ -\frac{\partial f}{\partial E} \right] dE\]

and:

\[\frac{1}{C_{\mathrm{total}}} = \frac{1}{C_Q} + \frac{1}{C_{\mathrm{EDL}}}\]

It also reminds the user that:

C_Q is obtained from the electronic DOS;
C_EDL depends on the electrolyte/interface;
C_EDL is not obtained from a bare electronic DOS calculation.

Quantum capacitance versus double-layer capacitance

Quantum capacitance

Quantum capacitance describes the electronic DOS limitation of charge storage in the electrode.

It depends strongly on:

DOS near the Fermi level;
electrode potential;
temperature;
electrode area;
spin-resolved electronic structure.

Double-layer capacitance

The electric-double-layer capacitance depends on the physical electrode / electrolyte interface.

It can depend on:

electrolyte composition;
ion concentration;
solvent;
surface charge;
dielectric screening;
pore structure;
electrode morphology;
interfacial structure.

The current VQkit module does not derive C_EDL from first-principles solvent/electrolyte simulations.

It must be supplied independently.

DOS convergence

Quantum capacitance can be highly sensitive to the DOS close to the Fermi level.

Before using the result, verify convergence with respect to:

k-point sampling;
smearing/broadening;
DOS energy-grid density;
plane-wave cutoff;
magnetic state;
exchange-correlation functional.

A coarse DOS near:

E_F

can produce an inaccurate C_Q(V) curve.

Energy window

The finite-temperature integral assumes that the available DOS energy range contains the relevant region around the shifted chemical potential.

If the requested potential moves the chemical potential close to or beyond the edge of the computed DOS window, the capacitance result can be unreliable.

Use a sufficiently broad DOS energy range.

Electrode area

The area used in the areal capacitance is:

\[A = |\mathbf{a}\times\mathbf{b}|\]

for automatic structure-based input.

This is appropriate when a and b define the physical electrode plane.

For a differently oriented slab, verify the area manually.

One-sided versus two-sided electrodes

The current code normalizes quantum capacitance by the supplied simulation-cell in-plane area.

It does not automatically divide or multiply by the number of exposed surfaces.

When comparing with an experimental areal capacitance, make sure the normalization convention is physically consistent.

Potential convention

The current VQkit potential coordinate is:

potential relative to E_F

and is used as an electronic chemical-potential shift in eV.

This is not automatically equivalent to an absolute electrochemical potential versus:

SHE;
RHE;
Li/Li+;
another experimental reference electrode.

Additional reference alignment is required for direct experimental-potential comparison.

Temperature dependence

Finite temperature broadens the DOS contribution around the chemical potential.

At higher temperature, a wider energy range contributes to:

\[D_{\mathrm{eff}}\]

through the Fermi-Dirac derivative.

The current module evaluates this electronic thermal broadening only.

It does not include structural or electrolyte temperature dependence.

Supercapacitor limitations

The current Supercapacitor module is focused on electronic quantum-capacitance analysis.

It does not automatically calculate:

Helmholtz-layer capacitance;
Gouy-Chapman capacitance;
pore capacitance;
pseudocapacitance;
redox capacitance;
ion adsorption free energies;
electrolyte decomposition;
charge-dependent electrode structure;
constant-potential electronic response;
full electrochemical impedance.

These require additional physical models or calculations.

Pseudocapacitive materials

For transition-metal oxides, hydroxides, MXenes, conducting polymers, and related pseudocapacitive materials, experimentally measured capacitance may contain substantial Faradaic contributions.

The DOS-derived C_Q should therefore not be interpreted as the complete experimental capacitance.

It is an electronic-structure descriptor.

Recommended workflow

For a first-principles supercapacitor screening study:

Relax the electrode structure.
Perform a converged SCF calculation.
Generate a high-quality DOS.
Confirm the Fermi level.
Determine the correct in-plane electrode area.
Run Quantum capacitance from DOS.
Calculate C_Q(V) over the desired potential interval.
Inspect the DOS near every relevant chemical potential.
Obtain C_EDL from experiment or an independent interface model if available.
Combine C_Q and C_EDL with the series relation.
State the potential and area normalization conventions clearly.
Avoid interpreting C_Q alone as full device capacitance.

Connection to other VQkit modules

Electronic DOS

The quantum-capacitance workflow uses the common VQkit DOS backend.

For detailed DOS analysis, use:

Post-processing
-> Electronic structure
-> Total density of states

Structural area

The automatic electrode area is derived from the structure using the same structure-reading infrastructure used elsewhere in VQkit.

Charge redistribution

For interface-related electronic analysis, complementary tools include:

Charge-density difference
Bader charge
ELF
Work function

These quantities can help interpret why the DOS and electronic response change after functionalization or adsorption.

Generated files summary

Quantum capacitance data

VQkit_QUANTUM_CAPACITANCE.dat

Quantum capacitance summary

VQkit_QUANTUM_CAPACITANCE_summary.txt

Potential-dependent plot

VQkit_QUANTUM_CAPACITANCE.png
VQkit_QUANTUM_CAPACITANCE.pdf

Summary

The current VQkit Supercapacitor module provides:

Quantum capacitance
-> VASP/QE DOS input
-> finite-temperature Fermi broadening
-> single-potential or C_Q(V) analysis
-> automatic or manual electrode area

Total interfacial capacitance
-> C_Q
-> C_EDL
-> series combination

The workflow is intended as a transparent electronic-capacitance analysis rather than a complete model of an electrochemical supercapacitor device.