Graphene Electrode Quantum Capacitance

Electrode Potential

Gate / electrode voltage V 0.00 V

Electrolyte

Helmholtz (double-layer) capacitance C_H 18 µF/cm²

Ion concentration Medium

Temperature

Electrode temperature T 300 K

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C_Q(V) — quantum capacitanceV = 0.00

Quantum cap. C_Q

— µF/cm²

Double-layer C_H

— µF/cm²

Total (series) C_tot

— µF/cm²

Fermi level E_F

— meV

How it works

A graphene electrode's total interfacial capacitance is the series combination of two independent capacitors stacked at the electrolyte/electrode interface:

1/C_total = 1/C_Q + 1/C_H
C_total   = C_Q · C_H / (C_Q + C_H)

C_H is the classical Helmholtz/electric-double-layer capacitance from ions crowding the electrolyte side of the interface. C_Q is unique to atomically thin, linearly-dispersing materials: because graphene's Dirac cone gives it a density of states that vanishes at the neutrality (Dirac) point, adding charge there requires shifting the Fermi level through a nearly empty band — the electronic sheet itself resists charging. That finite-DOS penalty is graphene's quantum capacitance:

C_Q(V,T) = (2e²k_BT)/(π(ħv_F)²) · ln[2(1+cosh(eV/k_BT))]
v_F ≈ 1.0×10⁶ m/s   (graphene Fermi velocity)

The curve is V-shaped: near V = 0 (the Dirac point) C_Q is thermally smeared but small, then grows roughly linearly with |V| as more states become available to fill. Because the two capacitors are in series, the smaller one dominates — this is why real graphene supercapacitors underperform their huge geometric surface area suggests: quantum capacitance, not electrolyte ion crowding, is often the true bottleneck near the neutrality point.

Gate voltage — moves the electrode away from the Dirac point; watch C_Q rise and the shaded Fermi level climb up (or drop down) the Dirac cone.
C_H — represents electrolyte ion size/packing at the interface; a stiffer double layer bottlenecks the total capacitance instead.
Ion concentration — thins the visible electrolyte layer and adds more instanced ions near the sheet, illustrating (not computing) Debye screening.
Temperature — thermally broadens the C_Q minimum, exactly as in the formula above.

Carbon lattice (graphene) Anions Cations Fermi level (Dirac cone)

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