Coulomb Blockade

Coulomb blockade suppresses sequential electron transfer through a small island because changing its integer charge costs energy. A gate changes that cost; a source–drain bias makes some transfers energetically favorable. The resulting conductance peaks and bias thresholds let us measure charging energies and capacitance ratios.

We begin with a normal-metal island, opaque tunnel barriers and a dense single-particle spectrum. This orthodox model treats each transfer perturbatively and describes the island by its charge-state probabilities. Discrete dot levels, strong tunneling, superconductivity and an energy-exchanging environment require extensions.

Required background. Chemical potential explains addition energies and reservoir comparisons; quantum dots introduces confinement and charging; mesoscopic transport develops sequential rates and charge-state probabilities.

Integer charge and charging energy

The island has $n$ excess electrons and charge $-ne$. Let $C_j$ be its capacitance to electrode $j$ at electrical potential $V_j$, including source, drain, gate and any stray conductors. Define

$$\begin{aligned} C_\Sigma&=\sum_j C_j,\qquad E_C=\frac{e^2}{2C_\Sigma},\\ n_g&=\frac{\sum_j C_jV_j+Q_0}{e}. \end{aligned}$$

$Q_0$ is a static charge offset. At fixed electrode voltages, the charge-dependent electrostatic potential, including the capacitor response to voltage sources, is

$$F_n=E_C(n-n_g)^2$$

up to an $n$-independent term. Some authors call $e^2/C_\Sigma=2E_C$ the charging energy; compare the defining equation before comparing numbers. The orthodox description and its experimental realization are discussed by Averin and Likharev (1986) and Fulton and Dolan (1987).

The island’s next addition threshold is a difference of parabolas:

$$\begin{aligned} \mu_{n+1}^{\rm isl}&=F_{n+1}-F_n\\ &=2E_C(n+\tfrac12-n_g). \end{aligned}$$

At zero lead bias, neighboring charge states are degenerate at $n_g=n+1/2$. At an integer $n_g=n$, either adding or removing one electron costs $E_C$. The spacing between successive addition thresholds is instead $2E_C$: it is the second difference $F_{n+1}-2F_n+F_{n-1}$. These are distinct energies.

Changing only the gate gives the ideal period $\Delta V_g=e/C_g$. A change $\delta Q_0$ shifts the same pattern by $\delta V_g=-\delta Q_0/C_g$; slowly moving charges therefore cause offset drift.

Top: the lowest parabola selects the stable charge at zero bias. Bottom: differential conductance from the normal-metal sequential-tunneling rates below, with equal junction resistances, $C_S/C_\Sigma=0.7$, and $k_BT=0.08E_C$. The grayscale saturates at $(R_S+R_D)dI/dV=1$, so black also represents values above this limit. Here $x=(C_gV_g+Q_0)/e$, the drain is grounded, and $v=eV/(2E_C)$. Dashed lines are zero-temperature thresholds. The $n=0$ diamond tips occur at $(x,v)=(-0.2,1)$ and $(0.2,-1)$; the vertical cut at $x=0$ meets the edges earlier, at $|v|=1/1.4$. Thermal rounding softens the edges without changing the underlying electrostatic construction.

The figure uses the full charge-state rate equation, with charges $-4\leq n\leq4$; enlarging this range leaves the displayed map unchanged. It does not include cotunneling or an environmental energy-exchange spectrum.

The energy change of one tunneling event

Electrical potential and electron electrochemical potential have opposite signs: take $\mu_j=-eV_j$ relative to a common reference. If an electron enters the island from lead $j$, the lead loses an electron as the island gains one. The total free-energy change is therefore

$$\Delta F_j^+(n)=\mu_{n+1}^{\rm isl}-\mu_j.$$

For an electron leaving an $n$-electron island,

$$\Delta F_j^-(n)=-\mu_n^{\rm isl}+\mu_j.$$

The reverse of $n\to n+1$ is $n+1\to n$, so $\Delta F_j^-(n+1)=-\Delta F_j^+(n)$. Keeping the charge index is essential. A positive $\Delta F$ is uphill; a negative one releases energy.

For a normal-metal junction with resistance $R_j$, constant densities of states and the same electron temperature on both sides, the orthodox rate is

$$\gamma_j(\Delta F)=\frac{1}{e^2R_j} \frac{\Delta F}{e^{\Delta F/(k_BT)}-1}.$$

Its numerator and denominator have the same sign. The expression counts occupied initial states and empty final states; equivalently its energy factor is $\int d\epsilon\,f(\epsilon)[1-f(\epsilon-\Delta F)]$. This continuum phase space distinguishes a metallic junction from a single discrete dot level, whose rate involves a Fermi factor rather than this energy integral (Ingold and Nazarov, 1992; Beenakker, 1991).

Three limits help check the formula:

$$\begin{aligned} \gamma_j(0)&=\frac{k_BT}{e^2R_j},\\ \gamma_j(\Delta F<0)&\longrightarrow\frac{-\Delta F}{e^2R_j},\\ \gamma_j(\Delta F>0)&\longrightarrow0 \quad(T\to0). \end{aligned}$$

It also obeys local detailed balance,

$$\frac{\gamma_j(\Delta F)}{\gamma_j(-\Delta F)} =e^{-\Delta F/(k_BT)}.$$

A finite rate at degeneracy does not mean equilibrium current: forward and backward events balance.

When are these rates reliable?

For an opaque normal junction, each transmission eigenvalue is small and its total dimensionless conductance should satisfy $R_K/R_j\ll1$, where $R_K=h/e^2\simeq25.8\,\mathrm{k\Omega}$. This controls quantum charge fluctuations. Separately, resolving a deep blockade valley requires $k_BT\ll E_C$, while using thermally broadened sequential rates requires lifetime broadening $\hbar\gamma\ll k_BT$. These are controlled-regime conditions, not sharp boundaries; near a charge degeneracy the thermal populations are deliberately shared.

For example, an energy-releasing event with $-\Delta F=0.20\,\mathrm{meV}$ through $R_j=1.0\,\mathrm{M\Omega}$ has a low-temperature rate about $1.25\times10^9\,\mathrm{s^{-1}}$. Its energy width $\hbar\gamma$ is only $0.82\,\mu\mathrm{eV}$. Always distinguish a rate from an energy broadening.

Coulomb diamonds from four inequalities

Ground the drain and bias the source: $V_D=0$, $V_S=V$. Then $\mu_D=0$ and $\mu_S=-eV$. Write $\alpha_S=C_S/C_\Sigma$ and $\alpha_g=C_g/C_\Sigma$; any other electrode potentials are held fixed. For one island transition,

$$\mu^{\rm isl}=\mu^0-e\alpha_gV_g-e\alpha_SV.$$

An $n$-electron state blocks sequential transfer at zero temperature if adding from either lead and removing to either lead are all uphill. Compactly,

$$\begin{aligned} \mu_{n+1}^{\rm isl}&>\max(\mu_S,\mu_D),\\ \mu_n^{\rm isl}&<\min(\mu_S,\mu_D). \end{aligned}$$

Equality gives an edge. Drain alignment, $\mu^{\rm isl}=0$, and source alignment, $\mu^{\rm isl}=-eV$, have slopes

$$\begin{aligned} m_D&=\frac{dV}{dV_g}=-\frac{C_g}{C_S},\\ m_S&=\frac{dV}{dV_g}=\frac{C_g}{C_\Sigma-C_S}. \end{aligned}$$

The denominator in $m_S$ includes drain, gate and stray capacitances. Swapping the biased lead changes the formulas. Reversing the plotted bias reverses both slope signs.

Tip height versus a vertical cut

Let the addition spacing be $E_{\rm add}=\mu_{n+1}^{\rm isl}-\mu_n^{\rm isl}$, independent of voltage in the constant-capacitance model. At the positive tip, one transition aligns with each lead. Subtract those alignment equations:

$$eV_{\rm tip}=E_{\rm add}.$$

For the purely metallic charging model, $E_{\rm add}=2E_C$, so the maximum positive bias is $2E_C/e$ and the negative tip is at $-2E_C/e$. The full vertical extent is $4E_C/e$. This result is independent of capacitance division, although the gate positions of the two tips depend on it.

By contrast, a vertical cut at the zero-bias charge center has threshold

$$e|V|_{\rm center} =\frac{E_C}{\max(\alpha_S,1-\alpha_S)}.$$

It reaches the tip height only for $\alpha_S=1/2$. Saying merely “diamond height” can confuse these three quantities. Bias-dependent capacitances, changing spectra, superconducting leads or multiple islands require a new alignment calculation.

What the slopes determine

Suppose the gate period is $20.0\,\mathrm{mV}$ and the slopes are $m_D=-0.080$, $m_S=0.100$. The ideal metallic model gives

$$\begin{aligned} C_g&=8.01\,\mathrm{aF},\\ C_S&=C_g/0.080=100.1\,\mathrm{aF},\\ C_\Sigma-C_S&=C_g/0.100=80.1\,\mathrm{aF}. \end{aligned}$$

Thus $C_\Sigma=180.2\,\mathrm{aF}$, $\alpha_g=0.0444$ and $E_C=0.445\,\mathrm{meV}$. The remaining data give $C_D+C_0=72.1\,\mathrm{aF}$, not $C_D$ separately. Here $C_0$ collects fixed-potential stray capacitances. For a dot with substantial orbital addition energies, the gate spacing is not automatically $e/C_g$; use the addition spectrum in extracting capacitances.

Single-electron transistor

An island with two tunnel contacts and a gate is a single-electron transistor. Near a charge degeneracy, electrons can make a sequential cycle through it; between degeneracies, an uphill step suppresses that current. The gate therefore controls a periodic source–drain response.

For just two relevant charge states, $n$ and $n+1$, let $a_j$ be the addition rate from lead $j$ and $b_j$ the reverse rate. Set $A=a_S+a_D$ and $B=b_S+b_D$. The stationary probability of state $n+1$ is $p_1=A/(A+B)$, and $p_0=1-p_1$. With positive conventional current from source to drain,

$$I=e\frac{a_D b_S-a_S b_D}{A+B}.$$

This follows by subtracting electron entry from electron exit at the source; the drain gives the same steady current. The master-equation derivation extends this calculation to many states and current noise.

For a dense metallic spectrum and $k_BT\ll E_C$, expanding these rates about zero bias gives a conductance peak

$$G(\delta)=\frac{1}{R_S+R_D} \frac{\delta/(2k_BT)}{\sinh[\delta/(k_BT)]},$$

where $\delta=\mu_{n+1}^{\rm isl}$ at zero lead bias. The maximum is $1/[2(R_S+R_D)]$. This two-state metallic peak is different from the $\operatorname{sech}^2$ line shape of a thermally broadened nondegenerate discrete level (Beenakker, 1991; van Houten, Beenakker, and Staring, 1992). The continuum rate cannot be carried unchanged into a few-level dot.

A nearby charge shifts the gate operating point, so the steep side of a peak can serve as an electrometer. Charge sensitivity, bandwidth, pumps and transfer accuracy are developed in single-electron devices.

Beyond sequential charging

Inside a diamond, sequential current is suppressed rather than every possible transfer being forbidden. Virtual intermediate states allow elastic or inelastic cotunneling; their relative importance grows as thermal sequential current freezes out. An odd-occupation dot can develop a correlated Kondo resonance. Photons, vibrations and leakage paths offer other possibilities, distinguished by their bias, temperature and coupling dependence.

Additional physics	What must change in the model
Resolved dot levels	Include orbital, spin and valley addition energies
Pauli spin blockade	Include spin selection rules, even when charge transfer is energetically allowed
Superconducting island or leads	Include the gap, parity and possibly coherent Josephson tunneling
Dynamical Coulomb blockade	Include energy exchange with circuit modes, often through $P(E)$
Multiple islands	Use a capacitance matrix and several integer charges

Ingold and Nazarov (1992) explain the environmental extension. A dark region by itself does not determine which model applies. Lines parallel to a diamond edge can involve island excitations, vibrations or lead structure; their assignment needs controlled spectroscopy. Likewise, stable charge occupation is useful for a spin qubit but does not measure its coherent control.

Exercises

1. Separate three charging quantities

For $E_C=0.60\,\mathrm{meV}$ in this page’s convention, find $C_\Sigma$, the central addition cost, and the spacing between addition thresholds. If $C_g=4.0\,\mathrm{aF}$, find the gate period and its displacement when $Q_0$ increases by $0.15e$.

Solution

$C_\Sigma=e^2/(2E_C)=134\,\mathrm{aF}$. The cost at the center is $0.60\,\mathrm{meV}$; the threshold spacing is $1.20\,\mathrm{meV}$. The gate period is $e/C_g=40.1\,\mathrm{mV}$ and its shift is $-0.15$ periods, or $-6.01\,\mathrm{mV}$.

2. Shift every electrical potential

Raise all electrode potentials, including the gate and any stray-capacitance electrodes, by $V_0$. Show that a tunneling event’s free-energy change is unchanged.

Solution

$n_g$ increases by $C_\Sigma V_0/e$, so each island addition threshold decreases by $2E_CC_\Sigma V_0/e=eV_0$. Each reservoir electrochemical potential also decreases by $eV_0$. Their difference is unchanged. Shifting only source and drain while fixing the gate is a physical common-mode gate change, not this change of energy reference.

3. Locate a diamond tip

Take $E_C=0.50\,\mathrm{meV}$ and $\alpha_S=0.7$, with the drain grounded. For the $n=0$ diamond, find the positive tip bias and gate coordinate $x=(C_gV_g+Q_0)/e$, assuming other induced charges vanish. Compare with the threshold at $x=0$.

Solution

At positive bias the boundaries are $x=1/2-\alpha_Sv$ and $x=-1/2+(1-\alpha_S)v$, with $v=eV/(2E_C)$. They meet at $v=1$, $x=1/2-\alpha_S=-0.2$. Thus the positive tip is $V=1.00\,\mathrm{mV}$. At $x=0$, the first edge is reached at $v=1/(2\alpha_S)=0.714$, giving $V=0.714\,\mathrm{mV}$. The full tip-to-tip vertical extent is $2.00\,\mathrm{mV}$.

4. A finite rate at zero energy cost

For $R_j=1.0\,\mathrm{M\Omega}$ and $T=0.10\,\mathrm K$, find $\gamma_j(0)$. For an uphill cost $\Delta F=3k_BT$, find the ratio of forward to reverse rates. Explain why equilibrium current still vanishes.

Solution

The limit gives $\gamma_j(0)=k_BT/(e^2R_j)=5.38\times10^7\,\mathrm{s^{-1}}$. At cost $3k_BT$, the ratio is $e^{-3}=0.0498$. Equilibrium state probabilities obey $p_{n+1}/p_n=e^{-\Delta F/(k_BT)}$ at zero reservoir potential, so $p_n\gamma_j(\Delta F)=p_{n+1}\gamma_j(-\Delta F)$. Equal probability currents cancel despite nonzero event rates.

5. Compare metallic and discrete-level peak widths

Normalize the metallic peak to its maximum. Show that its full width at half maximum is about $4.35k_BT$ in the detuning $\delta$. Compare with a discrete nondegenerate level whose normalized conductance is $\operatorname{sech}^2[\delta/(2k_BT)]$.

Solution

The metallic normalized peak is $y/\sinh y$ with $y=\delta/(k_BT)$. Its positive half-height root solves $\sinh y=2y$ and is $y\simeq2.177$, giving full width $4.355k_BT$. The discrete-level half-height root is $|\delta|=2k_BT\operatorname{arcosh}\sqrt2$, giving width $3.525k_BT$. Convert either energy width to gate voltage using $e\alpha_g$. Using the wrong line shape biases a temperature or lever-arm estimate.

References

• D. V. Averin and K. K. Likharev, “Coulomb Blockade of Single-Electron Tunneling, and Coherent Oscillations in Small Tunnel Junctions,” Journal of Low Temperature Physics 62, 345–373 (1986), doi:10.1007/BF00683469.
• C. W. J. Beenakker, “Theory of Coulomb-Blockade Oscillations in the Conductance of a Quantum Dot,” Physical Review B 44, 1646–1656 (1991), doi:10.1103/PhysRevB.44.1646.
• T. A. Fulton and G. J. Dolan, “Observation of Single-Electron Charging Effects in Small Tunnel Junctions,” Physical Review Letters 59, 109–112 (1987), doi:10.1103/PhysRevLett.59.109.
• G.-L. Ingold and Yu. V. Nazarov, “Charge Tunneling Rates in Ultrasmall Junctions,” in Single Charge Tunneling, edited by H. Grabert and M. H. Devoret, pp. 21–107, Plenum Press (1992), doi:10.1007/978-1-4757-2166-9_2, author manuscript.
• H. van Houten, C. W. J. Beenakker, and A. A. M. Staring, “Coulomb-Blockade Oscillations in Semiconductor Nanostructures,” in Single Charge Tunneling, edited by H. Grabert and M. H. Devoret, pp. 167–216, Plenum Press (1992), doi:10.1007/978-1-4757-2166-9_5, author manuscript.