6. Lever arm theory

Definition of the lever arm for a single quantum dot

The lever arm \(\alpha_\mathrm G\) of a gate G on a quantum dot is the conversion factor between the potential \(\varphi_\mathrm{bias}^\mathrm G\) applied on G and the electrochemical potential \(\mu\) of the quantum dot, i.e.

(6.1)\[\mu = \mu_0-e\alpha_{\mathrm G} \varphi_\mathrm{bias}^\mathrm G,\]

where \(e>0\) is the elementary charge and \(\mu_0\) is the dot’s electrochemical potential when \(\varphi_\mathrm{bias}^\mathrm G=0.\)

The electrochemical potential for a transition between the \((N-1)\)-electron and \(N\)-electron ground states of a quantum dot is [HKP+07]

(6.2)\[\mu(N) \equiv E_\mathrm{tot}(N) - E_\mathrm{tot}(N-1),\]

where \(E_\mathrm{tot}(N)\) is the total energy of a dot in the \(N\)-electron ground state.

In particular, within the constant-interaction model, the electrochemical potential of a quantum dot modeled like a single-electron transistor is given by [Fuh03]

(6.3)\[\mu(N) = \epsilon_N + \frac{e^2}{C_\Sigma}\left(N-\frac12\right) + e\left(\frac{Q_{bg}}{C_\Sigma} + \sum_{G=1}^n \frac{C_{0G}}{C_\Sigma} \varphi^G_\mathrm{bias}\right),\]

where \(\epsilon_N\) is the sum of the single-particle energies occupied by \(N\) electrons and \(C_\Sigma\equiv C_{00}=-\sum_{G=1}^n C_{0G}\) is the self capacitance of the dot, with \(C_{0G}\) the capacitance between the dot and gate \(G\). Here, sums over \(G\) run over all gates \(G\;\in\;\{1,...,n\}\) that can influence the quantum dot. In addition, \(Q_{bg}\) is the charge that remains on the dot (index \(0`\)) if all gate potentials are set to zero. Finally, \(\varphi^G_\mathrm{bias}\) is the potential applied on gate \(G\).

Identifying Eq. (6.3) with Eq. (6.1), we see that, within the constant-interaction model, the lever arm of gate \(G\) on the dot is given by [Fuh03]

(6.4)\[\alpha_{\mathrm G} = -\frac{C_{0\mathrm G}}{C_\Sigma},\]

i.e., by the ratio of the capacitance between the dot and gate \(G\) and the self capacitance of the dot.

To calculate the lever arm in QTCAD, we do not need to explicitly calculate these capacitances because we use the full device geometry instead of a lumped-element circuit model. Instead, we may directly calculate the response of the electronic structure of the quantum dot upon gate bias changes and extract the lever arm from a linear fit of this response.

QTCAD thus contains a lever arm module which may be used to calculate \(\alpha_\mathrm{G}\) in the following way.

  1. Solve the non-linear Poisson equation for a range of biases \(\varphi^G_\mathrm{bias}\) on gate \(G\), with all other gate potentials fixed.

  2. For each gate bias, solve the single-electron Schrödinger equation to find the quantum dot eigenenergies.

  3. Perform a linear fit of the ground state energy with respect to gate bias; the slope of the best fit gives \(-e\alpha_\mathrm{G}\).

An example of application of this method is given in Calculating the lever arm of a quantum dot.

Applications of the lever arm

Experimentally, lever arms are typically measured from the slope of the lines separating stability regions in charge-stability diagrams obtained from differential conductance measurements in the Coulomb blockade regime [KKN+18].

The lever arm determines the extent by which a specific gate is able to tune the chemical potentials of a quantum dot. From the above definitions, lever arms are unitless. For ideal capacitive coupling, the lever arm is \(1\), meaning that a change of \(1\textrm{ V}\) in the gate voltage leads to a change of \(-1\textrm{ eV}\) in the energy levels. More realistically, the lever arm is between \(0\) and \(1\).

The lever arm plays an important role in determining the strength of the coupling between the quantum dot and its environment. As such, it is useful to calculate the frequency of Rabi oscillations under electric-dipole spin resonance, to estimate the impact of charge noise on spin-qubit coherence, or to evaluate the strength of the coupling between a quantum-dot system and a microwave resonator [BLQCPLadriere16, CSorensenL04]. In addition, when the dependence of single-particle levels on gate biases is linear, computational steps associated with solving the Schrödinger and Poisson equations may be skipped when sweeping biases (e.g., in the evaluation of a charge stability diagram), which can lead to significant savings in computation time.

Source and drain lever arms

In the case of source and drain reservoirs, the best way to model a bias in QTCAD is to shift the Fermi level \(E_F\) over the entire source or drain volume. This is because the source and drain may individually be considered in quasi-equilibrium with their own environments, with the Fermi level only varying in between. This model is appropriate to describe sequential tunneling through quantum dots since in this scenario: (1) the source and drain statistics are not significantly impacted by the presence of the quantum dot system and (2) the source–drain bias is sufficiently small to have negligible impact on quantum dot electrostatics.

We may then define a source or drain lever arm on the quantum dot levels from

(6.5)\[\mu = \mu_0-e\alpha_{r} \varphi_\mathrm{bias}^r,\]

where \(\mu\) and \(\mu_0\) are (as above) the electrochemical potential of the quantum dot with and without bias, and \(\varphi_\mathrm{bias}^r\equiv - E_F^r/e\) is the potential applied on reservoir \(r\;\in\;\{\mathrm{S}, \mathrm{D}\}\) that corresponds to the (quasi-)Fermi level \(E_F^r\), with \(\mathrm S\) (\(\mathrm D\)) labeling the source (drain). The source or drain lever arm is then defined as the coefficient \(\alpha_r\) that sets the linear relationship between the source or drain bias and the quantum dot electrochemical potential, in complete analogy with the definition of a gate lever arm.

Generalization to multiple-dot systems: Lever arm matrix

In a general multiple-dot system, multiple gate biases may be applied, each having a distinct effect on the energy level spectrum of the quantum dots.

In such a situation, a single scalar-valued number is insufficient to describe the linear response of the system to applied biases.

The simplest way to handle multiple-dot systems is to choose a reference gate bias configuration in which each single-electron eigenstate is mostly localized within a single dot. Let us denote the potential applied at each contact \(G\) in this configuration by \(\varphi_{\mathrm{bias},G}^0\). Expanding the single-particle energy levels \(E_i\) of the multiple-dot system to first order in the detunings \(\varphi_{\mathrm{bias},G}-\varphi_{\mathrm{bias},G}^0\) with respect to the reference configuration then leads us to

(6.6)\[E_{i} \approx E^{0}_{i} - e \sum_G \alpha_{iG} (\varphi_{\mathrm{bias},G}-\varphi_{\mathrm{bias},G}^0),\]

where \(E^0_i\) is \(i\)-th single-particle energy level in the reference configuration and \(\alpha_{iG}\) is the lever arm matrix, defined by

(6.7)\[\alpha_{iG}=-\frac{1}{e}\frac{\partial E_i}{\partial \varphi_\mathrm{bias,G}}.\]

In general, applying a bias on one gate may have an impact on the levels of a quantum dot below another gate, an effect that is captured to first order by the lever arm matrix.


Strong tunneling may invalidate the lever arm matrix approximation. This is easily seen in the case of a nearly symmetric double quantum dot, in which case the splitting between the ground and first excited states is approximately given by \(E_{01} = \sqrt{\varepsilon^2+t^2}\), where \(\varepsilon\) is the energy difference between the confinement potential minima of the two dots, and \(t\) is the tunnel splitting. When \(t\gg\varepsilon\), we have \(E_{01}\approx t + \varepsilon^2/2t\), which is quadratic in the plunger gate biases. Consequently, the lever arm matrix approximation breaks down when the tunnel splitting between two dots in the system is comparable to or larger than their energy detuning.