Cotunneling current through quantum dots with phonon-assisted spin-flip processes

We consider cotunneling through a quantum dot in the presence of spin-flip processes induced by the coupling to acoustic phonons of the surrounding. An expression for the phonon-assisted cotunneling current is derived by means of a generalized Schrieffer-Wolff transformation. The influence of the spin-phonon coupling on the heating of the dot is considered. The result is evaluated for the case of a parabolic semiconductor quantum dot with Rashba and Dresselhaus spin-orbit coupling and a method for the determination of the spin-phonon relaxation rate is proposed.

Figure 1

Sketch of the quantum dot coupled via the tunneling Hamiltonian $H_{\mathrm{T}}$ to two leads at electrochemical potentials ${\mu }_{\mathrm{L}}$ and ${\mu }_{\mathrm{R}}$ and to a phonon bath via the spin-phonon coupling Hamiltonian $H_{\mathrm{S}}$. The dot can be either empty (0), in one of the spin directions $\uparrow$ and $\downarrow$, which are split by the Zeeman energy ${\Delta }_{\downarrow \uparrow }$, or in the singlet state $S$.

Figure 2

Ratio $p_{\uparrow }∕p_{\downarrow }$ of the populations as a function of the bias voltage $V$ in the presence of an Ohmic bath with different coupling strengths $\gamma$. The parameters are ${\Delta }_{\downarrow \uparrow }=0.1E_0$, $T=0.01E_0$, and $\hslash {\Gamma }_{\mathrm{L}}=\hslash {\Gamma }_{\mathrm{R}}=0.02E_0$. Here, $E_0=\mid E_1-\mu \mid$ is the energy difference between the energy $E_1=E_{\uparrow }=E_{\downarrow }$ of the degenerate dot level and the Fermi energy $\mu ={\mu }_{\mathrm{L}}={\mu }_{\mathrm{R}}$ in the leads in the absence of both the magnetic field and the bias voltage. The dashed vertical lines indicate the onset of the heating regime defined by $W_{\uparrow \downarrow }^{\mathrm{flip}}=W_{\downarrow \uparrow }^{\cot }$.

Figure 3

Sketch of the procedure employed for the determination of the onset $V_{\mathrm{on}}$ and of the width $\mathrm{\Delta }{V}_{\mathrm{step}}$ of the inelastic-cotunneling step. The dashed line shows the extrapolation of the differential conductance curve below the step. The dash-dotted line corresponds to the linear approximation of the step shape.

Figure 4

Cotunneling conductance $g$ as a function of the bias voltage $V$ in the presence of an Ohmic bath with different coupling strengths $\gamma$. The parameters are ${\Delta }_{\downarrow \uparrow }=0.1E_0$, $T=0.01E_0$, and (a) $\hslash {\Gamma }_{\mathrm{L}}=\hslash {\Gamma }_{\mathrm{R}}=0.02E_0$, (b) $\hslash {\Gamma }_{\mathrm{L}}=0.2E_0$ and $\hslash {\Gamma }_{\mathrm{L}}=0.002E_0$ with $E_0$ as defined in the caption of Fig. 2. In the inset of panel (b), the relative difference between the differential conductance $g$ in the presence of the spin-phonon coupling and the differential conductance $g_0$ without spin-phonon coupling is depicted.

Figure 5

Step width $\mathrm{\Delta }{V}_{\mathrm{step}}=2g_{\mathrm{inel}}^{\prime }({\Delta }_{\downarrow \uparrow }∕e)∕g_{\mathrm{inel}}({\Delta }_{\downarrow \uparrow }∕e)$ as a function of the coupling strength $\gamma$ to an Ohmic bath for different ratios of the tunneling rates ${\Gamma }_{\mathrm{L},\mathrm{R}}$, where the product ${\hslash }^2{\Gamma }_{\mathrm{L}}{\Gamma }_{\mathrm{R}}=(0.02E_0{)}^2$ is kept constant. The other parameters are as defined in Fig. 4. The horizontal dashed line shows the width according to Eq. (45) for $\gamma =0$. In the inset, the behavior for larger $\gamma$ is shown together with the step width without heating according to Eq. (45) (dashed).

Figure 6

Differential conductance $g$ as a function of the bias voltage $V$ for the parameters given in the main text. The crosses show the result in the absence of the spin-phonon coupling. In the inset, the ratio of the differential conductance $g_1$ at $V_1=1.5{\Delta }_{\downarrow \uparrow }∕e$ (see arrow in the main panel) and the differential conductance $g_0$ at zero voltage for different values of the dot-lead coupling rates $\Gamma ={\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}$. The dashed vertical line designates the value given by Eq. (50) and the arrow the dot-lead coupling rate in the experiment (see Ref. 12).