Spin effects in transport through non-Fermi-liquid quantum dots

The current-voltage characteristic of a one-dimensional quantum dot connected via tunnel barriers to interacting leads is calculated in the region of sequential tunneling. The spin of the electrons is taken into account. Non-Fermi liquid correlations implying spin-charge separation are assumed to be present in the dot and in the leads. It is found that the energetic distance of the peaks in the linear conductance shows a spin-induced parity effect at zero temperature $T$. The temperature dependence of the positions of the peaks depends on the non-Fermi liquid nature of the system. For nonsymmetric tunnel barriers negative differential conductances are predicted, which are related to the participation in the transport of collective states in the quantum dot with larger spins. Without spin-charge separation the negative differential conductances do not occur. Taking into account spin relaxation destroys the spin-induced conductance features. The possibility of observing in experiment the predicted effects are briefly discussed.

Figure 1

Schematic representation of the Luttinger liquid quantum dot $\left(\mid x\mid <a∕2\right)$ connected via tunneling barriers to semi-infinite wires. The charge $\left(g_{\rho }\right)$ and spin $\left(g_{\sigma }\right)$ interaction parameters in the leads and the dot are assumed to be different. The bias voltage $V$ controls the chemical potential difference of the leads. The gate voltage $V_g$ controls the chemical potential quantum dot (cf. Fig. 2).

Figure 2

Equivalent circuit for the quantum dot with asymmetric tunnel barriers. Left and right tunnel junctions are parameterized by the capacitances $C_{L,R}$ and the resistances $R_{L,R}\equiv {\omega }_c^2∕\pi e^2{t}_{L,R}^2$ [cf. Eq. (25)], ${\omega }_c$ cutoff energy of the leads, $t_{L,R}$ transmission amplitudes of the barriers, $C_g$ gate capacitance, $V_g$ gate voltage, $V$ bias voltage.

Figure 3

Schematic examples of excited states with respect to a background charge $n_0=(2a{k}_F∕\pi )-1$ and spin $s_0=0$. (a) $n=1$ and $s=1$, ${\mathcal{U}}_1\equiv \mathcal{U}(1,1,0,0)$ [see Eq. (9)], (b) $n=2$ and $s=0$, ${\mathcal{U}}_2\equiv \mathcal{U}(2,0,0,0)$, (c) excited state with $s=2$ for $n=2$, (d) spin excitation with respect to the state (b).

Figure 4

Schematic plot of the transition rate ${\Gamma }^{\left(\lambda \right)}\left(E\right)$, in units of ${\Gamma }_0^{\left(\lambda \right)}$, as a function of the tunneling energy $E$, for $g_0=1$ and $T=0$. Full line: no interactions in the dot, $g_{\rho }=g_{\sigma }=1$; dashed line: with interactions, $g_{\sigma }>1$ and $g_{\rho }<1$. Excitation energies are indicated at the energy-axis.

Figure 5

The prefactor $\tilde{\varphi }=2\varphi \left(g_0\right)∕\log 2$ in Eq. (36) as a function of the interaction in the leads. Inset: low-temperature shift of a conductance peak for $g_0=1.0$ (solid), $g_0=0.8$ (dashed), $g_0=0.6$ (dotted), $g_0=0.4$ (dashed-dotted) (temperature units $E_{\sigma }∕k_B$).

Figure 6

Position of the conductance peak for even $n$, as a function of the temperature $T$ (units $E_{\sigma }∕k_B$) for $g_0=0.8$, $g_{\rho }=0.3$, and $g_{\sigma }=1.0$. At high temperatures the peak position approaches the spinless value $n+1∕2$. Inset: double logarithmic plot of the conductance maximum $G^{\max }\left(T\right)$ in arbitrary units. Crosses: analytic result from Eq. (40).

Figure 7

The differential conductance $G$ (arbitrary units) as a function of bias voltage $V$ (units $E_{\sigma }∕e$) and number of gate charges $n_g$. System parameters are $A=G_L∕G_R=50$, $k_{B}T={10}^{-2}{E}_{\sigma }$, $C_L=5C_R$, $C_g∕C_R=0.01$ $g_{\rho }=0.63$, $g_{\sigma }=1.15$, $g_0=1$, $E_{\rho }=5E_{\sigma }$. Top: gray scale.

Figure 8

The differential conductance $G$ (arbitrary units) as a function of the bias voltage (units $E_{\sigma }∕e$) and $n_g$, for symmetric barriers ($A=1$, $\eta =1∕2$) for $k_{B}T={10}^{-2}{E}_{\sigma }$, $E_{\rho }=25E_{\sigma }$, $g_0=0.9$, $g_{\rho }=0.8$, $g_{\sigma }=1.15$ corresponding to ${\epsilon }_{\sigma }∕{\epsilon }_0=0.99$, ${\epsilon }_{\rho }∕{\epsilon }_0=1.42$, and $2E_{\sigma }∕{\epsilon }_0=0.86$. Right: gray scale.

Figure 9

Scheme of transition lines in the differential conductance for symmetric barriers ($\eta =1∕2$, $A=1$). Shaded regions denote Coulomb blockade. Thick black lines: ground-state to ground-state transitions; dark-gray sequence of lines: transitions to higher-spin states (spin values of the states involved in the transition are indicated); dashed lines: SDW and CDW excited states. For simplicity we have dropped indices $n$, $n+1$ denoting the transitions by spin values $s$ only.

Figure 10

The differential conductance $G$ (arbitrary units) as a function of $V$ (units $E_{\sigma }∕e$) and $n_g$ for asymmetric barriers with $A=50$, $C_L=5C_R$, $C_g∕C_R=0.01$, $k_{B}T={10}^{-2}{E}_{\sigma }$, $E_{\rho }=25E_{\sigma }$, $g_0=0.9$, $g_{\rho }=0.8$, $g_{\sigma }=1.1$ corresponding to ${\epsilon }_{\sigma }∕{\epsilon }_0=0.995$, ${\epsilon }_{\rho }∕{\epsilon }_0=1.37$, $2E_{\sigma }∕{\epsilon }_0=0.9$. Several transitions associated with NDC (white lines) parallel to transition lines $\mid n+1,2s-1⟩\to \mid n,2s,\rho ,\sigma ⟩$ are observed. Right: gray scale.

Figure 11

The differential conductance $G$ (arbitrary units) as a function of $V$ (units $E_{\sigma }∕e$) and $n_g$ for $A=50$, $C_L∕C_R=5$, $C_g∕C_R=0.01$, $k_{B}T={10}^{-2}{E}_{\sigma }$, $E_{\rho }=25E_{\sigma }$, $g_0=0.9$, $g_{\rho }=g_{\sigma }=1.0$, $2E_{\sigma }={\epsilon }_{\sigma }={\epsilon }_{\rho }={\epsilon }_0$. Despite the strong asymmetry there is no NDC. Lines parallel to the transitions $\mid n+1,1⟩\to \mid n,0⟩$ tend to have very small intensity. Gray scale as in Fig. 10.

Figure 12

Occupation probabilities $P_{n,0}$ (dashed), $P_{n,2}+P_{n,-2}$ (dashed-dotted), $P_{n+1,1}+P_{n+1,-1}$ (dotted), and current $I$ (full line, units $e{\Gamma }_0^{\left(R\right)}$) as a function of the bias voltage $V$ (units $E_{\sigma }∕e$) for $n_g=0.457$; other parameters as in Fig. 10.

Figure 13

Transport without spin-charge separation. Left: transition lines for asymmetric barriers ($A>1$) and $g_{\rho }=g_{\sigma }=1$. Right: values for the current $I$ in units $I_0=e{\Gamma }_0^{\left(R\right)}$ for $A\to \infty$ and $T=0$, keeping $G_R$ finite.

Figure 14

Regions analyzed analytically with respect to the NDC: I, II, III correspond to $l_{\sigma }=l_{\rho }=0$; $l_{\sigma }=1$, $l_{\rho }=0$; $l_{\sigma }=0$, $l_{\rho }=1$, respectively, in the transition $\mid n,2⟩\to \mid n+1,1⟩$.

Figure 15

Differential conductance (units $e^2{\Gamma }_0^R∕E_{\sigma }$) a function of positive and negative voltages $V$ (units $E_{\sigma }∕e$) in the five-states region for $n_g=0.485$ for spin relaxation rates $w=0.2$ (solid), 0.3 (dashed), 0.5 (dotted), and 0.6 (dashed-dotted) in units ${\Gamma }_0^{\left(L\right)}{(E_{\sigma }∕{\omega }_{\mathrm{c}})}^{\alpha }$, for $k_{B}T={10}^{-2}{E}_{\sigma }$, $E_{\rho }=25E_{\sigma }$, $A=10$, $\eta =1∕2$, $g_0=0.9$, $g_{\rho }=0.7$, and $g_{\sigma }=1.25$.

Figure 16

Phase diagram for the critical value of the spin-flip relaxation rate $w$ (units ${\Gamma }_0^{\left(L\right)}{(E_{\sigma }∕{\omega }_c)}^{\alpha }$) in region I as a function of the asymmetry $A$ for $T=0$ and different interaction parameters in the leads: $g_0=1.0$ (solid), $g_0=0.9$ (dashed), $g_0=0.75$ (dotted), and $g_0=0.65$ (dashed-dotted).