Charge density wave breakdown in a heterostructure with electron-phonon coupling

Understanding the influence of vibrational degrees of freedom on transport through a heterostructure poses considerable theoretical and numerical challenges. In this work, we use the density-matrix renormalization group method together with local basis optimization to study the half-filled Holstein model in the presence of a linear potential, either isolated or coupled to tight-binding leads. In both cases, we observe a decay of charge density wave states at a sufficiently strong potential strength. Local basis optimization selects the most important linear combinations of local oscillator states to span the local phonon space. These states are referred to as optimal modes. We show that many of these local optimal modes are needed to capture the dynamics of the decay, that the most significant optimal mode on the initially occupied sites remains well described by a coherent state typical for small polarons, and that those on the initially empty sites deviate from the coherent-state form. Additionally, we compute the current through the structure in the metallic regime as a function of voltage. For small voltages, we reproduce results for the Luttinger parameters. As the voltage is increased, the effect of larger electron-phonon coupling strengths becomes prominent. Further, the most significant optimal mode remains almost unchanged when going from the ground state to the current-carrying state in the metallic regime.

Figure 1

Two leads connected to a Holstein-model structure. The leads have hopping amplitude $t_1$. In the structure, the electrons couple to the phonons with a coupling strength $\gamma$ and have a hopping amplitude $t_0$. The phonons have the frequency ${\omega }_0$ and there is a local gate voltage ${\epsilon }_b$. The electrons can tunnel between the leads and the structure with an tunneling amplitude $t_{\mathrm{hyb}}$. At time $t=0$, a voltage $V$ is applied, with a linear interpolation through the structure.

Figure 2

(a) Order parameter in the electron sector for the ground states of the Holstein model (open symbols) and the Holstein model coupled to leads (filled symbols) for different $\gamma /{\omega }_0$. $L$ refers to the system size of the Holstein model and $L_{\mathrm{s}}$ to the length of the structure, which is coupled to the leads. In both cases, we set $t_0/{\omega }_0=1,\tilde{\epsilon }=0, M=35$ for $\gamma /{\omega }_0\le 2$, and $M=50$ for $\gamma /{\omega }_0>2$. For the Holstein model coupled to leads, we further use a total $L=236,t_l/{\omega }_0=2$, and $\mathrm{\Gamma }/{\omega }_0=1$. The dashed line indicates $\gamma /{\omega }_0=2\sqrt{2}$, which is predominately used later in this work. (b) Order parameter in the phonon sector for the same parameters as in (a). Note that we do not show the data point for $L_{\mathrm{s}}=5$, and $\gamma /{\omega }_0=2.5$, since it is not converged conclusively with respect to the criteria in the main text.

Figure 3

Order parameters for the Holstein model with $L=9,\gamma /{\omega }_0=2\sqrt{2},\tilde{\epsilon }=0,t_0/{\omega }_0=1,M=50$, and different voltages $V/{\omega }_0$ ($\mathrm{\Delta }V$ is the potential difference between consecutive sites in the structure). For the calculations, we use ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ and ${\epsilon }_{\mathrm{bond}}={10}^{-7}$. (a) Order parameter in the fermion sector, see Eq. (9). (b) Order parameter in the bosonic sector, see Eq. (10). The dashed lines are the exact data for the Holstein dimer.

Figure 4

Slopes of the linear fit to the initial decay of the order parameters (see main text for details) in the Holstein model. We use $L=9,\gamma /{\omega }_0=2\sqrt{2},\tilde{\epsilon }=0,t_0/{\omega }_0=1,M=50$, and different voltages $V/{\omega }_0$. For the calculations, we use ${\epsilon }_{\mathrm{LBO}}={10}^{-7},{\epsilon }_{\mathrm{bond}}={10}^{-7}$. The error bars indicate the standard deviation of the errors with maximum values of the order of ${10}^{-2}$ and thereby can not be seen for most points on the scale of the figure. Note that the $L=9$ data are rescaled with the factor 5/4 to make the number of electrons participating in the CDW breakdown commensurate with the Holstein dimer.

Figure 5

(a) Average current ${\langle \widehat{j}\rangle }_{\mathrm{av}}$, see text for details, for the structure with $L_{\mathrm{s}}=9,L=236,\mathrm{\Gamma }/{\omega }_0=1,t_l/{\omega }_0=2,{\epsilon }_{\mathrm{bond}}={10}^{-8},{\epsilon }_{\mathrm{LBO}}={10}^{-7},\tilde{\epsilon }=0$, and different $\gamma /{\omega }_0$ in the metallic phase. We further show exact results for $\gamma /{\omega }_0=0$ (black solid line). The inset in (a) shows ${\langle \widehat{j}\rangle }_{\mathrm{av}}$ at fixed $V/{\omega }_0=0.6$ as a function of $\gamma /{\omega }_0$. (b) Same data as in (a) but divided by $V$. The inset in (b) shows the Luttinger-liquid parameter, see main text for details, together with values obtained with different methods. The plus signs are calculated from our data, the green triangles are from Ref. [106], and the blue squares from Ref. [100].

Figure 6

(a) Order parameter in the electron sector, see Eq. (9), of the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\gamma /{\omega }_0=1,\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0$, and $V/{\omega }_0=0.6$. For the calculations, we use ${\epsilon }_{\mathrm{LBO}}={10}^{-7},{\epsilon }_{\mathrm{bond}}={10}^{-8}$. (b) Selected local densities in the structure, see main text for details. (c) Current from Eq. (8) together with selected local currents in the structure, see main text for details. The black dashed lines are the average over local densities (b) and local currents (c).

Figure 7

(a) Diagonal elements of the reduced density matrix in the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\gamma /{\omega }_0=1,\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0,M=30$ at different sites and different times with $V/{\omega }_0=0.6$. Further, ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ and ${\epsilon }_{\mathrm{bond}}={10}^{-8}$ are used for the calculations. (b) Absolute value squared of the components of the most significant optimal-basis state. All data are from the block matrix with an electron and the times are $t{\omega }_0=0$ in (a) and (b), and $t{\omega }_0=20$ in (c) and (d).

Figure 8

Order parameters for the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\gamma /{\omega }_0=2\sqrt{2},\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0$, and voltage gradients with different $\mathrm{\Delta }V/{\omega }_0$. For the calculations we use ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ and ${\epsilon }_{\mathrm{bond}}={10}^{-7}$. (a) Order parameter in the fermion sector, see Eq. (9). (b) Order parameter in the phonon sector, see Eq. (10). The dashed lines show the exact data for the Holstein dimer.

Figure 9

Local densities for the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\gamma /{\omega }_0=2\sqrt{2},\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0,M=50$, and different local voltage differences $\mathrm{\Delta }V/{\omega }_0$. We use ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ and ${\epsilon }_{\mathrm{bond}}={10}^{-7}$ for the calculations. (a)–(c) Local phonon number for $\mathrm{\Delta }V/{\omega }_0=2,8,10$. (d)–(f) Local electron occupation for the same values of $\mathrm{\Delta }V$.

Figure 10

(a) Diagonal elements of the reduced density matrix in the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\gamma /{\omega }_0=2\sqrt{2},\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0,M=50$. (b) Components of the most significant local optimal-basis state. (c) Same as in (a) but at $t{\omega }_0=20$ and $\mathrm{\Delta }V/{\omega }_0=2$. (d) Same as in (a) but at $t{\omega }_0=6$ and $\mathrm{\Delta }V/{\omega }_0=10$. (d) Same as in (b) but at $t{\omega }_0=20$ and $\mathrm{\Delta }V/{\omega }_0=2$. (f) Same as in (b) but at $t{\omega }_0=6$ and $\mathrm{\Delta }V/{\omega }_0=10$. We use ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ and ${\epsilon }_{\mathrm{bond}}={10}^{-7}$ for the calculations. The black dashed line corresponds to the Poisson distribution from Eq. (18).

Figure 11

Current $\langle \widehat{j}\left(t\right)\rangle$ from Eq. (8) for the Holstein structure with $L_{\mathrm{s}}=9,L=236,V/{\omega }_0=0.6,t_l/{\omega }_0=2,\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0,M=30$, and $\gamma /{\omega }_0=1$. (a) Fixed ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ and different ${\epsilon }_{\mathrm{bond}}$. (b) Fixed ${\epsilon }_{\mathrm{bond}}={10}^{-8}$ and different values of ${\epsilon }_{\mathrm{LBO}}$.

Figure 12

Order parameters for the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\gamma /{\omega }_0=\sqrt{2},\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0,\mathrm{\Delta }V/{\omega }_0=10$, and different $M$. For the calculations, we use ${\epsilon }_{\mathrm{LBO}}={\epsilon }_{\mathrm{bond}}={10}^{-7}$. (a) Order parameter in the fermion sector, see Eq. (9). (b) Order parameter in the bosonic sector, see Eq. (10). The inset shows the eigenvalues of the reduced density matrix in the one-electron sector at time $t{\omega }_0=6$ on a log scale.

Figure 13

Eigenvalues of the reduced density matrices at different sites $i$ for the Holstein structure with $L_{\mathrm{s}}=9,L=236,t_l/{\omega }_0=2,\mathrm{\Gamma }/{\omega }_0=1,\tilde{\epsilon }=0,$ and different $\gamma /{\omega }_0$ and $M$ and at different times. (a) $t{\omega }_0=0,M=30$ and $\gamma /{\omega }_0=1$. (b) $t{\omega }_0=0,M=50$ and $\gamma /{\omega }_0=2\sqrt{2}$. (c) Same as (a) but at $t{\omega }_0=20$ and $V/{\omega }_0=0.6$. (d) Same as (b) but at $t{\omega }_0=6$ and $V/{\omega }_0=100 (\mathrm{\Delta }V/{\omega }_0=10)$. We use ${\epsilon }_{\mathrm{LBO}}={10}^{-7}$ in all plots and ${\epsilon }_{\mathrm{bond}}={10}^{-8}\left[{10}^{-7}\right]$ in (a) and (c) [(b) and (d)]. We only show $w_{\alpha }^1>{\epsilon }_{\mathrm{LBO}}$.