Grand-canonical Peierls theory for atomic wires on substrates

We present a generic grand-canonical theory for the Peierls transition in atomic wires deposited on semiconducting substrates such as In/Si(111) using a mean-field solution of the one-dimensional Su-Schrieffer-Heeger model. We show that this simple low-energy effective model for atomic wires can explain naturally the occurrence of a first-order Peierls transition between a uniform metallic phase at high temperature and a dimerized insulating phase at low temperature as well as the existence of a metastable uniform state below the critical temperature.

Figure 1

Schematic of the uniform lattice configuration (top) and the two dimerized configurations (bottom) in the SSH model. The disks represent the sites (atoms) and the linewidths show the strength of the bond order or hopping term.

Figure 2

Dispersion (12) of the single-electron eigenenergies for $\mathrm{\Delta }=2\delta t_0=0.8t_0$. The horizontal dashed lines show the four different relative positions of the chemical potential: $\mu =0$ (half-filling), $0>\mu >-\mathrm{\Delta }, -\mathrm{\Delta }>\mu >-2t_0$, and $-2t_0>\mu$ (see the discussion in Sec. 4a).

Figure 3

Grand-canonical potential (15) at $\mu =0$ (half-filling) [see Eq. (20)] as a function of the order parameter $\delta$ for $\lambda =0.18$ and four different temperatures (from bottom to top): $T\approx 0, T\approx T_{\text{P}}/2, T\approx T_{\text{P}}$, and $T\approx 2T_{\text{P}}$.

Figure 4

Order parameter $\delta$ as a function of temperature at half-filling for the electron-phonon coupling $\lambda =0.18$. The horizontal line shows the zero-temperature value ${\delta }_0$ obtained from the minimization of the ground-state energy (22) while the vertical line shows the critical temperature (31).

Figure 5

Renormalized phonon frequency $\mathrm{\Omega }/{\omega }_0$ for the amplitude mode [see Eq. (26)] as a function of temperature at half-filling with an electron-phonon coupling $\lambda =0.18$. The horizontal dashed and solid lines show the zero-temperature frequency (27) and bare phonon frequency, respectively. The vertical line shows the critical temperature $T_{\text{P}}$, see Eq. (31).

Figure 6

Grand-canonical potential (15) at zero temperature [see Eq. (34)] as a function of the order parameter $\delta$ for $\lambda =0.18$ and several values of the chemical potential from $\mu =-0.25{\mathrm{\Delta }}_0$ (bottom curve) to $\mu =-1.25{\mathrm{\Delta }}_0$ (top curve) in steps of $-0.25{\mathrm{\Delta }}_0$.

Figure 7

Grand-canonical potential (15) at zero temperature [see Eq. (34)] as a function of the order parameter $\delta$ around $\delta =0$ for $\lambda =0.18$ and several values of the chemical potential from $\mu =-0.485{\mathrm{\Delta }}_0$ (bottom curve) to $\mu =-0.505{\mathrm{\Delta }}_0$ (top curve) in steps of $-0.005{\mathrm{\Delta }}_0$.

Figure 8

Grand-canonical potential (15) at zero temperature [see Eq. (34)] as a function of the order parameter $\delta \ge 0$ for $\lambda =0.18$ and several values of the chemical potential from $\mu =-0.375{\mathrm{\Delta }}_0$ (bottom curve) to $\mu =-0.875{\mathrm{\Delta }}_0$ (top curve) in steps of $-0.125{\mathrm{\Delta }}_0$.

Figure 9

Phase diagram ($\mu ,T$) of the SSH model in the mean-field approximation for an electron-phonon coupling $\lambda =0.37$. The black solid and dashed lines show the phase boundary $T_{\text{P}}\left(\mu \right)$ between the dimerized phase (labeled D or ${\mathrm{D}}^{\prime }$) and the uniform phase (labeled U or ${\mathrm{U}}^{\prime }$). The uniform configuration is metastable in the sector labeled ${\mathrm{D}}^{\prime }$ while the dimerized state is metastable in the sector labeled ${\mathrm{U}}^{\prime }$. The solid line indicates the continuous phase transition between the sectors D and U while the dashed line indicates the first-order transition between the sectors ${\mathrm{D}}^{\prime }$ and ${\mathrm{U}}^{\prime }$ with metastable configurations. The dot-dashed line indicates the position of the first-order metal-insulator transition within the dimerized phase. Circles show the zero-temperature boundaries predicted by the weak-coupling analysis. Vertical lines indicate the values of $\mu$ and the temperature ranges considered in Figs. 10 to 12.

Figure 10

Grand-canonical potential (15) as a function of the order parameter $\delta \ge 0$ for $\lambda =0.37, \mu =-0.5{\mathrm{\Delta }}_0\approx -0.446t_0$, and several temperatures $k_{\text{B}}T=0.005{\mathrm{\Delta }}_0\approx 0.00446t_0$ (black solid line), $k_{\text{B}}T=0.225{\mathrm{\Delta }}_0\approx 0.201t_0$ (blue dashed line), and $k_{\text{B}}T=0.45{\mathrm{\Delta }}_0\approx 0.401t_0$ (red dotted line).

Figure 11

Grand-canonical potential (15) as a function of the order parameter $\delta \ge 0$ for $\lambda =0.37, \mu =-0.625{\mathrm{\Delta }}_0\approx -0.557t_0$, and several temperatures $k_{\text{B}}T=0.05{\mathrm{\Delta }}_0\approx 0.0446t_0$ (black solid line), $k_{\text{B}}T=0.15{\mathrm{\Delta }}_0\approx 0.134t_0$ (blue dashed line), and $k_{\text{B}}T=0.25{\mathrm{\Delta }}_0\approx 0.223t_0$ (red dotted line).

Figure 12

Grand-canonical potential (15) as a function of the order parameter $\delta \ge 0$ for $\lambda =0.37, \mu =-0.75{\mathrm{\Delta }}_0\approx -0.668t_0$, and several temperatures $k_{\text{B}}T=0.005{\mathrm{\Delta }}_0\approx 0.00446t_0$ (black solid line), $k_{\text{B}}T=0.1{\mathrm{\Delta }}_0\approx 0.0891t_0$ (blue dashed line), and $k_{\text{B}}T=0.2{\mathrm{\Delta }}_0\approx 0.178t_0$ (red dotted line).

Figure 13

Order parameter $\delta$ as a function of temperature for $\lambda =0.37$. The vertical lines indicate the critical temperature $k_{\text{B}}{T}_{\text{P}}\left(\mu \right)\approx 0.102t_0$ and $k_{\text{B}}{T}_{\text{D}}\left(\mu \right)\approx 0.165t_0$ for $\mu =-0.625{\mathrm{\Delta }}_0\approx -0.557t_0$. The solid black and red dashed lines show $\delta$ for the thermodynamically stable dimerized phase [$T<T_{\text{P}}\left(\mu \right)$] and the metastable dimerized state [$T_{\text{P}}\left(\mu \right)<T<T_{\text{D}}\left(\mu \right)$] at this value of $\mu$, respectively. The blue dotted line shows $\delta \left(T\right)$ at half-filling ($\mu =0$).

Figure 14

Renormalized phonon frequency $\mathrm{\Omega }/{\omega }_0$ [Eq. (26)] as a function of temperature for $\lambda =0.37$ and $\mu =-0.625{\mathrm{\Delta }}_0\approx -0.557t_0$. The vertical lines indicate the critical temperature $k_{\text{B}}{T}_{\text{P}}\left(\mu \right)\approx 0.102t_0$ and $k_{\text{B}}{T}_{\text{D}}\left(\mu \right)\approx 0.165t_0$. The solid black line shows $\mathrm{\Omega }$ in the thermodynamically stable phases with a jump at $T=T_{\text{P}}\left(\mu \right)$. The green dotted and red dashed lines show the frequency for the metastable uniform configuration below $T_{\text{P}}\left(\mu \right)$ and the metastable dimerized configurations for $T_{\text{P}}\left(\mu \right)<T<T_{\text{D}}\left(\mu \right)$, respectively.