Quantum corrections to thermopower and conductivity in graphene

The quantum corrections to the conductivity and thermopower in monolayer graphene are studied. We use the recursive Green's-function method to calculate numerically the conductivity and thermopower of graphene. We then analyze these weak-localization corrections by fitting with the analytical theory as a function of the impurity parameters and gate potential. As a result of the quantum corrections to the thermopower, we find large magnetothermopower, which is shown to provide a very sensitive measure of the size and strength of the impurities. We compare these analytical and numerical results with existing experimental measurements of magnetoconductance of single-layer graphene and find that the average size and strength of the impurities in these samples can thereby be determined. We suggest favorable parameter ranges for future measurements of the magnetothermopower.

Figure 1

${\tau }_{\varphi }/{\tau }_{*}$-${\tau }_{\varphi }/{\tau }_i$ diagram illustrating the transition from positive magnetoconductance (PMC) to negative magnetoconductance (NMC) as obtained from Eq. (30). The transition is indicated by the dashed black line, corresponding to $\Delta \sigma \left(B\right)=0$. This behavior is in good agreement with experiments [3].

Figure 2

Slicing of graphene shown for two honeycomb cells when the contacts are at zigzag edges. Red and green dots indicate the two sublattices A and B; the connecting bonds of the sites are solid lines. The vertical slices, on the right side, are marked with vertical gray lines. $a_0$ represents the lattice constant, indicated in blue.

Figure 3

$V_0$, $V_{\overrightarrow{k_0}}$, and $V_0^{\prime }$ as set by Eqs. (49) and (50). (a) Effective potentials $V_0,V_{k_0}$, and $V_0^{\prime }$ in dependence on $\xi$. (b) Potential terms $V_0+V_0^{\prime },V_0-V_0^{\prime }$, and $V_{k_0}$ in dependence on $\xi$. For short-range impurities, $V_0$ and $V_{\overrightarrow{k_0}}$ approach $V$ because the potential is localized only at the sites of one of the sublattices. Wider impurities also cover the neighboring sublattice and $V_0^{\prime }$ start to increase and $V_0$, $V_{k_0}$ decrease. Also visible is the faster decrease of $V_0-V_0^{\prime }$ than $V_{\overrightarrow{k_0}}$. $\xi$ in units of the lattice constant $a_0$.

Figure 4

Average conductance $G$, as calculated numerically with the recursive Green's-function method for dimensions $N=30$ and $M=72$. The Fermi energy $E_F$ is varied as indicated in the figure. (a) Positive magnetoconductance, weak-localization effect at small correlation length $\xi =1$. (b) Negative magnetoconductance, weak-antilocalization effect at larger correlation length $\xi =3$. (c) Change between positive and negative magnetoconductance by tuning $E_F$. The lines are a guide for the eye.

Figure 5

Samples of numerical results showing strongly nonmonotonic magnetoconductance.

Figure 6

Schematic plots of different types of magnetoconductance $\sigma \left(B\right)$. At $B_{\max }$, all weak-localization corrections are suppressed. If the magnetoconductance is monotonous as in cases (1), the amplitude is $\Delta \sigma \left(B_{\max }\right)$. Otherwise, the magnetoconductance has a maximum at $B=B_{\mathrm{ex}}$, as in case (2).

Figure 7

Numerical magnetoconductance sign phase diagrams calculated with Eq. (82): (a),(b) as a function of correlation length $\xi$ and impurity strength parameter $K_0$ at fixed Fermi energy, (c) as a function of $K_0$ and $E_F$ for fixed $\xi$, and (d) as a function of $\xi$ and $E_F$ for fixed $K_0$. Positive magnetoconductance is indicated by the red area, while negative magnetoconductance is yellow. The system size is fixed to $M=30$ and $N=72$.

Figure 8

$E_F-K_0-\xi$ phase diagram for $M=30$ and $N=72$, where the magnetoconductance sign is obtained from the numerical results with Eq. (82). The surface with vanishing magnetoconductance is displayed.

Figure 9

Magnetoconductance amplitude obtained from the numerical results with Eq. (84) in (a),(b) the $\xi -K_0$ for two $E_F$, (c) the $K_0-E_F$, and (d) the $\xi -E_F$ plane, respectively. The black line indicates vanishing magnetoconductance. Contour lines are in units of $2e^2/h$. $M=30$ and $N=72$.

Figure 10

Numerical results as a function of the parameters $E_F,K_0,\xi$ for the magnetoconductance amplitude, given by Eq. (84), for $M=30$ and $N=72$. Surfaces of value $0.3$ (red), 0 (orange), and $-0.3$ (yellow) are displayed, in units of $2e^2/h$. Each grid line crossing corresponds to a numerically calculated value.

Figure 11

Numerical results for (a) $dG/d{E}_F$ in units of $(2e^2/h)$/eV as a function of magnetic flux $\varphi$ ($M=30$, $N=72$, $K_0=2$). (b) Magnetothermopower in units of $\mu$V/K${}^2$ for the same system as (a). The corresponding conductivity is shown in Fig. 4.

Figure 12

Numerical results for (a) conductance, (b) $dG/d{E}_F$ in units of $(2e^2/h)$/eV, and (c) magnetothermopower in units of $\mu$V/K${}^2$ [(a)–(c): $K_0=0.5$, $\xi =2$].

Figure 13

Numerical results for (a) magnetothermopower amplitude $\Delta S/T$, given by Eq. (86), as a function of $E_F,K_0,\xi$. Surfaces with value $-0.05$ (red), $-0.1$ (orange), and $-0.2$ (yellow) in units of $\mu$V/K${}^2$ are displayed. (b) Magnetoconductance amplitude $\Delta \sigma$, given by Eq. (84), as a function of the parameters $E_F,K_0,\xi$. Surfaces of value 0 (red), $-0.3$ (orange), $-0.6$ (yellow), and $-1$ (green) in units of $2e^2/h$ are displayed. (The grid corresponds to the numerically calculated values.)

Figure 14

The numerical results for the magnetoconductance amplitude $\Delta G\left(\varphi \right)=G\left(\varphi \right)-G\left(0\right)$ for $M=30$, $N=72$ together with the fitting in terms of $1/{\tau }_i$, $1/{\tau }_z$ in Eq. (25). Red $\xi =1$, green $\xi =1.5$, blue $\xi =2$, $E_F=0.01$, $K_0=1$.

Figure 15

${\tau }_{\varphi }/{\tau }_i$-${\tau }_{\varphi }/{\tau }_z$ diagram obtained from fitting of numerical results. Empty symbols: weak localization. Filled symbols: weak antilocalization. Black line: obtained from $\Delta \sigma \left(B\right)=0$ in Eq. (30).

Figure 16

Magnetoconductance in the ${\tau }_{\varphi }/{\tau }_i$-${\tau }_{\varphi }/{\tau }_{*}$ diagram. The ratios of scattering rates and dephasing rates ${\tau }_{\varphi }/{\tau }_i$, ${\tau }_{\varphi }/{\tau }_{*}$ are calculated from the parameters $\xi$ (red dots), $V$ (blue squares), and $E_F$ (green triangles), which are varied in the range given in the inset. Arrows indicate the direction of increasing parameter. The variation with $V$ and $E_F$ is shown for $\xi /a_0=0.05$ and $\xi /a_0=1.15$; $\xi$ is varied at fixed $V/\mathrm{eV}=0.5$ and $E_F/t=0.04$. Dashed line: localization transition. White area: PMC. Yellow area: NMC.

Figure 17

Sign of MC as a function of $V$ and $\xi$, corresponding to Fig. 16 for two different Fermi energies. Red is PMC; yellow is NMC. Compare with the numerical results of Fig. 7. Black line: transition between positive and negative MC.

Figure 18

(a) MC amplitude analytically calculated, given by Eq. (84), in units of $2e^2/h$. Black line: transition between PMC and NMC. Positive numbers and red color: weak localization. (b) MC amplitude as a function of ${\tau }_{\varphi }/{\tau }_i$ and ${\tau }_{\varphi }/{\tau }_{*}$.

Figure 19

Analytically calculated magnetoconductance amplitude, given by Eq. (84), as a function of $\xi ,V$, and $E_F$. Surfaces with value $-1$ (red), 0 (orange), and 1 (yellow) in units of $2e^2/h$ are displayed. Grid lines are guides to the eye.

Figure 20

Analytically calculated quantum correction to the conductance at zero magnetic field, $\delta \sigma \left(B=0\right)$ in units of $2e^2/h$ for two different Fermi energies, (a) $E_F=0.01t$ and (b) $E_F=0.1t$. Black dashed line: transition from positive to negative quantum correction, $\delta \sigma \left(B=0\right)=0$. Black line: Transition from PMC to NMC. (c) Same as (a), but displayed as a function of ${\tau }_{\varphi }/{\tau }_i$ and ${\tau }_{\varphi }/{\tau }_{*}$.

Figure 21

The function $\vartheta$, given by Eq. (88), (a) for ${\tau }_{\varphi }/{\tau }_w=0$ and (b) for ${\tau }_{\varphi }/{\tau }_z=0$. The continuous black line indicates the sign change of the function $\vartheta$. Dashed line: transition between PMC and NMC.

Figure 22

$\vartheta$, given by Eq. (88), for two different Fermi energies as a function of $\xi$ and $V$. As in Figs. 20 and 20, the black dashed line indicates the transition from positive to negative quantum correction to the conductance, $\delta \sigma \left(B=0\right)=0$. Black line: transition from PMC to NMC.

Figure 23

Analytically calculated $\delta S/T$ obtained from Eq. (90) in units of $\mu$V/K${}^2$ for two different Fermi energies. Black line: transition from PMC to NMC. Dashed line: $\delta \sigma =0$. White area: weak-localization correction to conductance exceeds classical value.

Figure 24

Analytically calculated $\delta S/T$-$\delta \sigma$ diagram at zero magnetic field displaying the relation between the quantum corrections to thermopower and conductivity corrections for the phases of WAL, I and II, for various impurity parameters $\xi$ and $V$, for two different Fermi energies. Dotted black lines: same impurity strength for different $\xi$.

Figure 25

$\kappa (t_i,t_z,t_w)$ displayed for the two limiting cases $1/{\tau }_w=0$ and $1/{\tau }_z=0$. Dashed lines: NMC to PMC transition from Fig. 1. Positive numbers indicate positive magnetothermopower (PMT), an increase of $\delta S/T$ for small magnetic fields.

Figure 26

Magnetoconductance in the ${\tau }_{\varphi }/{\tau }_i$-${\tau }_{\varphi }/{\tau }_z$ diagram for $1/{\tau }_w=0$. The ratios of scattering rates and dephasing rates ${\tau }_{\varphi }/{\tau }_i$, ${\tau }_{\varphi }/{\tau }_z$ are calculated from the parameters $\xi$ (red dots), $V$ (blue squares), and $E_F$ (green triangles), which are varied in the range given in the inset. Arrows indicate the direction of increasing parameter. The variation with $V$ and $E_F$ is shown for $\xi /a_0=.05$; $\xi$ is varied at fixed $V/\mathrm{eV}=0.5$ and $E_F/t=.04$. Dashed line: localization transition. White area: PMC. Yellow area: NMC.