Quantum and Boltzmann transport in a quasi-one-dimensional wire with rough edges

We study electron transport in quasi-one-dimensional metallic wires. Our aim is to compare an impurity-free wire with rough edges with a smooth wire with impurity disorder. We calculate the electron transmission through the wires by the scattering-matrix method, and we find the Landauer conductance for a large ensemble of disordered wires. We first study the impurity-free wire whose edges have roughness with a correlation length comparable with the Fermi wavelength. Simulating wires with the number of the conducting channels ($N_c$) as large as $34$–$347$, we observe the roughness-mediated effects which are not observable for small $N_c$ ($~3$–$9$) used in previous works. First, we observe the crossover from the quasi-ballistic transport to the diffusive one, where the ratio of the quasi-ballistic resistivity to the diffusive resistivity is $~N_c$ independent of the parameters of roughness. Second, we find that transport in the diffusive regime is carried by a small effective number of open channels, equal to $~6$. This number is universal—independent of $N_c$ and of the parameters of roughness. Third, we see that the inverse mean conductance rises linearly with the wire length (a sign of the diffusive regime) up to the length twice larger than the electron localization length. We develop a theory based on the weak-scattering limit and semiclassical Boltzmann equation, and we explain the first and second observations analytically. For the impurity disorder we find a standard diffusive behavior. Finally, we derive from the Boltzmann equation the semiclassical electron mean free path and we compare it with the quantum mean free path obtained from the Landauer conductance. They coincide for the impurity disorder; however, for the edge roughness they strongly differ, i.e., the diffusive transport in the wire with rough edges is not semiclassical. It becomes semiclassical only for roughness with a large correlation length. The conductance then behaves like the conductance of the wire with impurities, also showing the conductance fluctuations of the same size.

Figure 1

Wire made of the 2D conductor of width $W$ and length $L$. The figure on the left depicts the wire with impurities positioned at random with random signs of the impurity potentials. The figure on the right depicts the impurity-free wire with rough edges, where $d(x)$ and $h(x)$ are the $y$ coordinates of the edges at $y=0$ and $y=W$, respectively, randomly fluctuating with $x$. The roughness amplitude is $\Delta$ and the step $\Delta x$ also means the correlation length (see the text).

Figure 2

The Q1D wire placed between two contacts. The bold arrows denote the wave amplitudes ${\mathcal{A}}^{+}$, ${\mathcal{B}}^{-}$ coming into the wire and the amplitudes ${\mathcal{A}}^{-}$, ${\mathcal{B}}^{+}$ coming out the wire.

Figure 3

Wire with the randomly positioned pointlike impurities. The $n$ impurities described by the scattering matrices $s_i$ divide the wire into the $n+1$ free regions described by the matrices $p_i$. Also shown are the wave amplitudes ${\mathcal{A}}^{\pm }$ and ${\mathcal{B}}^{\pm }$.

Figure 4

The impurity-free wire with $n$ edge steps described by the scattering matrices $s_j$ and with $n+1$ free regions of constant width $W_j$, described by the scattering matrices $p_j$.

Figure 5

The wire with a single edge step at $x=0$. The symbols in the figure are discussed in the text.

Figure 6

The mean resistance $\langle \rho \rangle$ as a function of the wire length $L$. Note that $\langle \rho \rangle$ is reduced by the contact resistance ${\rho }_c=1/N_c$ and $L$ is scaled by the mean free path $l$. The left panel shows the mean resistance of the wire with impurity disorder; the right one shows the mean resistance of the wire with rough edges. The wire width $W$ is fixed to the value $W/{\lambda }_F=17.4$, which means that $N_c=34$. The parameters of disorder are listed in the figure together with the resulting values of $l$. Determination of $l$ is demonstrated in the next figure. Note that for the Au wire ($E_F=5.6$ eV and $m=9.109\times {10}^{-31}$ kg) we have ${\lambda }_F=0.52$ nm and $W=17.3{\lambda }_F=9$ nm, with the smallest $\Delta$ being only $0.01$ nm. Such small $\Delta$ is obviously not realistic, but it is used to emulate the weak roughness limit.

Figure 7

The solid lines show the selected numerical data from the preceding figure. The dashed line in the left panel shows the linear fit $\langle \rho \rangle ={\rho }_c+{\rho }_{\mathrm{dif}}L/W$, while the dashed line in the right panel is the linear fit $\langle \rho \rangle ={\rho }_c^{\mathrm{eff}}+{\rho }_{\mathrm{dif}}L/W$, with ${\rho }_c^{\mathrm{eff}}$ being the effective contact resistance. In both cases the resistivity ${\rho }_{\mathrm{dif}}$ is a fitting parameter and $l$ is extracted from the Drude formula ${\rho }_{\mathrm{dif}}=2/(k_{F}l)$. To determine ${\rho }_c^{\mathrm{eff}}$, the dashed line in the right panel is extrapolated to $L=0$. We find ${\rho }_c^{\mathrm{eff}}\simeq 1/7.5$ while ${\rho }_c=1/34$, the inset shows the solid line from the right panel for $L/l\ll 1$, where the transport is quasi-ballistic. The dotted lines show the linear fit $\langle \rho \rangle ={\rho }_c+{\rho }_{\mathrm{qb}}L/W$, where the quasi-ballistic resistivity ${\rho }_{\mathrm{qb}}$ is a fitting parameter. The quasi-ballistic mean free path $l_{\mathrm{qb}}$ is given as $l_{\mathrm{qb}}=2/(k_F{\rho }_{\mathrm{qb}})$.

Figure 8

Typical conductance $\langle \ln g\rangle$, mean resistance $\langle \rho \rangle$, and mean conductance $\langle g\rangle$ versus $L/\xi$. The results for the wire with impurity disorder (left panels) are compared with the results for the wire with rough edges (right panels). The calculations were performed for various sets of the parameters, listed in the Fig. 6. The results for various parameter sets collapse to a single curve shown as a solid line (see the remarks in the text). The dashed lines in the top panels show the fit $\langle \ln g\rangle =-L/\xi$, from which we determine the localization length $\xi$. The resulting values of $\xi$ are shown.

Figure 9

The solid lines show our numerical data for the mean resistance $\langle \rho \rangle$, inverse mean conductance $1/\langle g\rangle$, and conductance fluctuations $\sqrt{\text{var}(g)}$. The dashed lines in the top panels are the linear fits $\langle \rho \rangle ={\rho }_c+{\rho }_{\mathrm{dif}}L/W$ (left panel) and $\langle \rho \rangle ={\rho }_c^{\mathrm{eff}}+{\rho }_{\mathrm{dif}}L/W$ (right panel). The dashed lines in the middle panels are the linear fits $1/\langle g\rangle ={\rho }_c+{\rho }_{\mathrm{dif}}L/W$ (left panel) and $1/\langle g\rangle ={\rho }_c^{\mathrm{eff}}+{\rho }_{\mathrm{dif}}L/W$ (right panel). Onset of strong localization is marked by arrows at the points, where the numerical data start to deviate from the linear fit remarkably. The relative deviation from the linear fit is shown in a separate figure [Fig. (10)]. The dotted line in the bottom panels shows the theoretical value ($0.365$) of the conductance fluctuations, predicted in the limit $l/\xi \ll L/\xi \ll 1$ for the white-noise disorder.

Figure 10

Numerical data for $\langle \rho \rangle$ and $1/\langle g\rangle$ from the preceding figure, presented as a relative deviation from the linear fit. The dashed line shows the weak-localization-mediated relative deviation $\frac{1}{3}{\rho }_{\mathrm{dif}}\frac{L}{W}$, shifted by replacement $L\to (L-8l)$ to obey the limit $L\gg l$.

Figure 11

Mean resistance $\langle \rho \rangle$, inverse mean conductance $1/\langle g\rangle$, and conductance fluctuations $\sqrt{\text{var}(g)}$ for various wire parameters.

Figure 12

Effective number of open channels, $N_c^{\mathrm{eff}}$, versus the roughness correlation length $\Delta x$ for various $N_c$ and various roughness amplitudes $\Delta$. The parameters of the edge roughness are scaled as $\Delta x/{\lambda }_F$ and $\Delta /W$. The results for $\Delta /W=1/18$ are shown by open circles; the dashed and solid lines show the results for $\Delta /W=1/180$ and $\Delta /W=1/900$, respectively.

Figure 13

The top panels show the transmission probability $\langle T_n\rangle$ versus $L/\xi$ for the channel indices $n=1,2,\dots ,N_c$, where $N_c=34$. For $n$ ordered increasingly, the resulting curves are ordered decreasingly: The top curve shows $\langle T_{n=1}\rangle$, the bottom one shows $\langle T_{n=N_c}\rangle$. The bottom panels show $\langle T_n\rangle$ versus $n$ for various $L/\xi$.

Figure 14

Top panel: Quasi-ballistic mean free path $l_{\mathrm{qb}}$ in the Au wire as a function of the wire width $W$ for various roughness amplitudes $\Delta$. The roughness correlation length is fixed at $\Delta x=0.125\hspace{0.5em}\mathrm{nm}$, the Au material parameters are $m=9.109\times {10}^{-31}$ kg and $E_F=5.6$ eV. The circles show our numerical data and the squares (connected by a solid line) are the data points obtained from expression (79). Bottom panel: The same calculations as in the top panel, but $W$ is fixed at $9$ nm and the Fermi energy is varied. To explore the small $\Delta$ limit reliably, we have to use the values of $\Delta$, which are too small to be realistic.

Figure 15

Quasi-ballistic mean free path $l_{\mathrm{qb}}$ and diffusive mean free path $l$ in the Au wire with rough edges, calculated as a function of the wire width $W$ and Fermi energy $E_F$. The thick lines show formulas (88) and (84), derived for the uncorrelated roughness. The thin lines show the formulas valid for an arbitrary correlation length $\Delta x$, namely the quasi-ballistic result of Eq. (79) and the semiclassical $l$ extracted from the Boltzmann conductivity of Eq. (59). The roughness amplitude $\Delta$ and roughness-correlation length $\Delta x$ are fixed as $\Delta =0.5$ nm and $\Delta x=0.125$ nm (the limit $\Delta x/{\lambda }_F\ll 1$ is fulfilled for all considered data). The Fermi energy used in the top panels is $E_F=5.6$ eV, the wire width in the bottom panels is $W=9\hspace{0.5em}\mathrm{nm}$. The oscillations with sharp minima appear whenever the Fermi energy approaches the bottom of the energy subband $n=N_c$.

Figure 16

Ratio $l/l_{\mathrm{qb}}$, where $l$ is the diffusive mean free path and $l_{\mathrm{qb}}$ is the quasi-ballistic mean free path, evaluated as a function of $N_c$ for various roughness amplitudes $\Delta$. The roughness-correlation length $\Delta x$ is fixed to the value $\Delta x/{\lambda }_F=0.24$, which reasonably emulates the uncorrelated roughness. The open symbols show the results of our quantum transport simulation, where $l$ and $l_{\mathrm{qb}}$ are calculated as in Fig. 7.

Figure 17

Top panel: Diffusive mean free path $l$ in the Au wire with impurity disorder as a function of the wire width $W$, calculated for three different sets of impurity parameters (listed in the bottom panel). The semiclassical mean free path obtained from the Boltzmann Q1D conductivity Eq. (62) is shown as a solid line: All three sets of parameters are intentionally chosen to give the same semiclassical result. The data shown by symbols are the quantum-transport results obtained from the quantum resistivity ${\rho }_{\mathrm{dif}}$ (by using the procedure described in Fig. 7). Bottom panel: The same calculations as in the top panel, but $W$ is fixed at $9$ nm and the Fermi energy is varied.

Figure 18

The right panel shows the diffusive mean free path $l$ in the Au wire with rough edges as a function of the wire width $W$, calculated for various roughness amplitudes $\Delta$. The left panel shows the same calculation, but $W$ is fixed at $9$ nm and the Fermi energy is varied. The roughness-correlation length is kept at the value $\Delta x=0.125$ nm, which means that $\Delta x/{\lambda }_F\lesssim 1$ for most of the presented data. The solid lines show the semiclassical mean free path obtained from the Boltzmann Q1D conductivity [Eqs. (59) and (64)]. The open circles show the quantum mean free path extracted from the quantum resistivity ${\rho }_{\mathrm{dif}}$ (by using the procedure described in Fig. 7). The open squares show the mean free path obtained by means of the classical scattering-matrix calculation (see the text), in which the localization is absent. To explore the small $\Delta$ limit reliably, we have to use values of $\Delta$ which are too small to be realistic.

Figure 19

Mean resistance $\langle \rho \rangle$ of the wire with rough edges in dependence on the ratio $L/\xi$. The thin solid line is the result of the quantum scattering-matrix calculation (the same data as in the top right panel of Fig. 9), now labeled as $\langle \rho {\rangle }_{\mathit{QM}}$. The thick solid line is the result of the classical scattering-matrix calculation, labeled as $\langle \rho {\rangle }_{\mathrm{clas}}$. The dashed lines show the linear fits $\langle \rho \rangle ={\rho }_c^{\mathrm{eff}}+{\rho }_{\mathrm{dif}}L/W$, from which we determine the effective contact resistance ${\rho }_c^{\mathrm{eff}}$ and the diffusive mean free path $l=2/(k_F{\rho }_{\mathrm{dif}})$. The resulting ${\rho }_c^{\mathrm{eff}}$ are shown in the figure. The inset shows the classical result again, but for larger $L/\xi$ in order to stress the linear raise and absence of the localization.

Figure 20

The same calculation and the same symbols as in Fig. 9, except that the edge roughness is considered in the limit $\Delta x/{\lambda }_F⋙1$. Specifically, the data obtained for the wire with impurities (left column) are the same as those in the left column of Fig. 9, while for the edge roughness (right column) we show the data obtained for $\Delta x/{\lambda }_F=19\hspace{0.5em}300$ and $\Delta /{\lambda }_F=0.096$ (the data for $\Delta /{\lambda }_F=0.019$ and $\Delta /{\lambda }_F=0.96$, not shown, look similar except for statistical noise). We recall that the localization length $\xi$ is determined from the dependence $\langle \ln g\rangle =-L/\xi$. For the edge roughness we now obtain $\xi /l\simeq 0.92N_c$, which is close to the impurity case: $\xi /l\simeq 0.88N_c$.

Figure 21

The same calculation and the same symbols as in Fig. 13, except that the edge roughness is considered in the limit $\Delta x/{\lambda }_F⋙1$ (we show the data for $\Delta x/{\lambda }_F=19\hspace{0.5em}300$).

Figure 22

Diffusive mean free path $l$ in the wire with rough edges as a function of the roughness-correlation length $\Delta x$ for various roughness amplitudes $\Delta$, with $l$, $\Delta x$, and $\Delta$ scaled by ${\lambda }_F$. The wire width $W$ is fixed to the value $W/{\lambda }_F=17.4$, which means that $N_c=34$. The solid lines show the semiclassical mean free path obtained from the Boltzmann conductivity [Eqs. (59) and (64)]. The dashed lines for $\Delta x/{\lambda }_F\lesssim 1$ show the uncorrelated limit of Eq. (84), the dashed lines for $\Delta x/{\lambda }_F\gtrsim 1$ show the correlated limit of Eqs. (102). The circles show the quantum mean free path extracted from the quantum resistivity ${\rho }_{\mathrm{dif}}$ (by using the procedure described in Fig. 7). The squares show the mean free path obtained by means of the classical scattering-matrix calculation in which the localization effect is absent (see Sec. 4d). Note that for the Au wire ($E_F=5.6\hspace{0.5em}\mathrm{eV}$ and $m=9.109\times {10}^{-31}$kg) we have ${\lambda }_F=0.52$ nm, i.e., the wire width is $W=17.4{\lambda }_F=9$ nm and the smallest $\Delta$ is only $0.01$ nm. Such small $\Delta$ is obviously not realistic, which demonstrates an important finding: The semiclassical Boltzmann result (solid line) reproduces the classical scattering-matrix calculation (squares) for any $\Delta x$ only if $\Delta$ is too small to be realistic.