In the two-dimensional (2D) metal-insulator transition (MIT) regime, both the Coulomb interactions between electrons and the disorder are expected to be strong, leading to the formation of an electron glass [@Review; @Rice]. Recent experiments in 2D electron systems have revealed changes in the characteristics and the amplitude of the conduction noise as the change density of the system is varied from a high charge density metallic phase to a lower charge density insulating phase [@Popovic; @Sn; @Noise]. This has been interpreted as evidence for an onset of glassy dynamics near the insulating phase. These studies also find that the noise has a $1/f^{\alpha}$ characteristic, with $\alpha = 1.0$ in the metallic phase changing over to $\alpha \approx 1.8$ in the glassy phase. Other experiments have found similar results with $\alpha = 0.75$ in the metallic phase and $\alpha =1.3$ near the insulating phase [@N2d]. The glassy phase is signified by a large increase in the noise power along with the change in $\alpha$ [@Popovic; @Sn]. In addition, the noise power was observed to [*decrease*]{} with temperature [@Sn; @Noise], in contrast to single electron models with thermally activated trapping [@Weissman] and other models [@Voss] that predict an increase in the noise power with $T$. This suggests the importance of electron-electron interactions at the MIT. Other recent experiments near the 2D MIT have also found $1/f^{\alpha}$ noise, strong increases in noise power with decreasing charge density, and decreasing noise with increasing $T$ [@N2d; @Tourbot]. Additionally, $1/f^{\alpha}$ noise fluctuations in thin granular films have been interpreted as evidence for a glassy electron state [@Wu].

In theoretical studies, it was proposed that at the 2D MIT a freezing from an electron liquid to a partially ordered Wigner glass [@Kivelson] or a more strongly disordered electron glass [@Thakur; @Glass] may occur. Other theories suggest that an intermediate metallic glass phase appears between the liquid and insulating phases [@Pastor]. It is also possible that the metallic glass phase may consist of solid phase insulating regions coexisting with string-like liquid regions. Studies of glassy systems often find cooperative string-like motions or dynamical heterogeneities [@Glotzer]. Such motions can give rise to correlated dynamics and large fluctuations near a glass transition. It is, however, unclear what the origin of such cooperativity would be in the electron glass systems.

The strong enhancement of the noise observed when passing from a liquid to a glassy phase in the 2D charge system has noticeable similarities to the noise found in studies of vortex matter in superconductors [@Higgins]. In the vortex system, the existence of a transition from a weakly pinned liquid phase to a more strongly pinned glassy phase has been established. Noise studies in low temperature superconductors have shown that, as the glassy phase is approached from the liquid state, the voltage noise has a $1/f^{\alpha}$ characteristic with increasing $\alpha$, corresponding to an increase in the noise power [@Higgins].

The vortex system studies suggest that similar physics may be occurring in the 2D electron systems when there is a transition from a liquid like phase with low noise power to a pinned glassy phase with high noise power. This is also consistent with the theoretical prediction that the electron liquid freezes into a 2D disordered solid. In this work we propose a simple model for a classical 2D electron system consisting of interacting electrons with random disorder and temperature. We monitor the fluctuations and noise characteristics of the current as a function of electron density or temperature. The advantage of our model is that a large number of interacting electrons can be conveniently simulated, while a full quantum mechanical model of similar size would be computationally prohibitive. Despite the limitations of this model, we show that this approach captures many of the key experimental observations. Additionally, although our primary focus is to gain insight into the physics near the 2D MIT, our model is also relevant for other classical charge systems undergoing transitions from glass to liquid states, such as charged colloids interacting with random disorder. Our model is similar to previous studies of 2D classical electron systems with disorder [@Fertig; @Reichhardt]; however, these previous studies focused on the microscopics of the defects in the lattice [@Fertig] or the sliding dynamics [@Reichhardt]. In the present work we focus on the noise fluctuations in the strongly disordered phase as it changes from a liquid to a frozen state as a function of electron density for a fixed amount of disorder.

Our model consists of a 2D system of $N_{s}$ interacting electrons with periodic boundary conditions in the $x$ and $y$-directions. There are also $N_{p}$ defect sites which attract the electrons. We assume the electron motion is at finite temperature and the time evolution occurs through Langevin dynamics. The damping on the electrons comes

from their interactions with phonons or small scattering sites. The equation of motion for a electron $i$ is $$\eta {\bf v}={\bf f}_{i}=-\sum_{j}^{N_{s}}\nabla U(r_{ij}) + 
{\bf f}_{i}^{s} + {\bf f}_{i}^{T} + {\bf f}_{d}$$ Here $\eta= 1$ is the damping constant and $U_{i}(r) = -q^2/r$, with $q=1$, is the electron-electron interaction potential, treated as in [@Jensen]. The term ${\bf f}_{i}^{s}$ comes from the $N_p$ randomly spaced defect sites modeled as parabolic traps of radius $r_{p}=0.2$ and strength $f_{p} = 1.0$. The thermal noise ${\bf f}_{i}^{T}$ arises from random Langevin kicks with $<f^{T}(t)> = 0$ and $<f^{T}_i(t)f^{T}_j(t^{\prime})> = 2\eta k_B T \delta_{ij}
\delta(t - t^{\prime})$. The driving term ${\bf f}_{d}=f_d \hat x$ comes from an applied voltage, and we take $f_d= 0.1$. We start at a high temperature where the charges are diffusing rapidly and cool to a lower temperature. We then wait for $10^4$ simulation time steps to reduce transient effects before appling the drive and measuring the average velocity $v$ of the electrons, which is proportional to the conduction or resistance. We do this for a series of electron densities at fixed disorder strength. We have considered samples with constant pin densities $n_{p}$ for different system sizes such that $N_{p}$ ranges from $317$ to $1200$.

We first consider the average electron velocity as a function of temperature and charge density for a system with a fixed $N_{p}=1200$. In Fig. 1 we show a series of conductance curves vs $T$ for charge density varied over nearly an order of magnitude, $0.4 \leq N_{s}/N_{p} \leq 2.67$. For high $N_{s}/N_{p}> 0.6$ the conductance is finite down to $T = 0$, while for $N_{s}/N_{p} \leq 0.6$, the electron velocity drops to zero within our resolution, indicating that all the electrons are strongly pinned in an insulating phase. As $N_{s}/N_{p}$ increases above 1.0, the downward curvature of $v$ at low $T$ decreases. These curves appear very similar

to those typically observed in 2D MIT studies [@Popovic]. One difference is that we do not find a charge density $n_{s}^{up}$ above which the slope of the velocities turns up slightly at low $T$, as in the experiments. This may be due to the fact that in our model we do not directly include phonons. In the experimental regime of interest to us here, the large noise increases occur at charge densities $n_s<n^{up}_s$, where the velocity curves bend down at low temperature.

We next consider the relative fluctuations in the velocities, $\delta v(t)=(v(t) - <v>)$ for varied $N_{s}/N_{p}$ at a fixed $T = 0.09$ for the system in Fig. 1. This analysis is similar to that performed in experiments [@Popovic; @Sn; @Noise]. In Fig. 2 we show the time traces of the relative velocity fluctuations for $N_{s}/N_{p} = 0.5, 0.7, 
1.05, 1.64$ and $2.67$. Here the fluctuations increase as $N_s$ drops, in agreement with the experiments [@Popovic]. For $N_{s}/N_{p} < 0.44$, the system is pinned and there are no fluctuations. It is possible that, over a longer time interval such as that accessible experimentally, there would be even larger fluctuations at these small $N_s$ values; however, this is beyond the time scale we can access with simulations. A similar series of time traces can be obtained for $\delta v$ at fixed $N_{s}/N_{p}$ for increasing temperature (not shown). Here the fluctuations are reduced at the higher temperatures, in agreement with experiments [@Popovic].

From the fluctuations $\delta v$ we measure the power spectrum $$S(\nu) = \left|\int \delta v(t)e^{-2\pi i\nu t}dt\right|^2  .$$ In Fig. 3 we plot $S(\nu)$ for two different charge densities. At $N_{s}/N_{p} = 0.5$ (upper curve), the spectrum shows a $1/f^{\alpha}$ characteristic with $\alpha=1.37$ over a few orders of

magnitude in the frequency. In contrast, for $N_{s}/N_{p} = 1.67$ (lower curve), the noise power at lower frequencies is considerably reduced and the spectra is white with $\alpha \approx 0$. We not that it is the lower frequcies which will be most readly accesible in experiment. For fixed $N_{s}/N_{p} = 0.5$, we find that the power spectrum becomes white upon increasing $T$. We note that our results differ quantitatively from the experimental noise measurements [@Popovic] which find $1/f^{\alpha}$ noise with $\alpha = 1.0$ in the metallic phase and $\alpha = 1.8$ in the glassy regime. Our exponent $\alpha=1.37$ is close to the $\alpha = 1.3$ found in the glassy regime in other experiments [@N2d], where $\alpha=0.75$ in the metallic regime. It is possible that the exponents are not universal but depend on the details of the disorder strength; nevertheless, our results are in qualitative agreement with the experiments.

In Fig. 4(a) we show the noise power $S_{0}$ integrated over the first octave vs $N_{s}/N_{p}$ for a fixed $T = 0.09$. The noise power increases by four orders of magnitude as $N_{s}/N_p$ is reduced. At low $N_{s}/N_{p}$, the noise power decreases almost exponentially with charge density and begins to saturate at high $N_{s}/N_p$. Both these observations are in agreement with the experimental results [@Sn; @Noise]. In the inset of Fig. 4(a) we plot the noise spectrum exponent $\alpha$ vs $N_{s}/N_{p}$. A large increase in $\alpha$ occurs near $N_{s}/N_{p} = 0.7$. A similar sharp increase in the exponent is also observed in experiments [@Sn; @Noise; @N2d] as a function of charge density and has been interpreted as the glassy freezing transition. In Fig. 4(b) we plot $S_{0}$ vs $T$ for fixed $N_{s}/N_{p} = 0.5$. Here the noise power drops exponentially over four orders of magnitude with increasing $T$, which is in agreement with the experiments [@Sn; @Noise]. We note that most single electron models predict an [*increase*]{} in the noise power with temperature [@Weissman; @Voss]. Other models in the hopping regime predict a noise power that is independent of $T$ or has a power law decrease with $T$ [@Kozub]. These discrepancies

suggest that the noise in the experiment is not due to single electron hopping events, but is instead caused by correlated electron motions. In the inset of Fig. 4(b) we plot $\alpha$ vs $T$, showing that a sharp increase in $\alpha$ occurs near $T = 0.125$ at the onset of the glassy freezing.

We have also measured the non-Gaussian nature of the noise. At high $N_{s}$ and $T$ the noise fluctuations are Gaussian; however, in the regions of high noise power we find non-Gaussian noise fluctuations with a skewed distribution. Experiments have also found evidence for non-Gaussian fluctuations in the glassy regimes [@Noise].

Next we show evidence that the large noise is due to correlated regions of string like electron flow, and that within these regions the electrons move in 1D or quasi 1D channels. Because of the reduced dimensionality the electron motion is more correlated. In Fig. 5(a) we show the trajectories of the electrons for a fixed period of time for a system with $T = 0.09$ at $N_{s}/N_{p} = 1.67$. Here the electrons can flow freely throughout the sample, although there are some areas where electrons become temporarily trapped by a defect site. In Fig. 5(b) at $N_{s}/N_{p} = 1.37$, where the noise power is larger than in the system shown in Fig. 5(a), larger pinned regions appear and the electron motion consists of a mixture of 2D and 1D regions. If the trajectories are followed over longer times,

motion occurs throughout the entire sample. In Fig. 5(c) at $N_{s}/N_{p} = 0.5$, where the noise power and $\alpha$ are both maximum, the electron motion occurs mostly in the form of 1D channels that percolate through the sample. There are also regions where the electron motion occurs in small rings. The channel structures change very slowly with time, with a channel occasionally shutting off while another emerges elsewhere. It is the intermittent opening of the 1D channels which gives rise to the large noise fluctuations in this regime. When a percolating 1D channel opens, all the electrons in that channel move in a correlated fashion leading to a large increase in the conduction. Conversely, if a percolating channel closes all the electrons in that channel cease to move. It is well known that fluctuations in 1D are much more strongly enhanced than in 2D. As $T$ or $N_{s}$ is increased, the motion becomes increasingly 2D in nature and the strong correlations of the electron motion are lost. We also note that the appearance of string like motions in the large noise regions is consistent with studies in glassy systems, where dynamical heterogeneities in the form of 1D stringlike motions of particles have been observed in conjunction with large noise [@Glotzer]. In Fig. 5(d) at $N_{s}/N_{p} = 0.3$, deep in the insulating regime, there are no channels. Instead, the infrequent motion of electrons occurs only by small jumps from defect to defect.

In conclusion, we have presented a simple model for the glassy freezing of interacting electrons in 2D with random disorder. For high electron density or high temperatures, the electrons form a 2D liquid state and we find low conduction noise power with a white spectra. As the density of the electrons is lowered for fixed temperature, or conversely, as the temperature is lowered for fixed low electron density, there is a crossover to a $1/f^{\alpha}$ noise with large low frequency power and $\alpha = 1.37$. In this glassy regime, the electrons move in 1D intermittent stringlike paths which percolate throughout the sample. Similar stringlike motions are also observed in other glass forming systems. For low electron density, all the electrons are frozen by the defect sites and the motion occurs only by single electron hopping events. We find that the noise power decreases exponentially with temperature, in agreement with experiment. Many of our results are in qualitative agreement with recent experiments on 2D electron systems near the metal insulator transition.

This work was supported by the US DoE under Contract No. W-7405-ENG-36.

For a review, see: E. Abrahams, S.V. Kravchenko, and M.P. Sarachik, Rev. Mod. Phys. [**73**]{}, 251 (2001).

J.H. Davies, P.A. Lee, and T.M. Rice, Phys. Rev. Lett. [**49**]{}, 758 (1982).

S. Bogdanovich and D. Popovi' c, Phys. Rev. Lett. [**88**]{}, 236401 (2002).

S. Bogdanovich and D. Popovi' c, Physica E [**12**]{}, 604 (2002).

J. Jaroszy' nski, D. Popovi' c, and T.M. Klapwijk, Phys. Rev. Lett. [**89**]{}, 276401 (2002).

S. Kar [*et al.*]{}, Phys. Rev. Lett. [**91**]{}, 216603 (2003).

M.B. Weissman, Rev. Mod. Phys. [**60**]{}, 537 (1988); P. Dutta [*et al.*]{}, Rev. Mod. Phys. [**53**]{}, 497 (1981).

R.F. Voss and J. Clarke, Phys. Rev. B [**13**]{}, 556 (1976); O. Cohen and Z. Ovadyahu, Phys. Rev. B [**50**]{}, 10442 (1994).

R. Leturcq [*et al.*]{}, Phys. Rev. Lett. [**90**]{}, 076402 (2003).

E. Bielejec and W. Wu, Phys. Rev. Lett. [**87**]{}, 256601 (2001).

S. Chakravarty, S. Kivelson, C. Nayak, and K. Voelker, Philos. Mag. B [**79**]{}, 859 (1999).

J.S. Thakur and D. Neilson, Phys. Rev. B [**59**]{}, R5280 (1999).

A.A. Pastor and V. Dobrosavljevic, Phys. Rev. Lett. [**83**]{}, 4642 (1999).

V. Dobrosavljevic, D. Tanaskovic, and A.A. Pastor, Phys. Rev. Lett. [**90**]{}, 016402 (2003).

For reviews, see: S.C. Glotzer, J. Non-Cryst. Solids [**274**]{}, 342 (2000); R. Richert, J. Phys. Cond. Mat. [**14**]{}, R703 (2002).

A.C. Marley, M.J. Higgins, and S. Bhattacharya, Phys. Rev. Lett. [**74**]{}, 3029 (1995).

M.-C. Cha and H.A. Fertig, Phys. Rev. Lett. [**73**]{}, 870 (1994).

M.-C. Cha and H.A. Fertig, Phys. Rev. B [**50**]{}, 14368 (1994); C. Reichhardt [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 4354 (2001).

N.Grønbech-Jensen, Int. J. Mod. Phys. C [**8**]{}, 1287 (1997).

B.I. Shklovskii, Solid State Commun. [**33**]{}, 273 (1980); V.I. Kozub, Solid State Commun. [**97**]{}, 843 (1996); K. Shtengel and C.C. Yu, Phys. Rev. B [**67**]{}, 165106 (2003).
