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Influence of Coulomb Interaction of Tunable Shapes on the Collective Transport of Ultradilute Two-Dimensional Holes Jian Huang* Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
L. N. Pfeiffer, and K. W. West Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA (Received 25 September 2013; published 22 January 2014) In high quality updoped GaAs field-effect transistors, the two-dimensional charge carrier concentrations can be tuned to very low values similar to the density of electrons on helium surfaces. An important interaction effect, screening of the Coulomb interaction by the gate, rises as a result of the large charge spacing comparable to the distance between the channel and the gate. Based on the results of the temperature (T) dependence of the resistivity from measuring four different samples, a power-law characteristic is found for charge densities ≤ 2 × 109 cm−2 . Moreover, the exponent exhibits a universal dependence on a single dimensionless parameter, the ratio between the mean carrier separation and the distance to the metallic gate that screens the Coulomb interaction. Thus, the electronic properties are tuned through varying the shape of the interaction potential. DOI: 10.1103/PhysRevLett.112.036803
PACS numbers: 73.40.-c, 71.27.+a, 73.20.Qt
The study of the charge transport of two-dimensional (2D) electron systems [1] provides a unique means of studying the interplay between disorder and electronelectron interactions [2], a fundamentally important subject that has generated renewed interests due to its importance in a broad range of modern (correlated) electron systems. While noninteracting 2D electrons are believed to form Anderson insulators [3,4], the situation is more complex with interaction effects [5–8]. In the very dilute charge concentration limit, the carriers are expected to form a Wigner crystal (WC) [9–11], which is a unique quantum state that can be utilized for future applications including quantum electronics and spintronics. However, there have been outstanding experimental challenges associated with demonstrating such interaction-driven states [12,13]. The capability of tuning interaction, especially the shape of the Coulomb potential, and linking it with the corresponding electronic properties is essential. Though the effective Coulomb interaction can be enhanced through lowering the charge densities, it alone, however, does not change the nature (or shape) of the interaction. This Letter presents a study of a tunable gate screening effect that modifies the interaction potential and even the shape of interaction. Interestingly, this “tuning knob” helps identify the role of interaction via nonactivated collective properties [14,15] in the vicinity and, especially, the insulating side of the metal-to-insulator transition (MIT) [16]. Because the interaction parameter rs, a ratio of the Coulomb energy EC to the kinetic energy EF, increases with lowering of the electron density (n), samples with most dilute carriers are best suited for studying the collective phenomena. However, reducing n increases 0031-9007=14=112(3)=036803(5)
the carrier separation ∝ n−1=2 that usually exceeds the single-particle localization length ξ determined by the disorder level, so that the interaction effects are overshadowed by single-particle localization. Thus, an ultraclean 2D environment is an important requirement. Indeed, experimental studies of the transport of 2D systems has been greatly influenced by the sample quality and the nature of the disorder. Earlier results of Si devices demonstrated activated transport consistent with the Anderson insulator [2]: the Arrhenius conductivity σ ∼ e−T A =T at high temperatures, and the variable-range hopping conductance, ν σ ∼ e−ðT =TÞ , at lower temperatures, with ν ¼ 1=3 and 1=2 corresponding to Mott [17] and Efros-Shklovskii [18] (Coulomb gap) scenarios. In the mid-1990s, experiments performed on cleaner 2D electrons in Si metal-oxidesemiconductor field-effect transistors demonstrated both metal-like (dσ=dT < 0) and insulatorlike (dσ=dT > 0) behaviors, depending on whether the density was above or below a certain critical value nc [16]. Though the origin of the metallic regime at T → 0 is still debated, the transport on the insulating side, nevertheless, remains activated, agreeing with Anderson localization. However, the transport characteristics in the insulating regime of the MIT has changed qualitatively with the adoption of higher purity devices—undoped GaAs heterojunction-insulated-gate field-effect transistors (HIGFETs). Recent experiments [19–21] demonstrated nonactivated conductivity, while preserving the “insulating” sign dσ=dT > 0. A close-to-linear dependence σ ∝ T was first observed in a p-GaAs HIGFET for densities down to p ¼ 1.5 × 109 cm−2 [19]. Subsequent experiments in the devices of the same kind not only confirmed this observation
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TABLE I. The barrier thickness d (distance to the metallic gate) for four different samples. Samples
No. 2
No. 3
No. 4
Noh et al.
Barrier d
600 nm
600 nm
250 nm
500 nm
but also revealed a more general power-law-like T dependence, σ ∝ T α , with a varying exponent 1 ≲ α ≲ 2, at sufficiently low temperatures [20,21]. Remarkably, such a behavior persists even for a record-low 2D hole density of p ¼ 7 × 108 cm−2 , well into the WC regime with a Coulomb energy more than 70 times the nominal Fermi liquid (FL) energy. In this Letter we report the evidence of an interactiondriven nature of this power law T dependence based on data collected from four different p-type HIGFET samples that only differ by the structural barrier thickness (the distance d from the 2D layer to the metal gate, Table I). The sample fabrication and the measurement details are described in Ref. [22] for the first three samples, while the data from the fourth sample are drawn from Ref. [19] for comparison. We note the excitation used for this measurement is between 0.5–1 nA, well within the linear response window. The fitting of the conductivity for the temperatures 35 < T ≲ 200 mK to σ=σ Q ¼ G0 þ ðT=T 0 Þα ;
σ Q ¼ e2 =ð2πℏÞ
(1)
yields a sample-dependent exponent αðpÞ that grows with decreasing hole density p, while the T-independent term remains negligible, G0 σ Q ≪ σðTÞ. Moreover, α depends on the single dimensionless parameter κ ¼ a=d, where a ¼ ðπpÞ−1=2 (where 2a is the mean carrier distance). Such a single-parameter dependence connects to a significant screening effect over ≃2d for the Coulomb interaction, rather than to the sample-specific localization length dictated by disorder. In Fig. 1, we show the log-log plots of σðTÞ obtained from four different samples: (a) No. 2 and (b) No. 3 are samples from the same wafer, cooled down to the lowest temperature of around 80 mK; (c) sample No. 4 is cooled down to 35 mK; and (d) is the data from Ref. [19] with the lowest T of 65 mK. Each panel in Fig. 1 contains curves for a number of hole densities. For p above the critical density, pc ≃ 4 × 109 cm−2 , the transport is metal-like [2]. Below we focus on the low-density “insulating” regime. There are three features common to all four samples. (i) Although dσ=dT > 0, the conductivity never becomes activated. This is demonstrated in Ref. [20] by comparing σðTÞ to the hopping conduction models. Here, the activated T dependence with a strong downward bending for T below the activation temperature is clearly absent. As reported in Ref. [21], this nonactivated transport is particularly fragile to even a slight increase of disorder that
FIG. 1. Conductivity temperature dependence in the log-log scale for a set of hole densities for the samples. (a) No. 2 (scattered points are the dc results for p ¼ 8 × 108 cm−2 ), (b) No. 3, (c) No. 4, and (d) the sample from Ref. [19]. Fermi temperature EF is estimated in (a) assuming m ∼ 0.3m0 .
actually restores the activated conductance. (ii) For low densities p < 2 × 109 cm−2 , d log σ=d log T is roughly T independent for T ≲ 200 mK and increases with decreasing density [23]. (iii) Finally, though nonactivated, the conductivity values are 1–2 orders below e2 =ð2πℏÞ. The p-dependence αðpÞ of the power-law exponent (slopes in the log-log plot) is obtained by fitting σðTÞ to the formula (1) for the low density curves, p ≤ 3 × 109 cm−2 . The fitting parameters G0 and T 0 are plotted in Fig. 2. The T-independent term G0 remains around zero up to p ≃ 2 × 109 cm−2 for the samples No. 2, No. 3, and No. 4, indicating a fairly good power-law dependence: σ ∝ T α . On the other hand, for the sample from Ref. [19], the constant G0 is more significant. We stress that, although the high-temperature behavior in the sample from Noh et al. looks quite linear, σ ≃ A þ BT as reported in Ref. [19], at T ≲ 200 mK the conductivity curve exhibits a noticeable bending at T < 200 mK as demonstrated in the linear scale plot in the inset of Fig. 1(d), and is best fitted to Eq. (1) with α ≠ 1 [see Fig. 3]. The parameter T 0 for all four samples has a trend of a slow decrease with increasing density from about 400 mK to about 300 mK for densities up to p ≃ 2 × 109 cm−2 .
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PHYSICAL REVIEW LETTERS Sample #2 Sample #3 Sample #4 Noh et al.
Sample #2 Sample #3 Sample #4 Noh et al
(a) 2a
FIG. 3 (color online). The exponent α as a function of average charge spacing for all four samples. The dotted lines are guides to the eye. rs varies from 50–90.
Sample #2 Sample #3 Sample #4 Noh et al
(b)
FIG. 2 (color online). Density dependence of the fitting parameters (a) G0 and (b) T 0 from Eq. (1). The rs value ranges from 50–90 for p from 0.7 to 3 × 109 cm−2 assuming m ∼ 0.3m0 .
Figure 3 shows how the exponent α behaves as a function of the average carrier spacing 2a. The results obtained from samples No. 2 and No. 3 fall approximately onto a single curve in which α varies from 1.1 to about 1.5. The results from both samples No. 4 and Ref. [19] qualitatively follow the same trend albeit the values of α are greater by about 0.6 and 0.1, respectively. This shift in αðpÞ relative to that for samples No. 2 and No. 3 motivates us to look into the structural difference between the samples, specifically the barrier thickness d. As shown in Table I, samples No. 2 and No. 3 have the same d, while the values of d for the other two samples are notably different. Moreover, the values of α increase with the decrease in d for a given density. This trend points at the role of the screening by the metallic gate. The metallic gate at the distance d from the 2D hole layer starts to screen the 1=r interaction towards 1=r3 when r ≳ 2d. For the lowest p, interaction becomes effectively short ranged, with the relevant parameter being the ratio κ ¼ a=d between the carrier spacing ≈2a and the screening radius 2d. For our measurements, this ratio can be continuously varied in the range 0.1 < κ < 0.8; the sample of Noh et al. is also in this range. The effect of the gate screening becomes apparent in Fig. 4, where all the
power-law exponents α are plotted as a function of κ. Remarkably, αðκÞ from all four samples fall onto a single curve within reasonable error bars. This curve tends to saturate for α ≳ 2, while it most strongly varies when α ≈ 1. Thus the linear dependence σ ∝ T reported in Ref. [19] is most probably a crossover into an entirely different transport regime, rather than a universal signature at low T. [In the inset we plot σ 1=α as a function of T=T 0 ðpÞ to illustrate the relative insignificance of the constant G0 in Eq. (1).] The rs value for p ¼ 1 × 109 cm−2 is 80 if one assumes m ¼ 0.3m0 . Despite the enormous rs , σðT; pÞ indicates a strongly correlated liquid, a melted WC, instead of a WC. First, as discussed in Ref. [10], the melting point T m is already significantly lower than the classical level (∼Ec =137 ∼ 50–100 mK for p from 0.7 to 3 × 109 cm−2 ) due to quantum fluctuations and signal excitations, as well as the presence of residual disorder. Now, the gate screening adds another twist since it tends to weaken the long-range Coulomb potential with decreasing p once approaches d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa by reducing it by a factor of 1=½1=2r − 1= ð2rÞ2 þ ð2dÞ2 . The scaling of α − κ captures the effect of the tunable shapes Sample #2 Sample #3 Sample #4 Noh et al.
Noh et al.
#2,#3,#4 To
FIG. 4 (color online). The exponent α as a function of the ratio κ ¼ a=d. Inset: σ 1=α as a function of T=T 0 ðpÞ.
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of interaction, which further weakens the long-range order and necessarily reduces T m . This notion is not entirely surprising and has been shown by the kinetic and thermodynamic properties of classical plasmas dependent on the same screening parameter κ [24], even though the classical arguments are insufficient for our case. Thus, we now understand that the temperature window for the observed power-law σðTÞ is mostly above T m where the system arrives at a strongly correlated semiquantum liquid state whose transport properties have not yet been explored. The only known relevant theoretical work on transport is the hydrodynamical flow of weakly interacting neutral particles developed by Andreev [25], which predicted a linear T-dependent viscosity (or resistivity) and has been considered for FL-WC mixtures [12,26]. Intuitively, strong interaction and disorder should result in a stronger T dependence, i.e., power laws. Now, tuning κ ¼ a=d, through varying a and d, can be utilized as a knob for modifying the range (or shape) of the Coulomb force and realizing continuously modified states, even with reducing effective interaction with decreasing p at sufficiently low densities (a ≳ d). This is why a possibility of a reentry into the FL was suggested [26]. More importantly, the screening effect can be used to directly probe a WC through the dc measurement of the threshold transport characteristics [27] of a pinned quantum WC as shown in Ref. [13]. The onset of the gate screening cuts down the necessary long-ranged Coulomb field for a WC, and eventually leads to melting. Thus, the solid-toliquid transition caused by an increasing screening effect can be detected through a vanishing threshold when a WC is melted. Therefore, such a study will help to determine the long-range order (of a WC) in response to a tunable shape of the Coulomb potential. We finally note that varying d, in a similar fashion of changing the interaction range, also modifies the correlation length ξdis of the disorder potential [28]. In the case where long-ranged disorder is dominated by the charge impurities in the bulk, the gate screens the disorder potential harmonics for length scales ≳d, so that ξdis ∼ d. In such situations it becomes difficult to differentiate the electron-electron interaction effects from those due to the change in the distribution of disorder [29,30]. It is clear, however, that the change in the disorder alone cannot account for the power-law exponent α smoothly varying with κ, and the interaction effects should play a major role. The peculiar insulating behavior (dσ=dT < 0) is undoubtedly different from an Anderson insulator. To summarize, by performing transport measurements of high quality dilute 2D holes with densities down to 7 × 108 cm−2 , we established the dependence of the powerlaw exponent α [Eq. (1)] on the ratio a=d between the Wigner-Seitz radius a and the distance to the metal gate d, thus confirming the screening effect of the Coulomb interaction by the gate. These results provide direct evidence
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of the role of the electron-electron interaction in the 2D transport and suggest that the transport is sensitive to both the strength and the shape of the interaction potential controlled by the screening distance d. By varying the ratio a=d, one realizes a tunable strongly correlated system that exhibits different transport properties. We thank D. Novikov for helpful discussions. We acknowledge the support of this work from NSF under DMR-1105183. The work at Princeton was partially funded by the Gordon and Betty Moore Foundation through Grant GBMF2719, and by the National Science Foundation MRSEC-DMR-0819860 at the Princeton Center for Complex Materials.
*
[email protected] [1] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). [2] See, e.g., the review by E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. 73, 251 (2001). [3] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [4] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). [5] B. L. Altshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered Systems, edited by A. L. Efros and M. Pollak (North-Holland, Amsterdam, 1985). [6] A. M. Finkelstein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [, Sov. Phys. JETP 57, 97 (1983)]; Z. Phys. B 56, 189 (1984). [7] F. Stern and S. Das Sarma, Solid State Electron. 28, 211 (1985); A. Gold and V. T. Dolgopolov, Phys. Rev. B 33, 1076 (1986); S. Das Sarma, Phys. Rev. B 33, 5401 (1986). [8] G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64, 214204 (2001). [9] E. P. Wigner, Phys. Rev. 46, 1002 (1934). [10] S. T. Chui, Physics of the Electron Solid (International Press of Boston, Boston, 1994). [11] E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, E. Paris, and B. Etienne, Phys. Rev. Lett. 60, 2765 (1988); H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 65, 633 (1990). [12] B. Spivak, S. V. Kravchenko, S. A. Kivelson, and X. P. A. Gao, Rev. Mod. Phys. 82, 1743 (2010). [13] Jian Huang, L. N. Pfeiffer, K. W. West, arXiv:1306.4196. [14] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. (Amsterdam) 321, 1126 (2006). [15] A. Punnoose and A. M. Finkelstein, Science 310, 289 (2005). [16] S. V. Kravchenko, G. V. Kravchenko, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B 50, 8039 (1994). [17] N. F. Mott, J. Non-Cryst. Solids 1, 1 (1968). [18] B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin, 1984). [19] Hwayong Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 68, 241308(R) (2003). [20] Jian Huang, D. S. Novikov, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 74, 201302(R) (2006).
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[21] Jian Huang, L. N. Pfeiffer, K. W. West, Phys. Rev. B 85, 041304(R) (2012). [22] Jian Huang, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Int. J. Mod. Phys. B, 21, 1219 (2007). [23] Above T ≃ 200 mK the temperature dependence σðTÞ begins to saturate, peaks at T ∼ 1 K, and crosses over to that dominated by phonon scattering. The high-T analysis will be reported elsewhere. [24] G. J. Kalman, P. Hartmann, Z. Donkó, and M. Rosenberg, Phys. Rev. Lett. 92, 065001 (2004); P. Hartmann, G. J. Kalman, Z. Donkó, and K. Kutasi, Phys. Rev. E 72, 026409 (2005).
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[25] A. F. Andreev, JETP Lett. 28, 557 (1978); A. F. Andreev and Yu. A. Kosevich, Sov. Phys. JETP 50, 1218 (1979). [26] B. Spivak and S. A. Kivelson, Phys. Rev. B 70, 155114 (2004). [27] V. M. Pudalov, M. D’Iorio, S. V. Kravchenko, and J. W. Campbell, Phys. Rev. Lett. 70, 1866 (1993). [28] We estimate the tunneling probability of a carrier to scatter off surface imperfections at the gate level to be exponentially small. [29] A. L. Efros, F. G. Pikus, and V. G. Burnett, Phys. Rev. B 47, 2233 (1993). [30] M. M. Fogler, Phys. Rev. B 69, 121409(R) (2004).
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Influence of Coulomb Interaction of Tunable Shapes on the Collective Transport of Ultradilute Two-Dimensional Holes Jian Huang* Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
L. N. Pfeiffer, and K. W. West Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA (Received 25 September 2013; published 22 January 2014) In high quality updoped GaAs field-effect transistors, the two-dimensional charge carrier concentrations can be tuned to very low values similar to the density of electrons on helium surfaces. An important interaction effect, screening of the Coulomb interaction by the gate, rises as a result of the large charge spacing comparable to the distance between the channel and the gate. Based on the results of the temperature (T) dependence of the resistivity from measuring four different samples, a power-law characteristic is found for charge densities ≤ 2 × 109 cm−2 . Moreover, the exponent exhibits a universal dependence on a single dimensionless parameter, the ratio between the mean carrier separation and the distance to the metallic gate that screens the Coulomb interaction. Thus, the electronic properties are tuned through varying the shape of the interaction potential. DOI: 10.1103/PhysRevLett.112.036803
PACS numbers: 73.40.-c, 71.27.+a, 73.20.Qt
The study of the charge transport of two-dimensional (2D) electron systems [1] provides a unique means of studying the interplay between disorder and electronelectron interactions [2], a fundamentally important subject that has generated renewed interests due to its importance in a broad range of modern (correlated) electron systems. While noninteracting 2D electrons are believed to form Anderson insulators [3,4], the situation is more complex with interaction effects [5–8]. In the very dilute charge concentration limit, the carriers are expected to form a Wigner crystal (WC) [9–11], which is a unique quantum state that can be utilized for future applications including quantum electronics and spintronics. However, there have been outstanding experimental challenges associated with demonstrating such interaction-driven states [12,13]. The capability of tuning interaction, especially the shape of the Coulomb potential, and linking it with the corresponding electronic properties is essential. Though the effective Coulomb interaction can be enhanced through lowering the charge densities, it alone, however, does not change the nature (or shape) of the interaction. This Letter presents a study of a tunable gate screening effect that modifies the interaction potential and even the shape of interaction. Interestingly, this “tuning knob” helps identify the role of interaction via nonactivated collective properties [14,15] in the vicinity and, especially, the insulating side of the metal-to-insulator transition (MIT) [16]. Because the interaction parameter rs, a ratio of the Coulomb energy EC to the kinetic energy EF, increases with lowering of the electron density (n), samples with most dilute carriers are best suited for studying the collective phenomena. However, reducing n increases 0031-9007=14=112(3)=036803(5)
the carrier separation ∝ n−1=2 that usually exceeds the single-particle localization length ξ determined by the disorder level, so that the interaction effects are overshadowed by single-particle localization. Thus, an ultraclean 2D environment is an important requirement. Indeed, experimental studies of the transport of 2D systems has been greatly influenced by the sample quality and the nature of the disorder. Earlier results of Si devices demonstrated activated transport consistent with the Anderson insulator [2]: the Arrhenius conductivity σ ∼ e−T A =T at high temperatures, and the variable-range hopping conductance, ν σ ∼ e−ðT =TÞ , at lower temperatures, with ν ¼ 1=3 and 1=2 corresponding to Mott [17] and Efros-Shklovskii [18] (Coulomb gap) scenarios. In the mid-1990s, experiments performed on cleaner 2D electrons in Si metal-oxidesemiconductor field-effect transistors demonstrated both metal-like (dσ=dT < 0) and insulatorlike (dσ=dT > 0) behaviors, depending on whether the density was above or below a certain critical value nc [16]. Though the origin of the metallic regime at T → 0 is still debated, the transport on the insulating side, nevertheless, remains activated, agreeing with Anderson localization. However, the transport characteristics in the insulating regime of the MIT has changed qualitatively with the adoption of higher purity devices—undoped GaAs heterojunction-insulated-gate field-effect transistors (HIGFETs). Recent experiments [19–21] demonstrated nonactivated conductivity, while preserving the “insulating” sign dσ=dT > 0. A close-to-linear dependence σ ∝ T was first observed in a p-GaAs HIGFET for densities down to p ¼ 1.5 × 109 cm−2 [19]. Subsequent experiments in the devices of the same kind not only confirmed this observation
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TABLE I. The barrier thickness d (distance to the metallic gate) for four different samples. Samples
No. 2
No. 3
No. 4
Noh et al.
Barrier d
600 nm
600 nm
250 nm
500 nm
but also revealed a more general power-law-like T dependence, σ ∝ T α , with a varying exponent 1 ≲ α ≲ 2, at sufficiently low temperatures [20,21]. Remarkably, such a behavior persists even for a record-low 2D hole density of p ¼ 7 × 108 cm−2 , well into the WC regime with a Coulomb energy more than 70 times the nominal Fermi liquid (FL) energy. In this Letter we report the evidence of an interactiondriven nature of this power law T dependence based on data collected from four different p-type HIGFET samples that only differ by the structural barrier thickness (the distance d from the 2D layer to the metal gate, Table I). The sample fabrication and the measurement details are described in Ref. [22] for the first three samples, while the data from the fourth sample are drawn from Ref. [19] for comparison. We note the excitation used for this measurement is between 0.5–1 nA, well within the linear response window. The fitting of the conductivity for the temperatures 35 < T ≲ 200 mK to σ=σ Q ¼ G0 þ ðT=T 0 Þα ;
σ Q ¼ e2 =ð2πℏÞ
(1)
yields a sample-dependent exponent αðpÞ that grows with decreasing hole density p, while the T-independent term remains negligible, G0 σ Q ≪ σðTÞ. Moreover, α depends on the single dimensionless parameter κ ¼ a=d, where a ¼ ðπpÞ−1=2 (where 2a is the mean carrier distance). Such a single-parameter dependence connects to a significant screening effect over ≃2d for the Coulomb interaction, rather than to the sample-specific localization length dictated by disorder. In Fig. 1, we show the log-log plots of σðTÞ obtained from four different samples: (a) No. 2 and (b) No. 3 are samples from the same wafer, cooled down to the lowest temperature of around 80 mK; (c) sample No. 4 is cooled down to 35 mK; and (d) is the data from Ref. [19] with the lowest T of 65 mK. Each panel in Fig. 1 contains curves for a number of hole densities. For p above the critical density, pc ≃ 4 × 109 cm−2 , the transport is metal-like [2]. Below we focus on the low-density “insulating” regime. There are three features common to all four samples. (i) Although dσ=dT > 0, the conductivity never becomes activated. This is demonstrated in Ref. [20] by comparing σðTÞ to the hopping conduction models. Here, the activated T dependence with a strong downward bending for T below the activation temperature is clearly absent. As reported in Ref. [21], this nonactivated transport is particularly fragile to even a slight increase of disorder that
FIG. 1. Conductivity temperature dependence in the log-log scale for a set of hole densities for the samples. (a) No. 2 (scattered points are the dc results for p ¼ 8 × 108 cm−2 ), (b) No. 3, (c) No. 4, and (d) the sample from Ref. [19]. Fermi temperature EF is estimated in (a) assuming m ∼ 0.3m0 .
actually restores the activated conductance. (ii) For low densities p < 2 × 109 cm−2 , d log σ=d log T is roughly T independent for T ≲ 200 mK and increases with decreasing density [23]. (iii) Finally, though nonactivated, the conductivity values are 1–2 orders below e2 =ð2πℏÞ. The p-dependence αðpÞ of the power-law exponent (slopes in the log-log plot) is obtained by fitting σðTÞ to the formula (1) for the low density curves, p ≤ 3 × 109 cm−2 . The fitting parameters G0 and T 0 are plotted in Fig. 2. The T-independent term G0 remains around zero up to p ≃ 2 × 109 cm−2 for the samples No. 2, No. 3, and No. 4, indicating a fairly good power-law dependence: σ ∝ T α . On the other hand, for the sample from Ref. [19], the constant G0 is more significant. We stress that, although the high-temperature behavior in the sample from Noh et al. looks quite linear, σ ≃ A þ BT as reported in Ref. [19], at T ≲ 200 mK the conductivity curve exhibits a noticeable bending at T < 200 mK as demonstrated in the linear scale plot in the inset of Fig. 1(d), and is best fitted to Eq. (1) with α ≠ 1 [see Fig. 3]. The parameter T 0 for all four samples has a trend of a slow decrease with increasing density from about 400 mK to about 300 mK for densities up to p ≃ 2 × 109 cm−2 .
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PHYSICAL REVIEW LETTERS Sample #2 Sample #3 Sample #4 Noh et al.
Sample #2 Sample #3 Sample #4 Noh et al
(a) 2a
FIG. 3 (color online). The exponent α as a function of average charge spacing for all four samples. The dotted lines are guides to the eye. rs varies from 50–90.
Sample #2 Sample #3 Sample #4 Noh et al
(b)
FIG. 2 (color online). Density dependence of the fitting parameters (a) G0 and (b) T 0 from Eq. (1). The rs value ranges from 50–90 for p from 0.7 to 3 × 109 cm−2 assuming m ∼ 0.3m0 .
Figure 3 shows how the exponent α behaves as a function of the average carrier spacing 2a. The results obtained from samples No. 2 and No. 3 fall approximately onto a single curve in which α varies from 1.1 to about 1.5. The results from both samples No. 4 and Ref. [19] qualitatively follow the same trend albeit the values of α are greater by about 0.6 and 0.1, respectively. This shift in αðpÞ relative to that for samples No. 2 and No. 3 motivates us to look into the structural difference between the samples, specifically the barrier thickness d. As shown in Table I, samples No. 2 and No. 3 have the same d, while the values of d for the other two samples are notably different. Moreover, the values of α increase with the decrease in d for a given density. This trend points at the role of the screening by the metallic gate. The metallic gate at the distance d from the 2D hole layer starts to screen the 1=r interaction towards 1=r3 when r ≳ 2d. For the lowest p, interaction becomes effectively short ranged, with the relevant parameter being the ratio κ ¼ a=d between the carrier spacing ≈2a and the screening radius 2d. For our measurements, this ratio can be continuously varied in the range 0.1 < κ < 0.8; the sample of Noh et al. is also in this range. The effect of the gate screening becomes apparent in Fig. 4, where all the
power-law exponents α are plotted as a function of κ. Remarkably, αðκÞ from all four samples fall onto a single curve within reasonable error bars. This curve tends to saturate for α ≳ 2, while it most strongly varies when α ≈ 1. Thus the linear dependence σ ∝ T reported in Ref. [19] is most probably a crossover into an entirely different transport regime, rather than a universal signature at low T. [In the inset we plot σ 1=α as a function of T=T 0 ðpÞ to illustrate the relative insignificance of the constant G0 in Eq. (1).] The rs value for p ¼ 1 × 109 cm−2 is 80 if one assumes m ¼ 0.3m0 . Despite the enormous rs , σðT; pÞ indicates a strongly correlated liquid, a melted WC, instead of a WC. First, as discussed in Ref. [10], the melting point T m is already significantly lower than the classical level (∼Ec =137 ∼ 50–100 mK for p from 0.7 to 3 × 109 cm−2 ) due to quantum fluctuations and signal excitations, as well as the presence of residual disorder. Now, the gate screening adds another twist since it tends to weaken the long-range Coulomb potential with decreasing p once approaches d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa by reducing it by a factor of 1=½1=2r − 1= ð2rÞ2 þ ð2dÞ2 . The scaling of α − κ captures the effect of the tunable shapes Sample #2 Sample #3 Sample #4 Noh et al.
Noh et al.
#2,#3,#4 To
FIG. 4 (color online). The exponent α as a function of the ratio κ ¼ a=d. Inset: σ 1=α as a function of T=T 0 ðpÞ.
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of interaction, which further weakens the long-range order and necessarily reduces T m . This notion is not entirely surprising and has been shown by the kinetic and thermodynamic properties of classical plasmas dependent on the same screening parameter κ [24], even though the classical arguments are insufficient for our case. Thus, we now understand that the temperature window for the observed power-law σðTÞ is mostly above T m where the system arrives at a strongly correlated semiquantum liquid state whose transport properties have not yet been explored. The only known relevant theoretical work on transport is the hydrodynamical flow of weakly interacting neutral particles developed by Andreev [25], which predicted a linear T-dependent viscosity (or resistivity) and has been considered for FL-WC mixtures [12,26]. Intuitively, strong interaction and disorder should result in a stronger T dependence, i.e., power laws. Now, tuning κ ¼ a=d, through varying a and d, can be utilized as a knob for modifying the range (or shape) of the Coulomb force and realizing continuously modified states, even with reducing effective interaction with decreasing p at sufficiently low densities (a ≳ d). This is why a possibility of a reentry into the FL was suggested [26]. More importantly, the screening effect can be used to directly probe a WC through the dc measurement of the threshold transport characteristics [27] of a pinned quantum WC as shown in Ref. [13]. The onset of the gate screening cuts down the necessary long-ranged Coulomb field for a WC, and eventually leads to melting. Thus, the solid-toliquid transition caused by an increasing screening effect can be detected through a vanishing threshold when a WC is melted. Therefore, such a study will help to determine the long-range order (of a WC) in response to a tunable shape of the Coulomb potential. We finally note that varying d, in a similar fashion of changing the interaction range, also modifies the correlation length ξdis of the disorder potential [28]. In the case where long-ranged disorder is dominated by the charge impurities in the bulk, the gate screens the disorder potential harmonics for length scales ≳d, so that ξdis ∼ d. In such situations it becomes difficult to differentiate the electron-electron interaction effects from those due to the change in the distribution of disorder [29,30]. It is clear, however, that the change in the disorder alone cannot account for the power-law exponent α smoothly varying with κ, and the interaction effects should play a major role. The peculiar insulating behavior (dσ=dT < 0) is undoubtedly different from an Anderson insulator. To summarize, by performing transport measurements of high quality dilute 2D holes with densities down to 7 × 108 cm−2 , we established the dependence of the powerlaw exponent α [Eq. (1)] on the ratio a=d between the Wigner-Seitz radius a and the distance to the metal gate d, thus confirming the screening effect of the Coulomb interaction by the gate. These results provide direct evidence
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of the role of the electron-electron interaction in the 2D transport and suggest that the transport is sensitive to both the strength and the shape of the interaction potential controlled by the screening distance d. By varying the ratio a=d, one realizes a tunable strongly correlated system that exhibits different transport properties. We thank D. Novikov for helpful discussions. We acknowledge the support of this work from NSF under DMR-1105183. The work at Princeton was partially funded by the Gordon and Betty Moore Foundation through Grant GBMF2719, and by the National Science Foundation MRSEC-DMR-0819860 at the Princeton Center for Complex Materials.
*
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