PRL 110, 249701 (2013)
Comment on ‘‘Connecting the Reentrant Insulating Phase and the Zero-Field Metal-Insulator Transition in a 2D Hole System’’ In a recent Letter [1], Qiu et al. reported an observation of the reentrant insulating phases (RIP) and the QHEinsulator (RQHE-I) transitions for a low-density p-GaAs quantum well. The observation of the RQHE-I transitions in dc resistivity in a state-of-art two-dimensional (2D) hole system brilliantly confirms results, observed earlier for 2D carrier systems in Si-metal-oxide-semiconductor fieldeffect transistors (Si-MOSFETs) [2], and p-Si-SiGe [3]. However, the enormous drop in measured capacitance [1] is 2 orders of magnitude greater than that observed in RIP [4] and is erroneously attributed in Ref. [1] to the formation of an incompressible state. The similarity and intimate relation between reentrant [2] and zero-field insulating phase [5] (that is the goal of Ref. [1]) has been proven on the basis of activated transport, threshold I-V characteristics [2,5], and their continuous transformation from the zero-field insulator to RQHE-I regime [6]. These features allowed the authors of Refs. [2,5,6] to attribute the low- and zero-field insulating phase to the pinned Wigner solid (WS), similar to the conclusions made in Ref. [1]. Although the details of the density-field phase diagram for different material systems might be different, they reflect the same physics. As expected, the capacitive measurements in Si-MOSFETS [4] in the RIP phase did show a compressible state. Indeed, although the WS by itself is incompressible [7], the neutralizing oppositely charged gate makes the thermodynamic (! ! 0, q ! 0) compressibility of the Wigner solid finite and negative [8]. Conclusions on the incompressibility in Ref. [1] are made by considering the sample as two capacitors in series, 1=C ¼ 1=C0 þ 1=Cq , where C0 is the geometric one and Cq ¼ e2 dp=d (all per unit area), and is the chemical potential of the 2D system. The total capacitance was determined as C ¼ Iy =2fVx , with the recharging current I ¼ Ix þ iIy and the ac excitation Vx . The observed giant (factor of 2) decrease in capacitance [Fig. 1(b)] is 2 orders of magnitude higher than that typical for the RIP phase in Si-MOSFETs [4]; in Ref. [1] it was
FIG. 1 (color online). (a) Magnetocapacitance color plot from Ref. [1]. (b) Examples of magnetoresistance (lines) and magnetocapacitance (symbols) plots from Ref. [1].
0031-9007=13=110(24)=249701(2)
week ending 14 JUNE 2013
PHYSICAL REVIEW LETTERS
attributed to the drop in Cq value, i.e., to the formation of an incompressible state. However, the existence of finite area incompressible domains in the field-density phase diagram for the 2D system with charge-neutralizing background is unphysical. Indeed, had the inequality Cq & C0 been fulfilled in a finite density range, the difference in chemical potentials of the 2D layer between, e.g., points A and B (or C and R D) in Fig. 1(a), would have been equal to ðD C Þ ¼ ppDC e2 dp=Cq * e2 ðpD pC Þ=C0 20 meV, which is much higher than EF , @!c 0:3 meV, and the WS binding energy (EL ES & 2 eV for rs * 40 [9]). The enormous drop in measured capacitance Cmeas may be caused, e.g., by slow charging or discharging at a given frequency f, because the phases marked in Ref. [1] as ‘‘incompressible’’ [colored in violet in Fig. 1(a)] are very poorly conductive (xx ! 0). The true resistance value (1=xx , or R) in recharging processes is likely to be much higher than that found from transport measurements in Ref. [1] [see Fig. 1(b)], because for the zero-field insulating phase and RIP, the I-V characteristics [2,5] have a threshold character already in the fA range. In this case, the Cmeas value will be frequency dependent and the recharging current will acquire a phase shift. In case the Cmeas value is frequency independent, its enormous drop may result from reduction in the charging area, i.e., C0 value [10,11]; indeed, because of the inevitable fluctuations of donors, the 2D gas becomes inhomogeneous at ultralow densities [10] because nonlinear screening leads to the formation of fully depleted areas. This work is supported by RFBR and the Ministry of Education and Science, RF (Grant No. 8375). A. Yu. Kuntsevich1 and V. M. Pudalov1,2 1
P. N. Lebedev Physical Institute of the Russian Academy of Sciences Moscow 119991, Russia 2 Moscow Institute of Physics and Technology Dolgoprudny 141700, Russia Received 19 February 2013; published 13 June 2013 DOI: 10.1103/PhysRevLett.110.249701 PACS numbers: 71.30.+h, 73.40.c, 73.63.Hs [1] R. L. J. Qiu, X. P. A. Gao, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 108, 106404 (2012). [2] M. D’Iorio, V. M. Pudalov, and S. G. Semenchinsky, Phys. Lett. A 150, 422 (1990); Phys. Rev. B 46, 15 992 (1992). [3] M. R. Sakr, M. Rahimi, S. V. Kravchenko, P. T. Coleridge, R. Williams, and J. Lapointe, Phys. Rev. B 64, 161308 (2001). [4] S. V. Kravchenko, J. A. A. J. Perenboom, and V. M. Pudalov, Phys. Rev. B 44, 13 513 (1991). [5] V. M. Pudalov, M. D’Iorio, S. V. Kravchenko, and J. W. Campbell, Phys. Rev. Lett. 70, 1866 (1993); V. M. Pudalov and S. T. Chui, Phys. Rev. B 49, 14 062 (1994). [6] V. M. Pudalov, in Physics of the Electron Solid, edited by S.-T. Chui (International Press, Cambridge, MA, 1994), Chap. 4.
249701-1
Ó 2013 American Physical Society
PRL 110, 249701 (2013)
PHYSICAL REVIEW LETTERS
[7] R. Chitra, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 65, 035312 (2001). [8] R. Chitra and T. Giamarchi, Eur. Phys. J. B 44, 455 (2005). [9] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).
week ending 14 JUNE 2013
[10] M. M. Fogler, Phys. Rev. B 69, 121409(R) (2004). [11] G. Allison, E. A. Galaktionov, A. K. Savchenko, S. S. Safonov, M. M. Fogler, M. Y. Simmons, and D. A. Ritchie, Phys. Rev. Lett. 96, 216407 (2006).
249701-2
Comment on ‘‘Connecting the Reentrant Insulating Phase and the Zero-Field Metal-Insulator Transition in a 2D Hole System’’ In a recent Letter [1], Qiu et al. reported an observation of the reentrant insulating phases (RIP) and the QHEinsulator (RQHE-I) transitions for a low-density p-GaAs quantum well. The observation of the RQHE-I transitions in dc resistivity in a state-of-art two-dimensional (2D) hole system brilliantly confirms results, observed earlier for 2D carrier systems in Si-metal-oxide-semiconductor fieldeffect transistors (Si-MOSFETs) [2], and p-Si-SiGe [3]. However, the enormous drop in measured capacitance [1] is 2 orders of magnitude greater than that observed in RIP [4] and is erroneously attributed in Ref. [1] to the formation of an incompressible state. The similarity and intimate relation between reentrant [2] and zero-field insulating phase [5] (that is the goal of Ref. [1]) has been proven on the basis of activated transport, threshold I-V characteristics [2,5], and their continuous transformation from the zero-field insulator to RQHE-I regime [6]. These features allowed the authors of Refs. [2,5,6] to attribute the low- and zero-field insulating phase to the pinned Wigner solid (WS), similar to the conclusions made in Ref. [1]. Although the details of the density-field phase diagram for different material systems might be different, they reflect the same physics. As expected, the capacitive measurements in Si-MOSFETS [4] in the RIP phase did show a compressible state. Indeed, although the WS by itself is incompressible [7], the neutralizing oppositely charged gate makes the thermodynamic (! ! 0, q ! 0) compressibility of the Wigner solid finite and negative [8]. Conclusions on the incompressibility in Ref. [1] are made by considering the sample as two capacitors in series, 1=C ¼ 1=C0 þ 1=Cq , where C0 is the geometric one and Cq ¼ e2 dp=d (all per unit area), and is the chemical potential of the 2D system. The total capacitance was determined as C ¼ Iy =2fVx , with the recharging current I ¼ Ix þ iIy and the ac excitation Vx . The observed giant (factor of 2) decrease in capacitance [Fig. 1(b)] is 2 orders of magnitude higher than that typical for the RIP phase in Si-MOSFETs [4]; in Ref. [1] it was
FIG. 1 (color online). (a) Magnetocapacitance color plot from Ref. [1]. (b) Examples of magnetoresistance (lines) and magnetocapacitance (symbols) plots from Ref. [1].
0031-9007=13=110(24)=249701(2)
week ending 14 JUNE 2013
PHYSICAL REVIEW LETTERS
attributed to the drop in Cq value, i.e., to the formation of an incompressible state. However, the existence of finite area incompressible domains in the field-density phase diagram for the 2D system with charge-neutralizing background is unphysical. Indeed, had the inequality Cq & C0 been fulfilled in a finite density range, the difference in chemical potentials of the 2D layer between, e.g., points A and B (or C and R D) in Fig. 1(a), would have been equal to ðD C Þ ¼ ppDC e2 dp=Cq * e2 ðpD pC Þ=C0 20 meV, which is much higher than EF , @!c 0:3 meV, and the WS binding energy (EL ES & 2 eV for rs * 40 [9]). The enormous drop in measured capacitance Cmeas may be caused, e.g., by slow charging or discharging at a given frequency f, because the phases marked in Ref. [1] as ‘‘incompressible’’ [colored in violet in Fig. 1(a)] are very poorly conductive (xx ! 0). The true resistance value (1=xx , or R) in recharging processes is likely to be much higher than that found from transport measurements in Ref. [1] [see Fig. 1(b)], because for the zero-field insulating phase and RIP, the I-V characteristics [2,5] have a threshold character already in the fA range. In this case, the Cmeas value will be frequency dependent and the recharging current will acquire a phase shift. In case the Cmeas value is frequency independent, its enormous drop may result from reduction in the charging area, i.e., C0 value [10,11]; indeed, because of the inevitable fluctuations of donors, the 2D gas becomes inhomogeneous at ultralow densities [10] because nonlinear screening leads to the formation of fully depleted areas. This work is supported by RFBR and the Ministry of Education and Science, RF (Grant No. 8375). A. Yu. Kuntsevich1 and V. M. Pudalov1,2 1
P. N. Lebedev Physical Institute of the Russian Academy of Sciences Moscow 119991, Russia 2 Moscow Institute of Physics and Technology Dolgoprudny 141700, Russia Received 19 February 2013; published 13 June 2013 DOI: 10.1103/PhysRevLett.110.249701 PACS numbers: 71.30.+h, 73.40.c, 73.63.Hs [1] R. L. J. Qiu, X. P. A. Gao, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 108, 106404 (2012). [2] M. D’Iorio, V. M. Pudalov, and S. G. Semenchinsky, Phys. Lett. A 150, 422 (1990); Phys. Rev. B 46, 15 992 (1992). [3] M. R. Sakr, M. Rahimi, S. V. Kravchenko, P. T. Coleridge, R. Williams, and J. Lapointe, Phys. Rev. B 64, 161308 (2001). [4] S. V. Kravchenko, J. A. A. J. Perenboom, and V. M. Pudalov, Phys. Rev. B 44, 13 513 (1991). [5] V. M. Pudalov, M. D’Iorio, S. V. Kravchenko, and J. W. Campbell, Phys. Rev. Lett. 70, 1866 (1993); V. M. Pudalov and S. T. Chui, Phys. Rev. B 49, 14 062 (1994). [6] V. M. Pudalov, in Physics of the Electron Solid, edited by S.-T. Chui (International Press, Cambridge, MA, 1994), Chap. 4.
249701-1
Ó 2013 American Physical Society
PRL 110, 249701 (2013)
PHYSICAL REVIEW LETTERS
[7] R. Chitra, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 65, 035312 (2001). [8] R. Chitra and T. Giamarchi, Eur. Phys. J. B 44, 455 (2005). [9] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).
week ending 14 JUNE 2013
[10] M. M. Fogler, Phys. Rev. B 69, 121409(R) (2004). [11] G. Allison, E. A. Galaktionov, A. K. Savchenko, S. S. Safonov, M. M. Fogler, M. Y. Simmons, and D. A. Ritchie, Phys. Rev. Lett. 96, 216407 (2006).
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