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Mobility of a 2D Wigner crystal over liquid 4He and 3He – 4He solution
Physica B 249—251 (1998) 648—651
Mobility of a 2D Wigner crystal over liquid 4He and 3He — 4He solution V.V. Dotsenko*, V.E. Sivokon, Yu.Z. Kovdrya, V.N. Grigor’ev B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin avenue, Kharkov, Ukraine
Abstract Spectrum of the coupled electron—ripplon oscillations in the Wigner crystal over liquid helium with surface electron density 1.3]109 cm~2 in the temperature range 70—600 mK is studied. The experimental data of resonance width are shown not to be explained by capillary waves attenuation. The consideration is performed using both volume and surface hydrodynamics. A temperature dependence of mobility of the crystal is obtained. Influence of 3He impurities with concentration 0.25% on frequencies and width of the coupled electron—ripplon resonances is studied. It is found that the presence of the 3He impurity causes a noticeable widening of the resonances and decreases the resonant frequencies. The frequency decrease is explained to be due to the change in the surface tension of liquid. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Liquid helium; 3He—4He solution; Wigner solid
Electrons localized over the liquid helium surface form a two-dimensional (2D) electron system of unique purity and homogeneity. Electrons moving along liquid helium interact in the main with atoms of helium vapour at temperatures above 0.8 K and with quantized capillary waves (ripplons) at lower temperatures. One of the most interesting effects found in the 2D electron system over the liquid helium is the formation of the Wigner solid (WS). The WS was observed for the first time by Grimes and Adams [1] who observed resonant peculiarities in a response of the electron layer to an RF signal. These * Corresponding author. Fax: #380-572-322370; e-mail: [email protected].
peculiarities were explained by the excitation of capillary waves with wave vectors corresponding to ones of the reciprocal lattice [2] and by the appearance of coupled phonon—ripplon (CPR) oscillations in the electron system [3]. Width of the CPR resonances was measured in a wide temperature range [4]. It was found to increase as temperature increases. But there is still no explanation for the dependencies obtained. Meanwhile investigation of this phenomena is of great interest because it can give information about the interaction of the electron crystal with the superfluid helium surface. In the present paper spectra of the coupled phonon—ripplon oscillations in the Wigner solid over liquid 4He and 3He—4He solutions have been
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 2 8 0 - 4
V.V. Dotsenko et al. / Physica B 249—251 (1998) 648—651
studied experimentally in the temperature range 70—600 mK and at electron density n "1.3] s 109 cm~2. Measurements were carried out in the frequency region 1—20 MHz using the experimental cell of Corbino geometry, described in detail in Ref. [5]. Typical frequency dependencies of amplitude of the cell response to the RF signal for pure 4He(1) and 0.25% solution 3He in 4He(2) at a temperature of 70 mK are shown in Fig. 1. Two resonant peaks can be easily seen for each of the experimental series. The peaks correspond to two harmonics of the phonon—ripplon standing waves in the experimental cell. One can see from Fig. 1 that the presence of 3He leads to decreasing frequencies of the resonances and widening of the resonance. It is necessary to note that data for the more weak solution, 0.025% of 3He, coincides with data for 4He within experimental error. The temperature dependence of resonant widths *f are shown in Fig. 2 for both pure 4He and 3He—4He solutions. The dependence are almost linear at temperatures lower than 300 mK. To analyse the data obtained we use the formula for conductance of the experimental cell G [6].
649
Fig. 1. Typical frequency dependence of amplitude response for the 4He (n) and 3He—4He (v) solutions.
= Re(G)"n e2 + A n s n/1 n e2u2s s 1 ] , (mu2 !n e2us )2 # (n e2us )2 p s 2 s 1
A
= Im(G)"u n e2 + A n s n/1
B
mu2 ! n e2us p s 2 ] #g , 0 (mu2!n e2us )2#(n e2us )2 p s 2 s 1 (1) where s "Re p~1, s "Im p~1, p is the conduct1 2 ivity of the electron layer, u are the resonant p frequencies of plasma oscillations, A are geometry n dependent coefficients, g is the conductance of the 0 experimental cell in the absence of electrons. The value s describes dynamic properties of WS, the 2 value s is determined by dissipation processes. To 1 find s and s we use the dimensionless function 1 2
Fig. 2. Resonant peak width as a function of temperature for the 4He (n) and 3He—4He (v) solutions.
Z obtained in Ref. [7] u2(u2!u2#4c2) = l !1, l Re(Z)" + C l l (u2!u2)2#4u2c2 l l l/1 u4 = 2c l Im(Z)" + C l , l u (u2!u2)2#4u2c2 l l l/1
(2)
650
V.V. Dotsenko et al. / Physica B 249—251 (1998) 648—651
here C are coefficients of electron—ripplon coupl ling, u are the resonant frequencies of the capillary l waves corresponding to reciprocal vectors of electron lattice, c are coefficients of attenuation of the l capillary waves with frequency u . Resistivity of the l electron layer can be written as mu s " Im Z , 1 ne2 1
mu s "! Re Z . 2 1 ne2
(3)
The amplitude response is proportional to conductance of the experimental cell DGD"J(Re G)2#(Im G)2.
(4)
These expressions enable us to compare the existing theory with our experimental data. In the present work the value DGD is calculated numerically as a function of frequency using Eqs. (1)—(4). The resonant peculiarities follow from these and resonant frequencies are found to be in good agreement with experimental ones. Decreasing of the resonant frequency for 3He—4He solution in comparison with pure 4He can be explained as a result of the change in the surface tension a. We use the value of a interpolated from data [8]. The resonant peak width is determined mainly by s , which is determined by the capillary wave 1 attenuation c [7]. Value of c is calculated by l 1 extrapolating data of Ref. [9] to wave vectors g&105 cm~1 which is typical for our experiments. Using these values of c , we obtain on calculation 1 the resonant peak width *f+8.5 kHz. That is two orders less than the experimental value. Eq. (2) was obtained in the framework of volume hydrodynamics. However, at temperatures low enough, capillary wave attenuation can be determined by surface hydrodynamics [10]. Calculations using surface hydrodynamics [7] and c from Ref. [9] lead to 1 peak width *f also much less than the experimental values. Thus neither volume nor surface hydrodynamics describe the experimental data. It enables us to conclude that capillary wave attenuation does not determine energy losses when the Wigner crystal moves along the liquid helium surface. The resonant width of coupled phonon—ripplon oscillations can be attributed to limited mobility of
the electron crystal determined by the relation s "(n ek)~1. (5) 1 s We calculate the dependence DG(u)D taking s from 2 Eq. (3) and s from Eq. (5) and fit k so that the 1 calculated resonant width coincides with the experimental one. Mobilities obtained in such a way for the Wigner crystal over 4He and 3He—4He solutions are plotted versus temperature in Fig. 3. Mobility can be seen to increase as temperature decreases below 300 mK for both pure 4He and the solution. Impurity of 3He noticeably decreases mobility. In the same figure the mobility of WS with electron density n "9.5]108 cm~2 is shown for s pure 4He [11]. Our data are in an agreement with data from Ref. [11]. So in the present paper, width and frequencies of the coupled phonon—ripplon oscillations have been investigated. Numerical analysis of the experimental peak width shows capillary wave attenuation not to be the main reason for energy losses in phonon—ripplon oscillations. Mobility of the Wigner cyrstal over superfluid 3He—4He solutions has been measured by studying the phonon—ripplon spectrum of the electron solid.
Fig. 3. Mobility for the Wigner solid over the 4He (n) and 3He—4He (v) solutions.
V.V. Dotsenko et al. / Physica B 249—251 (1998) 648—651
References [1] C.C. Grimes, G. Adams, Phys. Rev. Lett. 42 (1979) 795. [2] V.B. Shikin, JETP Lett. 19 (1974) 335. [3] D.S. Fisher, B.I. Halperin, P.M. Platzman, Phys. Rev. Lett. 42 (1979) 798. [4] V.E. Sivokon´, V.V. Dotsenko, Yu.Z. Kovdrya, V.N. Grigorev, Low. Temp. Phys. 22 (1996) 845. [5] V.E. Sivokon´, V.V. Dotsenko, S.S. Sokolov, Yu.Z. Kovdrya, V.N. Grigorev, Low. Temp. Phys. 22 (1996) 549.
651
[6] Yu.Z. Kovdrya, V.A. Nikolaenko, O.I. Kirichek, S.S. Sokolov, V.N. Grigore´v, J. Low Temp. Phys. 91 (1993) 371. [7] Yu.P. Monarkha, Sov. JLTP 6 (1980) 331. [8] D.O. Edvards, W.F. Saam, Progress in Low Temp. Phys. VIIa (1978) 283. [9] P. Roche, G. Deville, K.O. Keshishev, N.J. Appleyard, F.I.B. Williams, Phys. Rev. Lett. 75 (1995) 3316. [10] A.F. Andreev, D.A. Kompaneets, Sov. Phys. JETP 34 (1972) 1316. [11] R. Mehrotra, C.J. Guo, Y.Z. Ruan, D.B. Mast, A.J. Dahm, Phys. Rev. Lett. 29 (1984) 5239.
Mobility of a 2D Wigner crystal over liquid 4He and 3He — 4He solution V.V. Dotsenko*, V.E. Sivokon, Yu.Z. Kovdrya, V.N. Grigor’ev B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin avenue, Kharkov, Ukraine
Abstract Spectrum of the coupled electron—ripplon oscillations in the Wigner crystal over liquid helium with surface electron density 1.3]109 cm~2 in the temperature range 70—600 mK is studied. The experimental data of resonance width are shown not to be explained by capillary waves attenuation. The consideration is performed using both volume and surface hydrodynamics. A temperature dependence of mobility of the crystal is obtained. Influence of 3He impurities with concentration 0.25% on frequencies and width of the coupled electron—ripplon resonances is studied. It is found that the presence of the 3He impurity causes a noticeable widening of the resonances and decreases the resonant frequencies. The frequency decrease is explained to be due to the change in the surface tension of liquid. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Liquid helium; 3He—4He solution; Wigner solid
Electrons localized over the liquid helium surface form a two-dimensional (2D) electron system of unique purity and homogeneity. Electrons moving along liquid helium interact in the main with atoms of helium vapour at temperatures above 0.8 K and with quantized capillary waves (ripplons) at lower temperatures. One of the most interesting effects found in the 2D electron system over the liquid helium is the formation of the Wigner solid (WS). The WS was observed for the first time by Grimes and Adams [1] who observed resonant peculiarities in a response of the electron layer to an RF signal. These * Corresponding author. Fax: #380-572-322370; e-mail: [email protected].
peculiarities were explained by the excitation of capillary waves with wave vectors corresponding to ones of the reciprocal lattice [2] and by the appearance of coupled phonon—ripplon (CPR) oscillations in the electron system [3]. Width of the CPR resonances was measured in a wide temperature range [4]. It was found to increase as temperature increases. But there is still no explanation for the dependencies obtained. Meanwhile investigation of this phenomena is of great interest because it can give information about the interaction of the electron crystal with the superfluid helium surface. In the present paper spectra of the coupled phonon—ripplon oscillations in the Wigner solid over liquid 4He and 3He—4He solutions have been
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 2 8 0 - 4
V.V. Dotsenko et al. / Physica B 249—251 (1998) 648—651
studied experimentally in the temperature range 70—600 mK and at electron density n "1.3] s 109 cm~2. Measurements were carried out in the frequency region 1—20 MHz using the experimental cell of Corbino geometry, described in detail in Ref. [5]. Typical frequency dependencies of amplitude of the cell response to the RF signal for pure 4He(1) and 0.25% solution 3He in 4He(2) at a temperature of 70 mK are shown in Fig. 1. Two resonant peaks can be easily seen for each of the experimental series. The peaks correspond to two harmonics of the phonon—ripplon standing waves in the experimental cell. One can see from Fig. 1 that the presence of 3He leads to decreasing frequencies of the resonances and widening of the resonance. It is necessary to note that data for the more weak solution, 0.025% of 3He, coincides with data for 4He within experimental error. The temperature dependence of resonant widths *f are shown in Fig. 2 for both pure 4He and 3He—4He solutions. The dependence are almost linear at temperatures lower than 300 mK. To analyse the data obtained we use the formula for conductance of the experimental cell G [6].
649
Fig. 1. Typical frequency dependence of amplitude response for the 4He (n) and 3He—4He (v) solutions.
= Re(G)"n e2 + A n s n/1 n e2u2s s 1 ] , (mu2 !n e2us )2 # (n e2us )2 p s 2 s 1
A
= Im(G)"u n e2 + A n s n/1
B
mu2 ! n e2us p s 2 ] #g , 0 (mu2!n e2us )2#(n e2us )2 p s 2 s 1 (1) where s "Re p~1, s "Im p~1, p is the conduct1 2 ivity of the electron layer, u are the resonant p frequencies of plasma oscillations, A are geometry n dependent coefficients, g is the conductance of the 0 experimental cell in the absence of electrons. The value s describes dynamic properties of WS, the 2 value s is determined by dissipation processes. To 1 find s and s we use the dimensionless function 1 2
Fig. 2. Resonant peak width as a function of temperature for the 4He (n) and 3He—4He (v) solutions.
Z obtained in Ref. [7] u2(u2!u2#4c2) = l !1, l Re(Z)" + C l l (u2!u2)2#4u2c2 l l l/1 u4 = 2c l Im(Z)" + C l , l u (u2!u2)2#4u2c2 l l l/1
(2)
650
V.V. Dotsenko et al. / Physica B 249—251 (1998) 648—651
here C are coefficients of electron—ripplon coupl ling, u are the resonant frequencies of the capillary l waves corresponding to reciprocal vectors of electron lattice, c are coefficients of attenuation of the l capillary waves with frequency u . Resistivity of the l electron layer can be written as mu s " Im Z , 1 ne2 1
mu s "! Re Z . 2 1 ne2
(3)
The amplitude response is proportional to conductance of the experimental cell DGD"J(Re G)2#(Im G)2.
(4)
These expressions enable us to compare the existing theory with our experimental data. In the present work the value DGD is calculated numerically as a function of frequency using Eqs. (1)—(4). The resonant peculiarities follow from these and resonant frequencies are found to be in good agreement with experimental ones. Decreasing of the resonant frequency for 3He—4He solution in comparison with pure 4He can be explained as a result of the change in the surface tension a. We use the value of a interpolated from data [8]. The resonant peak width is determined mainly by s , which is determined by the capillary wave 1 attenuation c [7]. Value of c is calculated by l 1 extrapolating data of Ref. [9] to wave vectors g&105 cm~1 which is typical for our experiments. Using these values of c , we obtain on calculation 1 the resonant peak width *f+8.5 kHz. That is two orders less than the experimental value. Eq. (2) was obtained in the framework of volume hydrodynamics. However, at temperatures low enough, capillary wave attenuation can be determined by surface hydrodynamics [10]. Calculations using surface hydrodynamics [7] and c from Ref. [9] lead to 1 peak width *f also much less than the experimental values. Thus neither volume nor surface hydrodynamics describe the experimental data. It enables us to conclude that capillary wave attenuation does not determine energy losses when the Wigner crystal moves along the liquid helium surface. The resonant width of coupled phonon—ripplon oscillations can be attributed to limited mobility of
the electron crystal determined by the relation s "(n ek)~1. (5) 1 s We calculate the dependence DG(u)D taking s from 2 Eq. (3) and s from Eq. (5) and fit k so that the 1 calculated resonant width coincides with the experimental one. Mobilities obtained in such a way for the Wigner crystal over 4He and 3He—4He solutions are plotted versus temperature in Fig. 3. Mobility can be seen to increase as temperature decreases below 300 mK for both pure 4He and the solution. Impurity of 3He noticeably decreases mobility. In the same figure the mobility of WS with electron density n "9.5]108 cm~2 is shown for s pure 4He [11]. Our data are in an agreement with data from Ref. [11]. So in the present paper, width and frequencies of the coupled phonon—ripplon oscillations have been investigated. Numerical analysis of the experimental peak width shows capillary wave attenuation not to be the main reason for energy losses in phonon—ripplon oscillations. Mobility of the Wigner cyrstal over superfluid 3He—4He solutions has been measured by studying the phonon—ripplon spectrum of the electron solid.
Fig. 3. Mobility for the Wigner solid over the 4He (n) and 3He—4He (v) solutions.
V.V. Dotsenko et al. / Physica B 249—251 (1998) 648—651
References [1] C.C. Grimes, G. Adams, Phys. Rev. Lett. 42 (1979) 795. [2] V.B. Shikin, JETP Lett. 19 (1974) 335. [3] D.S. Fisher, B.I. Halperin, P.M. Platzman, Phys. Rev. Lett. 42 (1979) 798. [4] V.E. Sivokon´, V.V. Dotsenko, Yu.Z. Kovdrya, V.N. Grigorev, Low. Temp. Phys. 22 (1996) 845. [5] V.E. Sivokon´, V.V. Dotsenko, S.S. Sokolov, Yu.Z. Kovdrya, V.N. Grigorev, Low. Temp. Phys. 22 (1996) 549.
651
[6] Yu.Z. Kovdrya, V.A. Nikolaenko, O.I. Kirichek, S.S. Sokolov, V.N. Grigore´v, J. Low Temp. Phys. 91 (1993) 371. [7] Yu.P. Monarkha, Sov. JLTP 6 (1980) 331. [8] D.O. Edvards, W.F. Saam, Progress in Low Temp. Phys. VIIa (1978) 283. [9] P. Roche, G. Deville, K.O. Keshishev, N.J. Appleyard, F.I.B. Williams, Phys. Rev. Lett. 75 (1995) 3316. [10] A.F. Andreev, D.A. Kompaneets, Sov. Phys. JETP 34 (1972) 1316. [11] R. Mehrotra, C.J. Guo, Y.Z. Ruan, D.B. Mast, A.J. Dahm, Phys. Rev. Lett. 29 (1984) 5239.