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Pulse Fourier transform method to study coupled plasmon-ripplon mode
Surface Science 170 (1986) 75-79 North-Holland, Amsterdam
75
P U L S E FOURIER T R A N S F O R M M E T H O D T O STUDY C O U P L E D PLASMON-RIPPLON MODE Kimitoshi KONO Hyogo University of Teacher Education, 942-1 Shimokume, Yashiro, Hyogo 673-14, Japan Received 18 July 1985; accepted for publication 13 September 1985
We have developed a new method to study the dynamic properties of a 2D Wigner crystal formed on liquid helium-4. Our new method utilizes the Fourier Transform (FT) of the pulse response of the electron sheet to obtain the frequency response function of the sheet, With this method we have succeeded in observing the coupled plasmon-ripplon mode resonance. The temperature dependences of the resonance position, width and amplitude are presented.
1. Introduction It is well known that two-dimensional (2D) electrons formed on liquid helium are ideal for studying the properties of the 2D electron crystal. Grimes and Adams (GA) [1,2] first succeeded in observing the RF resonance absorption due to the electron crystal formed on liquid helium. Fisher, Halperin and Platzman (FHP) [3] assigned those absorption peaks to the coupled plasmon-ripplon modes. After this experiment by GA, many experiments have been performed [4-9] so far to elucidate the properties of the crystal and the liquid to crystal phase transition. In this work, we have developed a new method to extend the GA experiment, in which a fast signal averager and microcomputer are used instead of a swept-frequency RF spectrometer. With the ability of the data analysis using a microcomputer, the present experimental system is powerful for the study of the line shape of the resonance.
2. Experimental and results Fig. 1A shows the schematic diagram of our apparatus. A pair of two concentric electrodes are placed parallel to each other and separated by 5 mm. The diameter of the upper inner electrode is 10 mm and that of the outer electrode is 25 mm. Those of the lower inner and outer electrodes are 12 and 20 mm, respectively. The separation gaps between the inner and outer elec0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
K. Kono / Pulse FT method to study coupled plasmon-ripplon mode
76
(R)
(B) T=O.513K n =9.5xl@8cm -d
33
........
! ,4,
]
!
S
QZ v
Z 0 H
8_ rr 0 U] m GZ
L.
.
.
.
- L
.
.
.
.
I
0
@
6
12 FREQUENCY
18
24
38
(MHz)
Fig. 1. (A) Schematic drawings of electrode assembly and electronics. Inside of broken line denotes the low temperature part. (B) Typical trace of power absorption oJ lrn X(~).
trodes are about 0.3 mm. This ratio of the electrode diameters is selected to achieve a maximum coupling to the basic modes of k = 3.83 cm ~. where k is the wavenumber of the coupled p l a s m o n - r i p p l o n modes (3). Electrons are supplied by briefly heating a thin Th W filament while the lower electrodes are kept at a positive bias voltage V,,. When the exciting pulse voltage V is applied to the upper inner electrode, the 2D electron sheet begins the radial motion. This motion causes a variation of the potential of the lower inner electrode, which is proportional to the number of electrons N ( t ) located just above the inner electrode. The signal is first amplified by a F E T source follower set close to the electrode assembly. It is then amplified by a wide-band pulse amplifier, digitally averaged and then transferred to a microcomputer to calculate the Fourier Transforms. The surface electron density was determined by using the relation n~ = CK,,, where C is the areal capacitance between the electron sheet and the lower electrode. The value of C is estimated from the depth of the liquid helium, which we can measure with an accuracy of 10%. After checking the validity of this relation at low V,, below 20 V, where we can measure n, directly by integrating the escape current [10], we have applied it for V,,, - 400 V. which is used in the experiment. The Fourier transform X(~0) of N(t) is obtained by the following process. The signal from the apparatus schematically drawn in fig. 1A includes not only the signal from the surface electrons, but also that from the electrode assembly
K. Kono / Pulse FT method to study coupledplasmon-ripplon mode
77
and a cross-talk from the other part of the experimental system. We m a y reasonably suppose the F T of the signal as
E(co) F(co)-aE(co) F(o~) X(co)+E(co) C(co),
a>O,
(I)
where E(co) is the FT of Vex, F(co) the frequency response function of the electrode assembly, and C(co) the response function of the other part which contributes to the cross-talk. On the other hand, when there are no electrons on the liquid helium, we obtain E(co) F(co)+ E(co) C(co).
(2)
Lastly, if we turn off the powers to the F E T follower, we obtain E(co) C(co).
(3)
C o m b i n i n g eqs. (1)-(3), we obtain aX(¢0) = [eq.(1) - e q . ( 2 ) ] / [ e q . ( 2 ) - eq.(3)]. Since we have used an exciting pulse with a width of 25 ns, E(co) has zeros at multiples of 40 MHz. Therefore, X(co) around 40 M H z contained a considerable noise. But in the frequency range below 30 MHz, we obtained a good signal to noise ratio. Fig. 1B shows a typical trace of co Im X(co) which is proportional to the power absorption. T w o absorption peaks are clearly observed which are assigned to the coupled p l a s m o n - r i p p l o n resonance. T o determine the positions, amplitudes and widths of the absorption peaks, a least square fitting was employed with the fitting formula /z(X(co)) = Z o + Z, co +
C, Co + co2 _ co2 + 2i~0co co~ _ w2 + 2i?~1c o '
(4)
where the parameters C o, coo, ?~o, C1, col and ?~, were assumed to be positive and real, and coo < col; Zo and Z, are complex numbers which were introduced to fit unexpected offsets and base line drifts. The last two terms of eq. (4) represent Lorentzian type line shapes with the resonance frequencies coo and co, and the d a m p i n g coefficients Xo and ~,. These parameters were determined to minimize the sum of the squares ~lx(co)-~(X(co))l
2.
¢o
In fig. 2, the temperature dependences of the resonance position COo, width )% and amplitude of the absorption peak C0/2?~ 0 are pl(,tt~d.
3. Discussion As shown in fig. 2, we see that the resonance position, width and amplitude of the absorption peak depend linearly on temperature at lower temperatures.
K. Kono / Pulse F T method to study coupled plasmon-ripplon mode
78
25
N
20
3: z:
15 u Z
L~
10
OO
mlm
o w
X
XX X
L~
X •
•
¢I
•
%x
X
mm
x
,4
.5
.6 TEMPERRTURE
x
i
i
.7
.8
x
.9
(K)
Fig. 2. Temperature dependence of the peak position, widths and amplitudes of the absorption peaks for electrons with n s = 9.5 × 10 s cm -2. Solid circles are the positions, solid square the widths and crosses the amplitudes C0/2X o. Scale for the amplitudes is arbitrary.
They have kinks at T1 - 0 . 7 1 5 K, then abruptly increase or decrease as temperature increases until the signal diminishes at T2 < 0.75 K. The value of 12 ( F = at T2 is estimated to be 121, which agrees with that obtained at other institutes as the critical value of F for the liquid to crystal phase transition. If we try T1, we get F~ = 127. The temperature dependence of the resonance position agrees qualitatively with the experiment of G A [1,2] and theories by F H P [3] and Namaizawa [11]. However, the calculated peak position using eq. (7) of ref. [3] is found to be 50% higher than the value which we obtained at the lowest temperature, where the Debye-Waller factor W 1 of 0.3 is used according to the ref. [11]. If we neglect this discrepancy, the observed resonances shown in fig. 1B can be assigned to W and X of those GA observed [1], namely the coupled plasmon-ripplon mode resonances. It is interesting that there appeared something like kinks at T~ in the temperature dependence of the resonance position. Morf [12] gave the temperature dependence of the shear modulus, supposing that the melting transition was a Kosterlitz-Thouless [13] type. According to his calculation, the shear modules decrease linearly with increasing temperature until it reaches 0.8 of the melting temperature, where it begins to decrease more rapidly. We can reasonably expect that near the transition temperature, the temperature dependence of the resonance position has the same type anomaly as that of the shear modulus. If this is the case, the kink will not appear in the temperature dependence of the resonance position, because the shear modulus is a smooth function of T in this region. Therefore, we are interested in this temperature
e2~,/kBT)
K. Kono / Pulse FT method to study coupled plasmon-ripplon mode
79
d e p e n d e n c e of the resonance position. A more detailed experiment a n d analysis are needed to elucidate this point. Lastly, we describe other qualitative features of this resonance. (1) At the lowest temperatures, the line shapes deviate from the Lorentzian. As a result, the peak frequency of the fitted curve is a b o u t 10% lower than that of the raw data. At present, we do n o t have a n y e x p l a n a t i o n for this. (2) W h e n Vex exceeds a b o u t 0.2 V, the coupled p l a s m o n - r i p p l o n resonance disappears. It m a y be attributed to the fact that the electron velocities exceed the phase velocities of ripplons [3].
Acknowledgements I thank Professors K. Kajita a n d W. Sasaki for interesting discussions a n d critical reading of the manuscript. I thank Professor S. K o b a y a s h i for stimulating discussions. I also t h a n k Professor G. K u w a b a r a for c o n t i n u o u s encouragem e n t a n d valuable discussions. This work was supported in part b y a G r a n t - i n Aid for Scientific Research from the Ministory of Education.
References [1] [2] [3] [4] [5] [6] [7]
C.C. Grimes and G. Adams, Phys. Rev. Letters 42 (1979) 795. C.C. Grimes and G. Adams, Surface Sci. 98 (1980) 1. D.S. Fisher, B.I. Halperin and P.M. Platzman, Phys. Rev. Letters 42 (1979) 798. D. Marty, J. Poitrenaud and F.I.B. Williams, J. Physique Lettres 41 (1980) L311. A.S. Rybalko, B.N. Esel'son and Yu.Z. Kovdrya, Soviet J. Low Temp. Phys. 5 (1979) 450. R. Mehrotra, B.M. Guenin and A.J. Dahm, Phys. Rev. Letters 48 (1982) 641. D.B. Mast, C.J. Guo, M.A. Stan, R. Mehrotra, Y.Z. Ruan and A.J. Dahm, Surface Sci. 142 (1984) 100. [8] F. Gallet, G. Deville, A. Valdes and F.I.B. Williams, Phys. Rev. Letters 49 (1982) 212. [9] G. Deville, A. Valdes, E.Y. Andrei and F.I.B. Williams, Phys. Rev. Letters 53 (1984) 588. [10] K. Kono, K. Kajita, S. Kobayashi and W. Sasaki, J. Low Temp. Phys. 46 (1982) 195. [11] H. Namaizawa, Solid State Commun. 34 (1980) 607. [12] R.H. Morf, Phys. Rev. Letters 43 (1979) 931. [13] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181.
75
P U L S E FOURIER T R A N S F O R M M E T H O D T O STUDY C O U P L E D PLASMON-RIPPLON MODE Kimitoshi KONO Hyogo University of Teacher Education, 942-1 Shimokume, Yashiro, Hyogo 673-14, Japan Received 18 July 1985; accepted for publication 13 September 1985
We have developed a new method to study the dynamic properties of a 2D Wigner crystal formed on liquid helium-4. Our new method utilizes the Fourier Transform (FT) of the pulse response of the electron sheet to obtain the frequency response function of the sheet, With this method we have succeeded in observing the coupled plasmon-ripplon mode resonance. The temperature dependences of the resonance position, width and amplitude are presented.
1. Introduction It is well known that two-dimensional (2D) electrons formed on liquid helium are ideal for studying the properties of the 2D electron crystal. Grimes and Adams (GA) [1,2] first succeeded in observing the RF resonance absorption due to the electron crystal formed on liquid helium. Fisher, Halperin and Platzman (FHP) [3] assigned those absorption peaks to the coupled plasmon-ripplon modes. After this experiment by GA, many experiments have been performed [4-9] so far to elucidate the properties of the crystal and the liquid to crystal phase transition. In this work, we have developed a new method to extend the GA experiment, in which a fast signal averager and microcomputer are used instead of a swept-frequency RF spectrometer. With the ability of the data analysis using a microcomputer, the present experimental system is powerful for the study of the line shape of the resonance.
2. Experimental and results Fig. 1A shows the schematic diagram of our apparatus. A pair of two concentric electrodes are placed parallel to each other and separated by 5 mm. The diameter of the upper inner electrode is 10 mm and that of the outer electrode is 25 mm. Those of the lower inner and outer electrodes are 12 and 20 mm, respectively. The separation gaps between the inner and outer elec0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
K. Kono / Pulse FT method to study coupled plasmon-ripplon mode
76
(R)
(B) T=O.513K n =9.5xl@8cm -d
33
........
! ,4,
]
!
S
QZ v
Z 0 H
8_ rr 0 U] m GZ
L.
.
.
.
- L
.
.
.
.
I
0
@
6
12 FREQUENCY
18
24
38
(MHz)
Fig. 1. (A) Schematic drawings of electrode assembly and electronics. Inside of broken line denotes the low temperature part. (B) Typical trace of power absorption oJ lrn X(~).
trodes are about 0.3 mm. This ratio of the electrode diameters is selected to achieve a maximum coupling to the basic modes of k = 3.83 cm ~. where k is the wavenumber of the coupled p l a s m o n - r i p p l o n modes (3). Electrons are supplied by briefly heating a thin Th W filament while the lower electrodes are kept at a positive bias voltage V,,. When the exciting pulse voltage V is applied to the upper inner electrode, the 2D electron sheet begins the radial motion. This motion causes a variation of the potential of the lower inner electrode, which is proportional to the number of electrons N ( t ) located just above the inner electrode. The signal is first amplified by a F E T source follower set close to the electrode assembly. It is then amplified by a wide-band pulse amplifier, digitally averaged and then transferred to a microcomputer to calculate the Fourier Transforms. The surface electron density was determined by using the relation n~ = CK,,, where C is the areal capacitance between the electron sheet and the lower electrode. The value of C is estimated from the depth of the liquid helium, which we can measure with an accuracy of 10%. After checking the validity of this relation at low V,, below 20 V, where we can measure n, directly by integrating the escape current [10], we have applied it for V,,, - 400 V. which is used in the experiment. The Fourier transform X(~0) of N(t) is obtained by the following process. The signal from the apparatus schematically drawn in fig. 1A includes not only the signal from the surface electrons, but also that from the electrode assembly
K. Kono / Pulse FT method to study coupledplasmon-ripplon mode
77
and a cross-talk from the other part of the experimental system. We m a y reasonably suppose the F T of the signal as
E(co) F(co)-aE(co) F(o~) X(co)+E(co) C(co),
a>O,
(I)
where E(co) is the FT of Vex, F(co) the frequency response function of the electrode assembly, and C(co) the response function of the other part which contributes to the cross-talk. On the other hand, when there are no electrons on the liquid helium, we obtain E(co) F(co)+ E(co) C(co).
(2)
Lastly, if we turn off the powers to the F E T follower, we obtain E(co) C(co).
(3)
C o m b i n i n g eqs. (1)-(3), we obtain aX(¢0) = [eq.(1) - e q . ( 2 ) ] / [ e q . ( 2 ) - eq.(3)]. Since we have used an exciting pulse with a width of 25 ns, E(co) has zeros at multiples of 40 MHz. Therefore, X(co) around 40 M H z contained a considerable noise. But in the frequency range below 30 MHz, we obtained a good signal to noise ratio. Fig. 1B shows a typical trace of co Im X(co) which is proportional to the power absorption. T w o absorption peaks are clearly observed which are assigned to the coupled p l a s m o n - r i p p l o n resonance. T o determine the positions, amplitudes and widths of the absorption peaks, a least square fitting was employed with the fitting formula /z(X(co)) = Z o + Z, co +
C, Co + co2 _ co2 + 2i~0co co~ _ w2 + 2i?~1c o '
(4)
where the parameters C o, coo, ?~o, C1, col and ?~, were assumed to be positive and real, and coo < col; Zo and Z, are complex numbers which were introduced to fit unexpected offsets and base line drifts. The last two terms of eq. (4) represent Lorentzian type line shapes with the resonance frequencies coo and co, and the d a m p i n g coefficients Xo and ~,. These parameters were determined to minimize the sum of the squares ~lx(co)-~(X(co))l
2.
¢o
In fig. 2, the temperature dependences of the resonance position COo, width )% and amplitude of the absorption peak C0/2?~ 0 are pl(,tt~d.
3. Discussion As shown in fig. 2, we see that the resonance position, width and amplitude of the absorption peak depend linearly on temperature at lower temperatures.
K. Kono / Pulse F T method to study coupled plasmon-ripplon mode
78
25
N
20
3: z:
15 u Z
L~
10
OO
mlm
o w
X
XX X
L~
X •
•
¢I
•
%x
X
mm
x
,4
.5
.6 TEMPERRTURE
x
i
i
.7
.8
x
.9
(K)
Fig. 2. Temperature dependence of the peak position, widths and amplitudes of the absorption peaks for electrons with n s = 9.5 × 10 s cm -2. Solid circles are the positions, solid square the widths and crosses the amplitudes C0/2X o. Scale for the amplitudes is arbitrary.
They have kinks at T1 - 0 . 7 1 5 K, then abruptly increase or decrease as temperature increases until the signal diminishes at T2 < 0.75 K. The value of 12 ( F = at T2 is estimated to be 121, which agrees with that obtained at other institutes as the critical value of F for the liquid to crystal phase transition. If we try T1, we get F~ = 127. The temperature dependence of the resonance position agrees qualitatively with the experiment of G A [1,2] and theories by F H P [3] and Namaizawa [11]. However, the calculated peak position using eq. (7) of ref. [3] is found to be 50% higher than the value which we obtained at the lowest temperature, where the Debye-Waller factor W 1 of 0.3 is used according to the ref. [11]. If we neglect this discrepancy, the observed resonances shown in fig. 1B can be assigned to W and X of those GA observed [1], namely the coupled plasmon-ripplon mode resonances. It is interesting that there appeared something like kinks at T~ in the temperature dependence of the resonance position. Morf [12] gave the temperature dependence of the shear modulus, supposing that the melting transition was a Kosterlitz-Thouless [13] type. According to his calculation, the shear modules decrease linearly with increasing temperature until it reaches 0.8 of the melting temperature, where it begins to decrease more rapidly. We can reasonably expect that near the transition temperature, the temperature dependence of the resonance position has the same type anomaly as that of the shear modulus. If this is the case, the kink will not appear in the temperature dependence of the resonance position, because the shear modulus is a smooth function of T in this region. Therefore, we are interested in this temperature
e2~,/kBT)
K. Kono / Pulse FT method to study coupled plasmon-ripplon mode
79
d e p e n d e n c e of the resonance position. A more detailed experiment a n d analysis are needed to elucidate this point. Lastly, we describe other qualitative features of this resonance. (1) At the lowest temperatures, the line shapes deviate from the Lorentzian. As a result, the peak frequency of the fitted curve is a b o u t 10% lower than that of the raw data. At present, we do n o t have a n y e x p l a n a t i o n for this. (2) W h e n Vex exceeds a b o u t 0.2 V, the coupled p l a s m o n - r i p p l o n resonance disappears. It m a y be attributed to the fact that the electron velocities exceed the phase velocities of ripplons [3].
Acknowledgements I thank Professors K. Kajita a n d W. Sasaki for interesting discussions a n d critical reading of the manuscript. I thank Professor S. K o b a y a s h i for stimulating discussions. I also t h a n k Professor G. K u w a b a r a for c o n t i n u o u s encouragem e n t a n d valuable discussions. This work was supported in part b y a G r a n t - i n Aid for Scientific Research from the Ministory of Education.
References [1] [2] [3] [4] [5] [6] [7]
C.C. Grimes and G. Adams, Phys. Rev. Letters 42 (1979) 795. C.C. Grimes and G. Adams, Surface Sci. 98 (1980) 1. D.S. Fisher, B.I. Halperin and P.M. Platzman, Phys. Rev. Letters 42 (1979) 798. D. Marty, J. Poitrenaud and F.I.B. Williams, J. Physique Lettres 41 (1980) L311. A.S. Rybalko, B.N. Esel'son and Yu.Z. Kovdrya, Soviet J. Low Temp. Phys. 5 (1979) 450. R. Mehrotra, B.M. Guenin and A.J. Dahm, Phys. Rev. Letters 48 (1982) 641. D.B. Mast, C.J. Guo, M.A. Stan, R. Mehrotra, Y.Z. Ruan and A.J. Dahm, Surface Sci. 142 (1984) 100. [8] F. Gallet, G. Deville, A. Valdes and F.I.B. Williams, Phys. Rev. Letters 49 (1982) 212. [9] G. Deville, A. Valdes, E.Y. Andrei and F.I.B. Williams, Phys. Rev. Letters 53 (1984) 588. [10] K. Kono, K. Kajita, S. Kobayashi and W. Sasaki, J. Low Temp. Phys. 46 (1982) 195. [11] H. Namaizawa, Solid State Commun. 34 (1980) 607. [12] R.H. Morf, Phys. Rev. Letters 43 (1979) 931. [13] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181.