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Electrical conductivity of highly correlated 2D electrons formed on solid neon
88
Surface Science 170 (1986) 88-93 North-Holland, Amsterdam
ELECTRICAL CONDUCTIVITY OF HIGHLY CORRELATED 2D E L E C T R O N S F O R M E D O N S O L I D N E O N Koji KAJITA, Yutaka NISHIO and Wataru SASAKI Department of Physics, Toho University, Miyama 2-2-1, Funabashi 274, Japan Received 20 July 1985; accepted for publication 13 September 1985
We have studied the transport phenomena of highly correlated 2D electrons formed on a solid neon. The contributions of the surface roughness scattering and the helium gas atom scattering to the electron resistivity are separately determined as functions of the electron correlation in the region 10 < F<160. When the electrons undergo the Wigner crystal transition, a qualitative change in the transport phenomenon has been observed.
1. Introduction T h e d e n s i t y of 2 D electrons f o r m e d on the surface of solid neon ranges from 1 0 8 / c m 2 to 3 x 101°/cm2, which covers the region of F from 7 to 400 when the experiments are p e r f o r m e d at liquid h e l i u m t e m p e r a t u r e s [1,2]. Here, F represents the strength of electron correlation of a classical electron system which is related to the t e m p e r a t u r e T a n d the electron density N e as F = (~rNe)l/ZeZ/kT. This system is one of the few examples of the strongly c o u p l e d o n e - c o m p o n e n t p l a s m a ( S C O C P ) which are realized in the l a b o r a t o r y . So, the e x p e r i m e n t a l investigation on this system will p r o v i d e valuable i n f o r m a t i o n which will aid to u n d e r s t a n d SCOCP. In this work, we have studied the effect of electron correlation on the d y n a m i c a l p r o p e r t i e s of t w o - d i m e n s i o n a l electrons f o r m e d on the surface of solid neon. In a previous p a p e r [2], we have shown that the electron correlation affects the t r a n s p o r t of electrons when the p a r a m e t e r F exceeds 10. It a p p e a r s as a decrease of c o n d u c t i v i t y with increasing N e a n d a decrease of c o n d u c t i v i t y with decreasing T. A c c o r d i n g to the expression of F, we see that the strength of the correlation can be varied either through the change in the electron d e n s i t y or the temperature. So, we have p e r f o r m e d two series of experiments. In the first series, the t e m p e r a t u r e has been fixed a n d the electron d e n s i t y has been varied, while in the second series, the electron density has been fixed and the 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) a n d Y a m a d a Science F o u n d a t i o n
K. Kafita et al. / Electrical conductivity of highly correlated 2D electrons
89
temperature has been varied. In each experiment, the electrical conductivity o at a frequency 8.2 M H z was measured as a function of helium gas atom density N G which was introduced into the experimental cell. As was pointed out in ref. [1], the present system is a unique one in which the density of scattering centers for electrons can be varied during an experimental run. Details of the experiment are described in ref. [2].
2. Experimental results Fig. 1 shows the inverse of the "electron mobility", #-1 _ (o/Nee)-l, for various values of the electron density N e at 4.2 K. We find that #-1 can be written as
# ,=#~ +#~, =#~1 +~Nc, as far as N G is not #-1 to N 6 = O) and roughness scattering Eq. (1), if written
(1)
too high. Here, # s 1 (which is obtained by extrapolating #G 1 ( = a N 6 ) , represent the contributions of the surface and the gas atom scattering, respectively. in terms of the resistivity p = o -1, expresses the well-known
I
T
Ne : 10 o239x10' 2 cm
T = 4.2 K
o
o
/
r~
v2
%
/o/
./o ~ ,"d° / o / °
1
~ 0
Ol.56×101C o
/f/ o
o o7.3×109
o°
~
o
~7×I0 1
2 N G ( c m-3 )
8 x102 ~ 3
Fig. 1. Inverse of electron mobility # - I = ( o / N e e ) - 1 plotted against the helium gas density NO. The values of "r are 18.4 for Ne = 6.7 × 108/cm 2, 61.1 for 7.3 × 109/cm 2, 89.2 for 1.56 × 10m/cm 2 and 110.5 for 2.39 × 101°/cm 2.
90
K. Kajita et al. / Electrical conductivity of highly correlated 2D electrons
Matthiessen rule that the resistivity is the sum of resistivities due to several scattering mechanisms. In refs. [1] and [2], we reported a similar p h e n o m e n o n observed in a non-correlated electron system with low N e. In that case, eq. (1) is replaced by a relation which says that the scattering rate for an electron, ~--], is the sum of the scattering rate due to the surface roughness, zs~, and that due to the helium gas atoms, ~r~ 1, as
~-* = % * +~c7 ~ = , s ~ + # N ~ . For the strongly correlated electron system we are studying, such a one-electron picture is not relevant to use in discussing the transport. But, eq. (1) states that a similar relation holds in the wide range of electron correlations if we replace the term "scattering rate" with the "resistivity" or the "inverse of mobility" In fig. 2, we have plotted/~s ~ and a as functions of F. This shows how the electron correlation affects the electron conduction. We first recognize a gradual rise in a in the region 20 < F < 40, which indicates that the correlation becomes effective. Following this region is the intermediate region. Here, if we write/~s ~, a ac F", we find that the index n has a value between 1.0 and 2.0. In the high F region near y = 135, where the electron system is expected to change from the gaseous state to the crystalline state [3], abrupt increases both in /~s ~ and a take place. Such a qualitative change in the transport phenome-
o[
T=4.2K
10
d
o°~" o//•
o~
0/
~d
I,
0/ 0/
oP"
~o~
•
"-I
?[/3 :::L 1 F=I 3..B
I
I
~ I l llll
10
I
I
100
C Fig. 2. /~s 1 a n d a transition occurs.
against
F. A
line is depicted which indicates
F =
135 where the Wigner crystal
K. Kafita et al. / Electrical conductivity of highly correlated 2D electrons
91
non should be discussed in connection with the transition of the electron system to a Wigner crystal, which is left for a future study. Returning to fig. 1, we find that the relation expressed in eq. (1) does not hold in the high NG region, where /z-1 increases superlinearly with NG. Judging from the same type of anomaly observed in non-correlated electron systems [1,2], we have concluded that this is a precursor of the electron localization in the highly random potential due to the dense helium gas atoms. We have examined if the correlation affects the electron localization. Finding that the correlation suppresses the electron motion significantly, we may reasonably expect that the correlation might enhance the electron localization. However, in the region of F up to 130, we could not find a remarkable deviation in the critical No, above which a nonlinear behavior of if- 1 appears. We thus conclude that although the correlation effect is so strong that both ffs 1 and a increase by about one order of magnitude with the change in F from 10 to 130, it is not so strong as to promote the electron localization. The results of the second experiment are plotted in fig. 3. Here, we also find r
r
I
I
I
i
T = 124 K 1.91
/
2.19
/ 2.63 D
' /
ru
2
Ne= 8.72x109cr~ 2 0
3.08
0
v
'/S
S
do/~
I0
•
'6
7
Ten SL
~s
eO
9
Ne = 8.72.10 crn 2
0l
I
0
I
I
0.2
I
0.4
NG (cm-3)
I
I
"=135 l
0.6 020
-I
1
I
l llJli
50
4
100
~
J
500
r
Fig. 3. p.- ] against the helium gas density NG. The values of F are 66.7 for 4.2 K, 91.0 for 3.08 K, 107 for 2.63 K, 128 for 2.19 K, 147 for 1.91 K and 161 for 1.74 K. Fig. 4. # s 1 and a against F.
92
K. Kajita et al. / Electrical conductivity of highly correlated 2D electrons
that eq. (1) is valid at low N(~. So we can separate /.ts I and c~, which are depicted in fig. 4. We find that a c~ F" where n = 1.0, while the slope for ~s ~ seems to be less than 1. Here too, we have recognized that abrupt increases in ~s ~ and a coincide with the Wigner crystal transition. In conclusion, we have observed qualitatively the same behavior of/.t s ~ and in two different types of experiments. However, it is noticed that the slopes of both curves in fig. 2 are steeper than those in fig. 4. Especially, while ~s ~ in the first experiment is a strong function of F, that in the second experiment depends weakly on F. To understand such discrepancies between two experiments, we have to check the experimental conditions. In the first experiment, the electron correlation has been varied through the change in N~. However, this process inevitably causes the change in the pressing electric field which is exerted on the surface electron, because it is proportional to the electron density [4]. So, the higher the electron density, the stronger the electrons are pressed towards the surface. The increasing pressing electric field causes the shrinkage of the electron wave function normal to the plane and this shrinkage, in turn, causes the increase in the interaction between electrons and scattering centers [5]. Moreover, for ~ s l we have to consider another direct effect of pressing field, because the interaction of electrons with the surface roughness increases with increasing pressing field [5], while in the second experiment, the pressing electric field stays constant throughout the experiment, and the confusing situation included in the first experiment is avoided. Such a difference in the situation explains the discrepancy between the data in fig. 2 and those in fig. 4, especially a significant differences of F dependences in /~s 1.
3. Conclusions We have found that, in a wide range of the correlations, the inverse of the mobility/~ 1 is written as the sum of the terms due to the surface roughness /~s i and due to the helium gas atoms ~(~, which is proportional to N(; if it is not too high. Each term is an increasing function of the electron correlation. If we write /~s 1, a ~ F", we find n = 1-2. Passing through the phase transition from the gaseous state to a Wigner crystalline state, a qualitative change in the transport p h e n o m e n a occurs. A theoretical approach to this problems is now awaited.
Acknowledgments This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education.
K. Kajita et al. / Electrical conductivity of highly correlated 2D electrons
References [1] [2] [3] [4] [5]
K. Kajita and W. Sasaki, Surface Sci. 113 (1982) 419. K. Kajita, Surface Sci. 142 (1984) 86. C.C. Grimes and G. Adams, Phys. Rev. Letters 42 (1979) 795. K. Kajita, J. Phys. Soc. Japan 52 (1983) 372. M. Saitoh, J. Phys. Soc. Japan 42 (1977) 201.
93
Surface Science 170 (1986) 88-93 North-Holland, Amsterdam
ELECTRICAL CONDUCTIVITY OF HIGHLY CORRELATED 2D E L E C T R O N S F O R M E D O N S O L I D N E O N Koji KAJITA, Yutaka NISHIO and Wataru SASAKI Department of Physics, Toho University, Miyama 2-2-1, Funabashi 274, Japan Received 20 July 1985; accepted for publication 13 September 1985
We have studied the transport phenomena of highly correlated 2D electrons formed on a solid neon. The contributions of the surface roughness scattering and the helium gas atom scattering to the electron resistivity are separately determined as functions of the electron correlation in the region 10 < F<160. When the electrons undergo the Wigner crystal transition, a qualitative change in the transport phenomenon has been observed.
1. Introduction T h e d e n s i t y of 2 D electrons f o r m e d on the surface of solid neon ranges from 1 0 8 / c m 2 to 3 x 101°/cm2, which covers the region of F from 7 to 400 when the experiments are p e r f o r m e d at liquid h e l i u m t e m p e r a t u r e s [1,2]. Here, F represents the strength of electron correlation of a classical electron system which is related to the t e m p e r a t u r e T a n d the electron density N e as F = (~rNe)l/ZeZ/kT. This system is one of the few examples of the strongly c o u p l e d o n e - c o m p o n e n t p l a s m a ( S C O C P ) which are realized in the l a b o r a t o r y . So, the e x p e r i m e n t a l investigation on this system will p r o v i d e valuable i n f o r m a t i o n which will aid to u n d e r s t a n d SCOCP. In this work, we have studied the effect of electron correlation on the d y n a m i c a l p r o p e r t i e s of t w o - d i m e n s i o n a l electrons f o r m e d on the surface of solid neon. In a previous p a p e r [2], we have shown that the electron correlation affects the t r a n s p o r t of electrons when the p a r a m e t e r F exceeds 10. It a p p e a r s as a decrease of c o n d u c t i v i t y with increasing N e a n d a decrease of c o n d u c t i v i t y with decreasing T. A c c o r d i n g to the expression of F, we see that the strength of the correlation can be varied either through the change in the electron d e n s i t y or the temperature. So, we have p e r f o r m e d two series of experiments. In the first series, the t e m p e r a t u r e has been fixed a n d the electron d e n s i t y has been varied, while in the second series, the electron density has been fixed and the 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) a n d Y a m a d a Science F o u n d a t i o n
K. Kafita et al. / Electrical conductivity of highly correlated 2D electrons
89
temperature has been varied. In each experiment, the electrical conductivity o at a frequency 8.2 M H z was measured as a function of helium gas atom density N G which was introduced into the experimental cell. As was pointed out in ref. [1], the present system is a unique one in which the density of scattering centers for electrons can be varied during an experimental run. Details of the experiment are described in ref. [2].
2. Experimental results Fig. 1 shows the inverse of the "electron mobility", #-1 _ (o/Nee)-l, for various values of the electron density N e at 4.2 K. We find that #-1 can be written as
# ,=#~ +#~, =#~1 +~Nc, as far as N G is not #-1 to N 6 = O) and roughness scattering Eq. (1), if written
(1)
too high. Here, # s 1 (which is obtained by extrapolating #G 1 ( = a N 6 ) , represent the contributions of the surface and the gas atom scattering, respectively. in terms of the resistivity p = o -1, expresses the well-known
I
T
Ne : 10 o239x10' 2 cm
T = 4.2 K
o
o
/
r~
v2
%
/o/
./o ~ ,"d° / o / °
1
~ 0
Ol.56×101C o
/f/ o
o o7.3×109
o°
~
o
~7×I0 1
2 N G ( c m-3 )
8 x102 ~ 3
Fig. 1. Inverse of electron mobility # - I = ( o / N e e ) - 1 plotted against the helium gas density NO. The values of "r are 18.4 for Ne = 6.7 × 108/cm 2, 61.1 for 7.3 × 109/cm 2, 89.2 for 1.56 × 10m/cm 2 and 110.5 for 2.39 × 101°/cm 2.
90
K. Kajita et al. / Electrical conductivity of highly correlated 2D electrons
Matthiessen rule that the resistivity is the sum of resistivities due to several scattering mechanisms. In refs. [1] and [2], we reported a similar p h e n o m e n o n observed in a non-correlated electron system with low N e. In that case, eq. (1) is replaced by a relation which says that the scattering rate for an electron, ~--], is the sum of the scattering rate due to the surface roughness, zs~, and that due to the helium gas atoms, ~r~ 1, as
~-* = % * +~c7 ~ = , s ~ + # N ~ . For the strongly correlated electron system we are studying, such a one-electron picture is not relevant to use in discussing the transport. But, eq. (1) states that a similar relation holds in the wide range of electron correlations if we replace the term "scattering rate" with the "resistivity" or the "inverse of mobility" In fig. 2, we have plotted/~s ~ and a as functions of F. This shows how the electron correlation affects the electron conduction. We first recognize a gradual rise in a in the region 20 < F < 40, which indicates that the correlation becomes effective. Following this region is the intermediate region. Here, if we write/~s ~, a ac F", we find that the index n has a value between 1.0 and 2.0. In the high F region near y = 135, where the electron system is expected to change from the gaseous state to the crystalline state [3], abrupt increases both in /~s ~ and a take place. Such a qualitative change in the transport phenome-
o[
T=4.2K
10
d
o°~" o//•
o~
0/
~d
I,
0/ 0/
oP"
~o~
•
"-I
?[/3 :::L 1 F=I 3..B
I
I
~ I l llll
10
I
I
100
C Fig. 2. /~s 1 a n d a transition occurs.
against
F. A
line is depicted which indicates
F =
135 where the Wigner crystal
K. Kafita et al. / Electrical conductivity of highly correlated 2D electrons
91
non should be discussed in connection with the transition of the electron system to a Wigner crystal, which is left for a future study. Returning to fig. 1, we find that the relation expressed in eq. (1) does not hold in the high NG region, where /z-1 increases superlinearly with NG. Judging from the same type of anomaly observed in non-correlated electron systems [1,2], we have concluded that this is a precursor of the electron localization in the highly random potential due to the dense helium gas atoms. We have examined if the correlation affects the electron localization. Finding that the correlation suppresses the electron motion significantly, we may reasonably expect that the correlation might enhance the electron localization. However, in the region of F up to 130, we could not find a remarkable deviation in the critical No, above which a nonlinear behavior of if- 1 appears. We thus conclude that although the correlation effect is so strong that both ffs 1 and a increase by about one order of magnitude with the change in F from 10 to 130, it is not so strong as to promote the electron localization. The results of the second experiment are plotted in fig. 3. Here, we also find r
r
I
I
I
i
T = 124 K 1.91
/
2.19
/ 2.63 D
' /
ru
2
Ne= 8.72x109cr~ 2 0
3.08
0
v
'/S
S
do/~
I0
•
'6
7
Ten SL
~s
eO
9
Ne = 8.72.10 crn 2
0l
I
0
I
I
0.2
I
0.4
NG (cm-3)
I
I
"=135 l
0.6 020
-I
1
I
l llJli
50
4
100
~
J
500
r
Fig. 3. p.- ] against the helium gas density NG. The values of F are 66.7 for 4.2 K, 91.0 for 3.08 K, 107 for 2.63 K, 128 for 2.19 K, 147 for 1.91 K and 161 for 1.74 K. Fig. 4. # s 1 and a against F.
92
K. Kajita et al. / Electrical conductivity of highly correlated 2D electrons
that eq. (1) is valid at low N(~. So we can separate /.ts I and c~, which are depicted in fig. 4. We find that a c~ F" where n = 1.0, while the slope for ~s ~ seems to be less than 1. Here too, we have recognized that abrupt increases in ~s ~ and a coincide with the Wigner crystal transition. In conclusion, we have observed qualitatively the same behavior of/.t s ~ and in two different types of experiments. However, it is noticed that the slopes of both curves in fig. 2 are steeper than those in fig. 4. Especially, while ~s ~ in the first experiment is a strong function of F, that in the second experiment depends weakly on F. To understand such discrepancies between two experiments, we have to check the experimental conditions. In the first experiment, the electron correlation has been varied through the change in N~. However, this process inevitably causes the change in the pressing electric field which is exerted on the surface electron, because it is proportional to the electron density [4]. So, the higher the electron density, the stronger the electrons are pressed towards the surface. The increasing pressing electric field causes the shrinkage of the electron wave function normal to the plane and this shrinkage, in turn, causes the increase in the interaction between electrons and scattering centers [5]. Moreover, for ~ s l we have to consider another direct effect of pressing field, because the interaction of electrons with the surface roughness increases with increasing pressing field [5], while in the second experiment, the pressing electric field stays constant throughout the experiment, and the confusing situation included in the first experiment is avoided. Such a difference in the situation explains the discrepancy between the data in fig. 2 and those in fig. 4, especially a significant differences of F dependences in /~s 1.
3. Conclusions We have found that, in a wide range of the correlations, the inverse of the mobility/~ 1 is written as the sum of the terms due to the surface roughness /~s i and due to the helium gas atoms ~(~, which is proportional to N(; if it is not too high. Each term is an increasing function of the electron correlation. If we write /~s 1, a ~ F", we find n = 1-2. Passing through the phase transition from the gaseous state to a Wigner crystalline state, a qualitative change in the transport p h e n o m e n a occurs. A theoretical approach to this problems is now awaited.
Acknowledgments This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education.
K. Kajita et al. / Electrical conductivity of highly correlated 2D electrons
References [1] [2] [3] [4] [5]
K. Kajita and W. Sasaki, Surface Sci. 113 (1982) 419. K. Kajita, Surface Sci. 142 (1984) 86. C.C. Grimes and G. Adams, Phys. Rev. Letters 42 (1979) 795. K. Kajita, J. Phys. Soc. Japan 52 (1983) 372. M. Saitoh, J. Phys. Soc. Japan 42 (1977) 201.
93