- Email: [email protected]
Negative low-field magnetoresistance of electrons on liquid helium
365
Surface Science 229 ( 1990) 365-367 North-Holland
NEGATIVE LOW-FIELD A.M.L. JANSSEN,
MAGNETORESISTANCE
R.W. van der HEIJDEN,
A.T.A.M.
OF ELECTRONS
ON LIQUID HELIUM
de WAELE, H.M. GIJSMAN
Department of Physics, Eindhoven University of Technology, P.0. Box 5 I3, 5600 MB Eindhoven. The Netherlands
and F.M. PEETERS Department of Physics, University ofAntwerp, B-2610 Antwerpen, Belgium Received 11July 19X9;accepted for publication 14 September 1989
A negative low-field magnetoresistance is reported for the nondegenerate two-dimensional electron gas on the surface of bulk liquid helium. The behaviour is observed at temperatures of about 2 K and is interpreted as a manifestation of weak localisation. Preliminary results indicate that the effect becomes larger for lower electron densities, suggesting that electron-electron scattering contributes to the dephasing mechanism.
1. Introduction Weak localisation effects have received much attention in the past decade. They have been studied extensively in thin metal films [ 1 ] and in two-dimensional electron gases (2DEG) in semiconductors [ 2 1. Recently, weak localisation was reported in the magnetoresistance (MR) of the nondegenerate 2DEG on solid hydrogen [ 31. The study of nondegenerate systems is of interest as the important parameter in localisation theory ki, (k is the electron wave vector, 1, is the electron mean free path), is tunable in a wide range, e.g. by varying temperature, thus making both weak (kl, >> 1) and strong localisation regimes (k&s 1) accessible for experimental investigation [ 3-5 1. Fu~hermore, the absence of a lattice and magnetic impurities in such a system facilitates the interpretation of the experimental results in terms of weak localisation. When compared with the electron system on solid hydrogen, there are several reasons to justify a search for weak localisation of electrons on bulk liquid helium. First, surface scattering is negligible at temperatures above 1 K. Second, the electron system can only interact with the vapour atoms. Because of the large mass difference, electron-He atom scattering is ~039-6028/90/$03.50 (Nosh-Holland)
0 Elsevier Science Publishers B.V.
quasi-elastic and so one might expect to derive both the dephasing time and the elastic scattering time from the same scattering process. Finally, the extent of the electron wave function perpendicular to the surface is five times larger for electrons on helium than for electrons on hydrogen. Recently it was suggested [ 51 that the extent of the wave function may play an important role in weak localisation. In this work. a negative magnetoresistance at low magnetic fields is reported for electrons on bulk liquid helium at temperatures close to 2 K. This is taken as evidence for weak localisation and the data are analysed with the theoretical expression for the nondegenerate case reported in ref. [ 31. Estimating 1, from the mobility, the localisation parameter is found to be kl,- 5.
2. Experimental
method
The experimental method is a variation of the Sommer and Tanner technique [ 6,7] and is the same as in ref. [ 8 f . The rectangular geometry if preferred over the Corbino geometry, as the strong positive MR of the latter might obscure the negative MR. The electrode geometry, which is positioned approxi-
366
(cl 1.0
xp2Ec In-’
2 (>
[y(cx+$)-w(a’+i)l
I
. (1)
04 _
b
1.0
02
.
10% @-y;0.4
0.6
0.8
1.0
1.2
1.4
B(T) Fig. 1. Normalised magnetoresistance at three different temperatures. Note the different scales. Full lines: calculation with eq. (I) with optimum p and 7,/z,,: (a) T=2.05 K, P= 1.4 m’/V-s and z,/r,,=4.9; T=2.00 K. j~=1.7 m’/V-s and 7Jr0=4.6; (c) T= 1.95 K,p=2.5 mz/V*sand 5,/z-,=2.3.
mately 0.5 mm below the helium surface, is sketched in the inset in fig. 1. A standard capacitance bridge was connected to electrodes A and B. The data are interpreted with an equivalent lumped impedance circuit [ 7 1. As the imaginary part of the admittance is nearly constant, the real part is proportional to the resistance of the electron layer. This real part was recorded as a function of magnetic field.
3. Results and discussion Fig. 1 shows three traces of the resistance R(B) as a function of magnetic field B, normalised to the resistance at B=O, R( 0), for three different temperatures. The negative low-field MR is visible. The positive going MR is a result of the formation of Landau levels [ 8 1. The accuracy of the experiment increases with increasing resistance. Therefore, to find the (small) negative MR, we searched mainly at large resistance values, which implies low electron densities and high temperatures (but below the lambda point 2.17 K). The density corresponding to the applied electric field is of the order of 10’ electrons cm-2. It was deliberately lowered below the saturation value. The unsaturated value is not precisely known but may be an order of magnitude less. To analyse the data, the expression as given by Adams and Paalanen [ 31 is used:
Eq. (I ) describes the field dependence of the conductivity with the weak localisation correction for the nondegenerate 2DEG. Here crQ is the classical Drude conductivity and /3= (ks7’-’ with k, the Boltzmann constant and T the temperature. r. and r, are the elastic scattering and phase coherence times. E, is a lower energy cut-off accounting for strongly localised electrons given by Ec=fze/ (2nmp)ln( rii/ro) [ 3,9], with @ the mobility. In our case this cut-off is almost zero, &~0.2 K. u, is the digamma function and o and fy’ are related through cy= (ri/ro)~’ where a!= fie/(4mBEp’). To calculate the resistivity P.~~~=a,,/ ( azX + & ), which corresponds to the experimental situation, the classical uncorrected values for o,, ( = o:,) and aXVwere used: atX = nep/ (1 +p2B2) and a,,= ,uBo$ (n is the electron density). It is assumed that p=ez,/m (e is the elementary charge, m the eiectron mass). It turns out that puB is too large to be neglected. When p and ri/zb are adjusted, and assuming pxx(B)/p,(0)=R(B)/R(O), the low-field data are precisely described by eq. ( 1) (see full line in fig. 1). The full behaviour is not observed because it is obscured by level quantization effects [ 81. The optimised fi values have the magnitudes expected from the DC-transport mobility [ 71. The values of r,/rO are comparable to the values found for electrons on hydrogen [ 3,5]. The accuracy in determining~ and Ti/ T* is estimated to be of the order of 20%. Absolute values for the transport mobility p cannot be obtained from the present experiments, because the geometric factor and the density are not known. The variation of the optimised h’s with the temperature is, however, consistent with the observed variation in zero-field resistance. Although there is considerable noise on the experimental data for AR/R, the data in fig. 1 suggest a temperature dependence of r,/rO. If only electronhelium atom scattering were involved, a constant ratio si/r,, would be expected. In fig. 2, data for a fixed temperature but different electron densities (the rel-
A.M.L. Janssen et al/Negative low-field magnetoresistance of electrons on liquid helium
367
( _N5X lo-l2 s), the value for Ziis found to be of the order of lo-” s. The timescale for electron-electron scattering is set by w; ’ , where wp is the two-dimensional plasma frequency at wavevector k-=2n/a with a the inter-electron distance. For a density nr IO7 cmT2, it follows 0; ’ 2 lo- ” s, close to the value found for 4.
Acknowledgement
One of us (F.M.P.) acknowledges NFWO for financial support. 1
I
I
0.1
0.2
I
1
1
0.3 0.4
I
0.5
B(T)
Fig. 2. Normalised magnetoresistance for three different electron densities at T= 1.85K. The density increases from lower to upper curve in the ratio 1:2.2:3.6. Note the different scales. Full lines: calculation with eq. ( 1) with optimum ,u and ri/rp For the upper curve, the possible negative MR is within the noise, the theoretical curve, which would have 7J7,, close to 1, has therefore not been drawn.
ative change being determined from the zero-field resistance) are displayed. It shows that the negative MR becomes smaller as the density increases. This would suggest that electron-electron scattering contributes to the dephasing mechanism. From the obtained ratio ri/ro and estimating ~~from the mobility
References [I] G. Bergmann, Phys. Rep. 101 (1984) 1. [2] D.J. Newson, M. Pepper and T.J. Thornton, Philos. Mag. B 56 (1987) 715. [ 31 P.W. Adams and M.A. Paalanen, Phys. Rev. Lett. 58 (1987) 2106. [ 4 ] P. W. Adams and M.A. Paalanen, Phys. Rev. Lett. 6 1 ( 1988 ) 451. [S] P.W. Adams and M.A. Paalanen, Phys. Rev. B 39 ( 1989) 4733. [ 61 W.T. Sommer and D.J. Tanner, Phys. Rev. Lett. 27 ( 197 1) 1345. [7] Y. Iye, J. Low Temp. Phys. 40 (1980) 441. [ 81 R.W. van der Heijden, H.M. Gijsman and F.M. Peelers, J. Phys. C (Solid State Phys.) 21 (1988) L1165. [ 91 M.J. Stephen, Phys. Rev. B 36 ( 1987) 5663.
Surface Science 229 ( 1990) 365-367 North-Holland
NEGATIVE LOW-FIELD A.M.L. JANSSEN,
MAGNETORESISTANCE
R.W. van der HEIJDEN,
A.T.A.M.
OF ELECTRONS
ON LIQUID HELIUM
de WAELE, H.M. GIJSMAN
Department of Physics, Eindhoven University of Technology, P.0. Box 5 I3, 5600 MB Eindhoven. The Netherlands
and F.M. PEETERS Department of Physics, University ofAntwerp, B-2610 Antwerpen, Belgium Received 11July 19X9;accepted for publication 14 September 1989
A negative low-field magnetoresistance is reported for the nondegenerate two-dimensional electron gas on the surface of bulk liquid helium. The behaviour is observed at temperatures of about 2 K and is interpreted as a manifestation of weak localisation. Preliminary results indicate that the effect becomes larger for lower electron densities, suggesting that electron-electron scattering contributes to the dephasing mechanism.
1. Introduction Weak localisation effects have received much attention in the past decade. They have been studied extensively in thin metal films [ 1 ] and in two-dimensional electron gases (2DEG) in semiconductors [ 2 1. Recently, weak localisation was reported in the magnetoresistance (MR) of the nondegenerate 2DEG on solid hydrogen [ 31. The study of nondegenerate systems is of interest as the important parameter in localisation theory ki, (k is the electron wave vector, 1, is the electron mean free path), is tunable in a wide range, e.g. by varying temperature, thus making both weak (kl, >> 1) and strong localisation regimes (k&s 1) accessible for experimental investigation [ 3-5 1. Fu~hermore, the absence of a lattice and magnetic impurities in such a system facilitates the interpretation of the experimental results in terms of weak localisation. When compared with the electron system on solid hydrogen, there are several reasons to justify a search for weak localisation of electrons on bulk liquid helium. First, surface scattering is negligible at temperatures above 1 K. Second, the electron system can only interact with the vapour atoms. Because of the large mass difference, electron-He atom scattering is ~039-6028/90/$03.50 (Nosh-Holland)
0 Elsevier Science Publishers B.V.
quasi-elastic and so one might expect to derive both the dephasing time and the elastic scattering time from the same scattering process. Finally, the extent of the electron wave function perpendicular to the surface is five times larger for electrons on helium than for electrons on hydrogen. Recently it was suggested [ 51 that the extent of the wave function may play an important role in weak localisation. In this work. a negative magnetoresistance at low magnetic fields is reported for electrons on bulk liquid helium at temperatures close to 2 K. This is taken as evidence for weak localisation and the data are analysed with the theoretical expression for the nondegenerate case reported in ref. [ 31. Estimating 1, from the mobility, the localisation parameter is found to be kl,- 5.
2. Experimental
method
The experimental method is a variation of the Sommer and Tanner technique [ 6,7] and is the same as in ref. [ 8 f . The rectangular geometry if preferred over the Corbino geometry, as the strong positive MR of the latter might obscure the negative MR. The electrode geometry, which is positioned approxi-
366
(cl 1.0
xp2Ec In-’
2 (>
[y(cx+$)-w(a’+i)l
I
. (1)
04 _
b
1.0
02
.
10% @-y;0.4
0.6
0.8
1.0
1.2
1.4
B(T) Fig. 1. Normalised magnetoresistance at three different temperatures. Note the different scales. Full lines: calculation with eq. (I) with optimum p and 7,/z,,: (a) T=2.05 K, P= 1.4 m’/V-s and z,/r,,=4.9; T=2.00 K. j~=1.7 m’/V-s and 7Jr0=4.6; (c) T= 1.95 K,p=2.5 mz/V*sand 5,/z-,=2.3.
mately 0.5 mm below the helium surface, is sketched in the inset in fig. 1. A standard capacitance bridge was connected to electrodes A and B. The data are interpreted with an equivalent lumped impedance circuit [ 7 1. As the imaginary part of the admittance is nearly constant, the real part is proportional to the resistance of the electron layer. This real part was recorded as a function of magnetic field.
3. Results and discussion Fig. 1 shows three traces of the resistance R(B) as a function of magnetic field B, normalised to the resistance at B=O, R( 0), for three different temperatures. The negative low-field MR is visible. The positive going MR is a result of the formation of Landau levels [ 8 1. The accuracy of the experiment increases with increasing resistance. Therefore, to find the (small) negative MR, we searched mainly at large resistance values, which implies low electron densities and high temperatures (but below the lambda point 2.17 K). The density corresponding to the applied electric field is of the order of 10’ electrons cm-2. It was deliberately lowered below the saturation value. The unsaturated value is not precisely known but may be an order of magnitude less. To analyse the data, the expression as given by Adams and Paalanen [ 31 is used:
Eq. (I ) describes the field dependence of the conductivity with the weak localisation correction for the nondegenerate 2DEG. Here crQ is the classical Drude conductivity and /3= (ks7’-’ with k, the Boltzmann constant and T the temperature. r. and r, are the elastic scattering and phase coherence times. E, is a lower energy cut-off accounting for strongly localised electrons given by Ec=fze/ (2nmp)ln( rii/ro) [ 3,9], with @ the mobility. In our case this cut-off is almost zero, &~0.2 K. u, is the digamma function and o and fy’ are related through cy= (ri/ro)~’ where a!= fie/(4mBEp’). To calculate the resistivity P.~~~=a,,/ ( azX + & ), which corresponds to the experimental situation, the classical uncorrected values for o,, ( = o:,) and aXVwere used: atX = nep/ (1 +p2B2) and a,,= ,uBo$ (n is the electron density). It is assumed that p=ez,/m (e is the elementary charge, m the eiectron mass). It turns out that puB is too large to be neglected. When p and ri/zb are adjusted, and assuming pxx(B)/p,(0)=R(B)/R(O), the low-field data are precisely described by eq. ( 1) (see full line in fig. 1). The full behaviour is not observed because it is obscured by level quantization effects [ 81. The optimised fi values have the magnitudes expected from the DC-transport mobility [ 71. The values of r,/rO are comparable to the values found for electrons on hydrogen [ 3,5]. The accuracy in determining~ and Ti/ T* is estimated to be of the order of 20%. Absolute values for the transport mobility p cannot be obtained from the present experiments, because the geometric factor and the density are not known. The variation of the optimised h’s with the temperature is, however, consistent with the observed variation in zero-field resistance. Although there is considerable noise on the experimental data for AR/R, the data in fig. 1 suggest a temperature dependence of r,/rO. If only electronhelium atom scattering were involved, a constant ratio si/r,, would be expected. In fig. 2, data for a fixed temperature but different electron densities (the rel-
A.M.L. Janssen et al/Negative low-field magnetoresistance of electrons on liquid helium
367
( _N5X lo-l2 s), the value for Ziis found to be of the order of lo-” s. The timescale for electron-electron scattering is set by w; ’ , where wp is the two-dimensional plasma frequency at wavevector k-=2n/a with a the inter-electron distance. For a density nr IO7 cmT2, it follows 0; ’ 2 lo- ” s, close to the value found for 4.
Acknowledgement
One of us (F.M.P.) acknowledges NFWO for financial support. 1
I
I
0.1
0.2
I
1
1
0.3 0.4
I
0.5
B(T)
Fig. 2. Normalised magnetoresistance for three different electron densities at T= 1.85K. The density increases from lower to upper curve in the ratio 1:2.2:3.6. Note the different scales. Full lines: calculation with eq. ( 1) with optimum ,u and ri/rp For the upper curve, the possible negative MR is within the noise, the theoretical curve, which would have 7J7,, close to 1, has therefore not been drawn.
ative change being determined from the zero-field resistance) are displayed. It shows that the negative MR becomes smaller as the density increases. This would suggest that electron-electron scattering contributes to the dephasing mechanism. From the obtained ratio ri/ro and estimating ~~from the mobility
References [I] G. Bergmann, Phys. Rep. 101 (1984) 1. [2] D.J. Newson, M. Pepper and T.J. Thornton, Philos. Mag. B 56 (1987) 715. [ 31 P.W. Adams and M.A. Paalanen, Phys. Rev. Lett. 58 (1987) 2106. [ 4 ] P. W. Adams and M.A. Paalanen, Phys. Rev. Lett. 6 1 ( 1988 ) 451. [S] P.W. Adams and M.A. Paalanen, Phys. Rev. B 39 ( 1989) 4733. [ 61 W.T. Sommer and D.J. Tanner, Phys. Rev. Lett. 27 ( 197 1) 1345. [7] Y. Iye, J. Low Temp. Phys. 40 (1980) 441. [ 81 R.W. van der Heijden, H.M. Gijsman and F.M. Peelers, J. Phys. C (Solid State Phys.) 21 (1988) L1165. [ 91 M.J. Stephen, Phys. Rev. B 36 ( 1987) 5663.