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Escape rates of electrons from the surface of liquid helium
38
Surface Science 196 (1988 ) 38-42 North-Holland, Amsterdam
ESCAPE RATES OF ELECTRONS FROM THE SURFACE OF LIQUID HELIUIVI J.M. GOODKIND, Gordon F. SAVILLE, Andrei E. RUCKENSTEIN Department of Physics, Unirersity of California at San Diego. San Diego, CA 92093, USA
and Phillip M. PLATZMAN AT&T Bell Labaratories, Murray Hill, New Jersey 07974, USA Received 23 June 1987; accepted for publication 23 June 1987
Electrons on the surface of liquid helium are confined to the plane of the surface by the image potential in the liquid. The confining field can be further increased by an external electric field normal to the surface. If these potentials are sufficiently weak, electrons can escape from the surface. As an electron moves off of the surface the potential barrier includes additional contributions from the electrostatic field of the electrons remaining on the surface and their correlations. These contributions depend on the electron density. Electrons escape by thermal activation over the barrier at high temperatures and/or small external field. At low temperatures and/or high applied field the electrons escape by quantum tunneling. We have measured the escape rate of electrons from the surface of liquid helium as a function of temperature and the electrostatic field normal to the surface. The results show the transition from thermal activation to tunneling and the increase in correlation energy with increasing density.
1. A description of the physical system The potential in which electrons are trapped on the surface of liquid helium results from the linage charge in the helium, the interactions between electrons and, in most experiments, an external applied "pressing" field, Vp. Microwave and infrared spectroscopy has been used to confirm the hydrogen-like nature of the potential, V(z), close to the surface [ 1-3]. At distances above the surface, z, of order of the interelectron spacing, V(z) is drastically modified by the electron-electron interactions. A crude treatment of the Hartree potential [4] shows that, with increasing density the effective barrier increases and moves ;lightly closer to the surface. Electrons which escape from the surface must pass through this potential. Consequently escape rates can provide information about electron-electron correlations in the two-dimensional electron gas as well as about the 0039-6028/88/$ 03.50 (e) Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
J.M. Goodkind et al./Escape rates of electronsfrom fiquid helium
39
potential of single electron surface states (low densities). In addition they provide an entirely new possibility; to measure the effect of dissipation on the quantum mechanics of a one-dimensional potential. Previous experiments [4] have studied the escape rates of electrons from the surface, but they were restricted to situations dominated by low barrier energies. (The escape rates were measured by rapidly reducing the external field to zero, or actually reversing it.) Electrons escaped as a result of collisions with gas atoms at temperatures above 1 K and by thermal activation just below l K. The interesting regime of quantum tunneling was not accessible by this method. We have made measurements through the range of densities and temperature where thermal activation dominates and into the range where quantum tunneling dominates.
2. Experimental method The experimental cell has two horizontal parallel capacitor plates 2 cm in diameter, separated by 1 cm. Free electrons are obtained by a discharge through a small hole in the upper plate. A cylinder surrounding the plates, but insulated from them, is held at a negative potential, Vr, to provide a radial field to prevent the electrons from reaching the walls of the container. In this geometry the charge forms a circular disc of almost uniform density. The diameter of the disc depends on Vp, 1I, and the total charge, Q. The disc acts as an electronic shield between the two plates. The greater the diameter of the disc the greater the shielding effect. Thus the capacitance can be used as a measure of the disc diameter and consequently of the total charge on the surfce. For our measurements we limit the fractional change of the ~otai charge so that the relation between charge and capacitance is linear to suffi~ c ~ precision.
3. Experimental results Two types of measurements have been made: (1) the escape rate as a function of temperature at constant Vp; (2) the escape rate as a function of Vp at constant temperature. In the first method, the charged surface is cooled to about 0.5 K. The output of the capacitance bridge is then monitored as Vp is reduced in steps of I to 5 V. At high values of Vp the capacitance changes reversibly and no drift of the capacitance can be seen between steps. At a sharply defined Vc (which decreases with decreasing temperature) the capacitance starts to drift upward, indicating that charge is esca;fing from the surface. The rate of change of the capacitance bridge output, R, for ttlese small changes is proportional to the rate of change of the charge. Vp is set so as to allow a small but measurable escape rate. The temperature is then raised in :;1eps, allowing the sample chamber to reach thermal equilibrium at ech step. R
£ M. Goodkind et aL/Escape rates of electronsfrom liquid helium
40 --3.5
--4,-
--4..5
-5
-5.5
i
v C
-60 t3
-6,5
n
-7
-7.5
-
~
.1
~
=
1.3
1.5
,
,
,
1.7
1.9
1/T
13
Vp=E4 V
+
Vp=6# V reduced Q
o
Vp=51 V
Fig. 1. Log of escape rate versus I/T. Open squares and + ' s represent data for two different charge densities at the same Vn. Open diamonds represent data at low Vp and higher charge density. Lines are drawn by hand to indicate the qualitative behavior of the data.
is computed from the raw data by linear regression to the segments of data between temperature steps. Results of this type are shown in fig. I. At low temperatures, R is almost independent of temperature. In the second method, electrons are collected on the surface and held there by a large 1~ as the temperature is lowered. As V~is reduced, In R increases linearly with decreasing Vp for small R. The proportionality constant, d(ln R)ldVp, is determined for each temperature. Fig. 2 shows that it decreases with decreasing temperature.
4. The inte~retation of resu|ts The qualitative dependence of the escape rate on temperature (fig. 1) can be explained as thermal activation at high temperatures and tunneling at low tempcratures. The activation energy for the high temperature part is roughly 20 K and depends on Vp arid the electron density, n~. This is consistent with the hydrogenlike Hamiltonian with the additional potential, Vn, and the electrostatic field from the sheet of electrons at positions above the plane. The slight temperature dependence in the tunneling regime is, in part, due to ,he distribution of electrons over
J.M. Goodkind et al./Escape rates of electrons from liquid helium
41
0.7 CI E]
0.6
0.5
--
0 > 0.4 C
0 0
[]
[] []
0.3 [] 0.2
0."I
I
0.45
I
I
0.55 1:3
Fig.
I
0.65
I
I
0.75
I
I
0.85
TEMPERATURE(K)
2. Voltage d e r i v a t i v e o f the escape rate versus temperature, revealing the smaller dependence at l o w e r temperatures. Ring p o t e n t i a l 90 V.
the "bound" states from which they are tunneling. It should also be due, in part, to the interaction of the escaping electrons with the ripplon bath during the tunneling process. The voltage dependence of R is not as simple to interpret, since ns and the associated electric field changes with Vw However, the value of d(In R)/dVp at the lowest temperatures is small because it is determined by pure tunneling. At intermediate temperatures as Vp is decreased, the barrier height decreases. An increasing fraction of the electrons escape by the faster thermal activation process and d (In R)/d Vp becomes larger. An obvious experimental concern is whether the electrons are in thermal equilibrium with the liquid helium and the sample chamber connected to our thermometer. If the electrons simply did not cool below a certain temperature, then the escape rate would appear, artificially, to be independent of temperature. The data of fig. 1 rule out this possibility. This is because the inelastic electron-ripplon rate [5 ] (which is expected to dominate in our regime) increases with increasing pressing field. Thus the thermal contact would be best at the highest pressing field so that samples at high pressing fields wou~d reach lc,~er ~em_~c~u~. ©~ d~ other hand the data show that the transition to the temperature independent regime is at higher temperatures for larger pressing fields. The temperature dependence of d (In R)/d Vp provides additional evidence that
42
J.M. Goodkind et al./Escape rates of electronsfrom liquid helium
the electron temperature follows the temperature of the sample chamber. The value of d(ln R)/d Vp at the lowest temperatures is small because the only escape mechanism operating is tunneling and it is slow. At higher temperatures as Vp is decreased and the barrier height decreases, an increasing fraction of the electrons escape by the faster thermal activation process so that d (In R)/d Vpbecomes larger. A quantitative interpretation of the voltage dependence of R is complicated by the fact that n~ and the associated electric field change with V~.
Acknowledgement Work supported by NSF grant DMR-8645550.
References [ I ] C.C. Grimes, T.R. Brown, M.L. Burns and C.L. Zipfei, Phys. Rev. B13 (1976) 140. [2] D.K. Lambert aad P.L. Richards, Phys. Rev. Letters 44 (1980) 1427. [3] D.K. Lambert and P.L. Richards, Phys. Rev. B23 (li981) 3282. [4] Y. lye, K. Kono, K. Kajita and W. Sasaki, J. Low-Temp. Phys. 38 (1980) 293. [5] M. Saitoh, J. Phys. Soc. Japan 42 (1977) 201.
Surface Science 196 (1988 ) 38-42 North-Holland, Amsterdam
ESCAPE RATES OF ELECTRONS FROM THE SURFACE OF LIQUID HELIUIVI J.M. GOODKIND, Gordon F. SAVILLE, Andrei E. RUCKENSTEIN Department of Physics, Unirersity of California at San Diego. San Diego, CA 92093, USA
and Phillip M. PLATZMAN AT&T Bell Labaratories, Murray Hill, New Jersey 07974, USA Received 23 June 1987; accepted for publication 23 June 1987
Electrons on the surface of liquid helium are confined to the plane of the surface by the image potential in the liquid. The confining field can be further increased by an external electric field normal to the surface. If these potentials are sufficiently weak, electrons can escape from the surface. As an electron moves off of the surface the potential barrier includes additional contributions from the electrostatic field of the electrons remaining on the surface and their correlations. These contributions depend on the electron density. Electrons escape by thermal activation over the barrier at high temperatures and/or small external field. At low temperatures and/or high applied field the electrons escape by quantum tunneling. We have measured the escape rate of electrons from the surface of liquid helium as a function of temperature and the electrostatic field normal to the surface. The results show the transition from thermal activation to tunneling and the increase in correlation energy with increasing density.
1. A description of the physical system The potential in which electrons are trapped on the surface of liquid helium results from the linage charge in the helium, the interactions between electrons and, in most experiments, an external applied "pressing" field, Vp. Microwave and infrared spectroscopy has been used to confirm the hydrogen-like nature of the potential, V(z), close to the surface [ 1-3]. At distances above the surface, z, of order of the interelectron spacing, V(z) is drastically modified by the electron-electron interactions. A crude treatment of the Hartree potential [4] shows that, with increasing density the effective barrier increases and moves ;lightly closer to the surface. Electrons which escape from the surface must pass through this potential. Consequently escape rates can provide information about electron-electron correlations in the two-dimensional electron gas as well as about the 0039-6028/88/$ 03.50 (e) Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
J.M. Goodkind et al./Escape rates of electronsfrom fiquid helium
39
potential of single electron surface states (low densities). In addition they provide an entirely new possibility; to measure the effect of dissipation on the quantum mechanics of a one-dimensional potential. Previous experiments [4] have studied the escape rates of electrons from the surface, but they were restricted to situations dominated by low barrier energies. (The escape rates were measured by rapidly reducing the external field to zero, or actually reversing it.) Electrons escaped as a result of collisions with gas atoms at temperatures above 1 K and by thermal activation just below l K. The interesting regime of quantum tunneling was not accessible by this method. We have made measurements through the range of densities and temperature where thermal activation dominates and into the range where quantum tunneling dominates.
2. Experimental method The experimental cell has two horizontal parallel capacitor plates 2 cm in diameter, separated by 1 cm. Free electrons are obtained by a discharge through a small hole in the upper plate. A cylinder surrounding the plates, but insulated from them, is held at a negative potential, Vr, to provide a radial field to prevent the electrons from reaching the walls of the container. In this geometry the charge forms a circular disc of almost uniform density. The diameter of the disc depends on Vp, 1I, and the total charge, Q. The disc acts as an electronic shield between the two plates. The greater the diameter of the disc the greater the shielding effect. Thus the capacitance can be used as a measure of the disc diameter and consequently of the total charge on the surfce. For our measurements we limit the fractional change of the ~otai charge so that the relation between charge and capacitance is linear to suffi~ c ~ precision.
3. Experimental results Two types of measurements have been made: (1) the escape rate as a function of temperature at constant Vp; (2) the escape rate as a function of Vp at constant temperature. In the first method, the charged surface is cooled to about 0.5 K. The output of the capacitance bridge is then monitored as Vp is reduced in steps of I to 5 V. At high values of Vp the capacitance changes reversibly and no drift of the capacitance can be seen between steps. At a sharply defined Vc (which decreases with decreasing temperature) the capacitance starts to drift upward, indicating that charge is esca;fing from the surface. The rate of change of the capacitance bridge output, R, for ttlese small changes is proportional to the rate of change of the charge. Vp is set so as to allow a small but measurable escape rate. The temperature is then raised in :;1eps, allowing the sample chamber to reach thermal equilibrium at ech step. R
£ M. Goodkind et aL/Escape rates of electronsfrom liquid helium
40 --3.5
--4,-
--4..5
-5
-5.5
i
v C
-60 t3
-6,5
n
-7
-7.5
-
~
.1
~
=
1.3
1.5
,
,
,
1.7
1.9
1/T
13
Vp=E4 V
+
Vp=6# V reduced Q
o
Vp=51 V
Fig. 1. Log of escape rate versus I/T. Open squares and + ' s represent data for two different charge densities at the same Vn. Open diamonds represent data at low Vp and higher charge density. Lines are drawn by hand to indicate the qualitative behavior of the data.
is computed from the raw data by linear regression to the segments of data between temperature steps. Results of this type are shown in fig. I. At low temperatures, R is almost independent of temperature. In the second method, electrons are collected on the surface and held there by a large 1~ as the temperature is lowered. As V~is reduced, In R increases linearly with decreasing Vp for small R. The proportionality constant, d(ln R)ldVp, is determined for each temperature. Fig. 2 shows that it decreases with decreasing temperature.
4. The inte~retation of resu|ts The qualitative dependence of the escape rate on temperature (fig. 1) can be explained as thermal activation at high temperatures and tunneling at low tempcratures. The activation energy for the high temperature part is roughly 20 K and depends on Vp arid the electron density, n~. This is consistent with the hydrogenlike Hamiltonian with the additional potential, Vn, and the electrostatic field from the sheet of electrons at positions above the plane. The slight temperature dependence in the tunneling regime is, in part, due to ,he distribution of electrons over
J.M. Goodkind et al./Escape rates of electrons from liquid helium
41
0.7 CI E]
0.6
0.5
--
0 > 0.4 C
0 0
[]
[] []
0.3 [] 0.2
0."I
I
0.45
I
I
0.55 1:3
Fig.
I
0.65
I
I
0.75
I
I
0.85
TEMPERATURE(K)
2. Voltage d e r i v a t i v e o f the escape rate versus temperature, revealing the smaller dependence at l o w e r temperatures. Ring p o t e n t i a l 90 V.
the "bound" states from which they are tunneling. It should also be due, in part, to the interaction of the escaping electrons with the ripplon bath during the tunneling process. The voltage dependence of R is not as simple to interpret, since ns and the associated electric field changes with Vw However, the value of d(In R)/dVp at the lowest temperatures is small because it is determined by pure tunneling. At intermediate temperatures as Vp is decreased, the barrier height decreases. An increasing fraction of the electrons escape by the faster thermal activation process and d (In R)/d Vp becomes larger. An obvious experimental concern is whether the electrons are in thermal equilibrium with the liquid helium and the sample chamber connected to our thermometer. If the electrons simply did not cool below a certain temperature, then the escape rate would appear, artificially, to be independent of temperature. The data of fig. 1 rule out this possibility. This is because the inelastic electron-ripplon rate [5 ] (which is expected to dominate in our regime) increases with increasing pressing field. Thus the thermal contact would be best at the highest pressing field so that samples at high pressing fields wou~d reach lc,~er ~em_~c~u~. ©~ d~ other hand the data show that the transition to the temperature independent regime is at higher temperatures for larger pressing fields. The temperature dependence of d (In R)/d Vp provides additional evidence that
42
J.M. Goodkind et al./Escape rates of electronsfrom liquid helium
the electron temperature follows the temperature of the sample chamber. The value of d(ln R)/d Vp at the lowest temperatures is small because the only escape mechanism operating is tunneling and it is slow. At higher temperatures as Vp is decreased and the barrier height decreases, an increasing fraction of the electrons escape by the faster thermal activation process so that d (In R)/d Vpbecomes larger. A quantitative interpretation of the voltage dependence of R is complicated by the fact that n~ and the associated electric field change with V~.
Acknowledgement Work supported by NSF grant DMR-8645550.
References [ I ] C.C. Grimes, T.R. Brown, M.L. Burns and C.L. Zipfei, Phys. Rev. B13 (1976) 140. [2] D.K. Lambert aad P.L. Richards, Phys. Rev. Letters 44 (1980) 1427. [3] D.K. Lambert and P.L. Richards, Phys. Rev. B23 (li981) 3282. [4] Y. lye, K. Kono, K. Kajita and W. Sasaki, J. Low-Temp. Phys. 38 (1980) 293. [5] M. Saitoh, J. Phys. Soc. Japan 42 (1977) 201.