Evaporation induced instabilities of electrons trapped at a helium surface

~

Solid State Communications, Vol. 76, No. 5, pp. 617-619, 1990. Printed in Great Britain.

0038-1098/9053.00+.00 Pergamon Press plc

EVAPORATION INDUCED INSTABILITIES OF ELECTRONS TRAPPED AT A HELIUM SURFACE M. Ya Azbel School of Physics and Astronomy, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, ISRAEL P. M. Platzman AT&T

Bell Laboratories, Murray Hill, N e w Jersey 07974, USA (Received 28 June 1990 by S. Alexander)

W e analyze the recently observed density and elecuic field dependent instability for electrons trapped at a liquid helium surface.

the observed instability of the escape rates. The the.oretical picture is based on an instability of the initial state of the system to increases in the escape rate itself. It gives a semi-quantitative fit to the data and suggests that by looking at somewhat higher densities and lower temperatures, a new re,entrant regime for the stability boundary may be observed. The motion of an electron normal to the surface is governed by the image potential in the Helium,~

Tunneling and evaporation from a well characterized many body quantum mechanical system is an important and intriguing experimental and theoretical problem. 2-D electrons trapped at the bulk helium vacuum interface is such a system. There are strong coulomb correlations among the electrons whose density may be easily varied over a wide range. The strong correlations are manifested by a crystallization at sufficiently high densities and low temperatures.] Escape of the electrons from such a system is initiated by the application of external fields which are roughly a hundred volts per cm. In addition the system is extraordinarily clean and homogeneous, i.e., free of defects, local field hot spots, impurities, etc. Recent experiments performed on such system2-3 at temperatures near I"K and at densities n = 10s cm-2 show a variety of interesting phenomena. In particular there was a region of parameter space which showed a well characterized smoothly varying escape rate which was activated at high temperatures and was rather temperature independent (slight increase) at lower temperatures. At higher densities an extraordinarily rapid increase (more than four orders of magnitude) in the escape rate for a change of temperature as small as 30 milli degrees was observed. 3 In ref. (3) it was suggested that the instability was connected with a chain reaction initiated by electrons in an excited Rydberg state. This model did not fit the data very well. It is probably safe to say that there was no quantitative explanation of this rather remarkable instability. More recently two theoretical papers4-s have analyzed in some detail the tunneling and evaporation rates of such systems when the initial state of the many body system is itself stable to the small loss of electrons from the surface. The general conclusions of this work are; 1. True tunneling has not been observed; i.e. it is much too small to account for any of the observed escape rates. 2. Evaporation is complicated. It may, for example, have an activated regime, a temperature independent regime, and even a regime where the rate increases as the temperature is lowered (not yet convincingly observed). In this short note wc present a consistent picture of

-Ae 2 Vl(Z) = - - , ~ Z = V0(=I eV), z<0

(1)

with A = 0.007, and the external pressing field Ep which is required to maintain a finite density of electrons which has a potential eEvz. In addition, w e must include the Coulomb potential Uc of the other 21) electrons. Clearly we cannot simply treat the many particle Coulomb interaction as a one particle potential. In this highly correlated system it is however qualitatively correct to think of, an effective one electron potential which is for example the potential the escaping electron feels as it moves out from the surface of a 2-D triangular lattice of electrons. Since the neighboring electrons are at distances of ½ order n- the coulomb effects will only be important for z ~ n -~ i.e. approximately 103-104A for the system under consideration. To arrive at a specific estimate of rates etc., wc consider a simple model, suggested by Iye et al.,7 where for z < n -I/2 Uc(z) ------4v-le2nZ/2z2 and v = 1 in their model. The quantity v for the moment expresses our ignorance of the many body aspects of the evaporation process. W e will discuss this a bit more later on and it is discussed in some detail in L However, independent of such considerations, this formula for 13o may be seen as the expansion of the dimensionless selfconsistent Uc/e2n ½, which has a m a x i m u m at z = 0, with respect to the small dimensionless parameter zn½. At larger distances z > n-~ the effect of the other electrons is that of a uniform electric field F~ = 2x'ne. If 8 = Ep/Ec < 1 then tunneling and evaporation are allowed. 617

618

INSTABILITIES OF ELECTRONS TRAPPED AT A HELIUM SURFACE

Neglecting very weak coupling to ripplons we can write the total effective one body potential U for 0 < z < n-'~ as, U = - A e 2 / z + eEp Z -- 4e2n3/2z2/v.

(2)

When z < 0, then V0 in Eq. (1) is so large that we may assume U = ,~. The first two terms in Eq. (2) describe a Stark shifted 1D hy .d~ogen atom with the Bohr radius aB ~ If2/mAe2 - 76.2A and a Rydberg energy, ER(Ep)

= mA2e4/2if

2 --

1.5 eEpaB

.

Umax =

e2n½v82.

single and di-vacancies reads, cl = ( 1 - c r c 2 ) q 0 t + 2c2q21 - clql0 - 2c~q12 - ClW1 (6) ~2 = clWl + c~ql2 - c2~t.

(4)

The classical evaporation rate over such a barrier (for T,~ER/I~) is given by, s

q01 ci = ~ . qlo-W1

(8)

Since qol/qlo = exp(-El/kB T) < < l,Cl is small as long as the escape rate WI is siguificanfly less than the vacancy relaxation rate ql0 into regular sites. The latter is almost exclusively related to the diffusion to boundaries and is very low. When Wt gets very close to ql0 the cl goes from its tiny value to almost unity, i.e., we have a chain reaction. Then one must use the more accurate but still elementary solution of Eqs. (6, 7) to determine cl i.e., ci = (a2+l~)I/2 - a

W = T/If e -E'/k'T e -'178 62v/x

(7)

Here Chj,is the rate for the i-vacancy to j-vacancy change in the absence of any escape channel. As long as cl,c2 <<1 the concentration of regular sites (1-c1-c2) may be considered unity and,

(3)

The last term in Eq. (2) changes the Rydberg energy by 12 v-le2n 3/2 a 2, which is too small to be significant. The height of the barrier Umax is determined (z:~aB) by the last two terms in Eq. (3) i.e.,

Vol. 76, NO. 5

(9)

(5) where

where x -- T/TIn and Tm = e2 (xn)1/2/128 is the melting temperature of the electron solid} Since all the phenomena of interest occur at low temperatures relative to the coulomb energy it is very accurate to think of the electron liquid as a solid in the neighborhood of the electron which is escaping. In order to understand the instability of the lattice due to evaporation consider an electron escaping from the undisturbed lattice and creating a vacancy. Assuming it escapes adiabatically (the best situation), it will extract an energy 0.97e2n ~ - 0.23e2n ½ = 0.74e2n ½ from the lattice. The second term is the defect formation energy E I. 9 Since the energy to create a divacancy is almost the same, i.e. two vacancies are bound by the same energy 9 namely 0.25e2n ~, when an electron escapes from a favorable site (one on the boundaries of the single vacancy), creating a di-vacancy, it can in principle gain an energy of 0.97e2n ½ from the lattice. Since AEi = .23e2n ~ = 5" K, this leads to a much more rapid, up to -exp(AEl/kbT) times, evaporation rate. The most effective poly-vacaney is the one with the minimal defect energy per defect site. Meanwhile, the equilibrium concentration per lattice site c i - exp(-Ei/kbT) of an isite vacancy is related to its total defect energy El, which monotonically increases with i. The result is the generation of evaporation induced voids. Suppose, just for simplicity, that a single vacancy has the highest evaporation rate (Wl), while all other evaporation rates (including the one from a regular site W0) are negligible. Evaporation changes a vacancy concentration c 1 into a divacaney concentration c2. Since in equilibrium c2 is less than cl, it is a good approximation to assume that a divacancy only decays into two vacancies. Its complete relaxation to the regular lattice is assumed to be negligibly slow. Evaporation changes the two single vacancies into two divacancies, which in turn decay into four vacancies, etc. The balance equations (in this model) for destruction and creation of

a = q21 [qloeqo1-Wl (l--q01/~1)]/2q0iql2

(10)

and 13 = ~1/q12. Since W i changes exponentially with temperature, the width of the transition is extremely sharp. Of course, when the concentration of escape induced vacancies is high, the evaporating system can no longer even approximately be considered a crystal. Furthermore, a temperature increase will decrease the jump in the escape rate since,

W 1/W 0 - exp (A E 1IT).

(11)

The instability boundary is in turn given by, Wl qi-d = 1

(12)

If we now use Eq. (5) in Eq. (12), where W is Wl and the configuration dependence of the rate parametrized by v is called v 1 the rate for an electron to escape from the neighborhood of a single vacancy, we arrive at a phenomenological expression for the instability boundary, i.e., ER 178 82Vl ln(tlT/~) . . . . (13) kBT ,~ where ti is very roughly q~01. In Fig. 1, we have plotted ER/kBT + 178 52Vl/'~ versus % using the experimental data presented in ref.(3). Fig. l(a) chooses vl = 1. For T > T m ti is not activated so that In(tiT/if) will be, to a reasonable approximation, a constant over this range of temperature. If the model has the correct physics in it, all the data should reduce to a constant line parallel to the x axis. We see that t h e theory, Eq. (13) with vi = 1, is only a rough fit to the data. However we know physically that vl does decrease with temperature i.e., the higher the temperature the more uniform the charge distribution, and a more uniform charge distribution lowers the barrier height. If we arbitrarily take v i q = 1 + x (a not unreasonable guess) we get the results shown in Fig. l(b) a quite good fit to the data.

INSTABILITIES OF ELECTRONS TRAPPED AT A HELIUM SURFACE

Vol. 76, NO. 5 60~

(a) v t = t

5O %

~ 4o ÷

3o

~1%~,

2o

IO o 1.o

I

I

I

I

I

I

t2

t.4

t.6

1.8

2.0

2.2

2.4

T V,=I+T

entropy of that configuration. In addition in I it was shown that the evaporation problem could be mapped on to a kind of thermodynamic phase diagram. The function w($,x) for different fixed values 8 could in analogy with the liquid solid phase boundary and the Van der Walls m equation of state have a non-monotonic dependence, a point of inflection or a minimum and a maximum depending on the values of 8. For 5 = .42 in fact the experimental results (see I) give such a multivaiued type of curve. Since the physics of this type of behavior occurs because of a switchover from one type of local electronic configuration to another we believe that a good approximation to the "real" chain reaction problem is given by, Wteff = 1 ,

+

2°t5 E

1.0

1.2

t.4

Fig. !a-lb

t.6

t.8

2.0

2.2

2.4

r

Comparison of theory and experiment. The curves are

for three values of the pressing field: circles, Ep=186; triangles, Ep = 168; squares, F~ =204 V/cm and densities in the neighborhood of n= 3 × 108 cm-2. Our simple analysis of the chain reaction, even with a phenomenological vl(x) is one which only includes the kinetics of a single well defined configuration. It is not expected to be completely accurate. More generally the evaporation rate of electrons from the surface is given by," w

T e-ER/kRTe-W(8,x)/kBT

= -ff

.

(14)

In I it was shown that the function w(8,'c) could in a reasonable approximation be written as an average over different configurations and that the configuration which was favored as the temperature is changed was determined by a competition between the energctics and

619

(15)

where tae is some effective relaxation time of the dominant configuration. In this case consistent with the evaporation data we would predict that there are values of the applied field for which the instability boundary is itself multivalued. This will probably happen, (since we know from I that w(5, x) is multivalued for 8 = .42), when 8=.42 and the Rydberg energy ER is small (high fields, high density), Small Rydberg energy is a favorable situation simply because we want the nonmonotonic dependence of w(5, x) to dominate. In this case the system starting from stability will be unstable as one raises the temperature or as one lowers it, i.e. the stability boundary is reentrant. At lower temperatures, x less than some minimum temperature, we know that the escape will be dominated by tunneling and the behavior will again change in a way which we will not discuss in any detail. This is an extraordinarily simple system where strong Coulomb interaction effects play a dominant role. The observed collective instability is a true many body effect. It will be intriguing to examine the behavior of the instability boundaries over a wide range of dimensionless parameter space to see if it fits the rather simple model suggested in this short note. Acknowledgements - One of us P. M. Platzman, would like to thank J. Goodkind and G. Saville for several enlightning discussions regarding the data.

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