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Relation between bulk compressibility and surface energy of electron-hole liquids
Solid State Communications, Vol. 34, pp. 209-210. Pergamon Press Ltd. 1980. Printed in Great Britain. RELATION BETWEEN BULK COMPRESSIBILITY
AND SURFACE ENERGY OF ELECTRON-HOLE
LIQUIDS
K.S. Singwi* and M.P. Tosi International
Centre for Theoretical
Physics, Trieste, Italy
(Received 1 September 1979 by R. Fieschi) Attention is drawn to the existence of an empirical relation xu/alf - 1 between the compressibility, the surface energy and the excitonic radius in electron-hole liquids.
the various EHL’s, one does indeed see that the surface thickness is of the order of the excitonic Bohr radius in all these cases. A simple microscopic derivation of the relation (1) for an electron gas, which leads to an estimation of L, has been given by Brown and March [ 7 ] on the assumption that the electron gas near the surface can be treated by retaining only the lowest order term in the gradient expansion. A similar argument has already been given by Rice [5] for the EHL. The above simplifying assumption, coupled with the assumption that the energy density function E@) has the form
IT HAS BEEN KNOWN [ 1,2] for a long time that the product of the isothermal compressibility x and the surface tension 0 is roughly a constant for a variety of classical liquids ranging from liquid metals to insulating liquids and molten salts. This constant is of the order of 1 A for all these liquids near their triple point. The purpose of this communication is to stress that a similar relation also exists for quantum electron-hole liquids. However, the product ~a in the latter is, at low temperatures, of the order of the effective excitonic Bohr radius, i.e. of the order of 100 A in semiconductors like Ge and Si. In the table below we report those theoretical estimates [3,4] of x and u which have been obtained in a selfconsistent manner. One should bear in mind that these values are vastly different from those [2] of liquid metals or other atomic liquids. We also give the values of the product xa and of xa/ai , aI*,being the excitonic Bohr radius. It should be noted that while the ranges of variation of x and u span three orders of magnitude, the product xo/a$ is, within a factor of two, a constant close to unity. If we assume local charge neutrality everywhere in the EHL including the surface region (an approximation which has been found to be good [5] ), the general macroscopic considerations of Cahn and Hilliard [6] can be taken over to the present case. These authors consider the excess free energy associated with the surface as composed of a local elastic term and a nonlocal term expressed through the density gradient. They are then able to show that x0-L
[(f-J’- 2(g3]
e(P)= IEBIE,P
where ,oo is the bulk equilibrium density and EB is the binding energy of the liquid in units of the effective Rydberg E,, allows an analytic determination of the surface profile. This then leads to xc = &l;/@BI.
(1)
where L is the “surface thickness” loosely defined as the distance over which the density varies in going from its bulk value to its value in the vapour. If one examines the surface density profiles that have been reported [4] for * Permanent address: Department of Physics and Astronomy, Northwestern University, Evanston, 60201, U.S.A.
IL
209
(3)
Although the above expression still displays a dependence of xu/ai on the system through EB, it turns out in practice that the former is a constant of the order of 0.3-0.4 for the systems considered in Table 1. For the EHL in polar semiconductors considered by Beni and Rice [8], equation (3) leads to a value of xo/ai between 0.5 and 1 when one uses the experimental (or the theoretical) values of the binding energy. Although equation (3) has been derived on the assumption of a gradient expansion for the kinetic energy functional, we feel on the basis of the results displayed in Table 1 that the approximate constancy of the quantity xu/a; is of more general validity. This constant is of order unity, within a factor of two which is also typical of the range of variation of xu in liquid metals [2]. It would be interesting to have a more general proof of this conjecture. As it stands now, this relation may already prove to be of empirical usefulness.
210
COMPRESSIBILITY
AND SURFACE ENERGY OF ELECTRON-HOLE
Table 1. Bulk and surface properties of electron-hole Ge[lll] 00(cmm3) f$(meV cm6)a g(cm* dyn-‘) o(dyn cm-r)b ox(cm) axlaB
1.11 3.37 13.5 0.2 2.7 1.5
1o16 1o-33 10-a 1o-4 IO+
4.47 1.17 5.98 11.4 0.68 1.4
a P. Vashishta, S.G. Das & K.S. Singwi,Phys. b A.K. Kalia & P. Vashishta,Phys.
x x x x x
IO” 1o-35 lo+ 1o-4 10-6
2.20 1.88 3.14 3.7 1.2 0.7
Rev. Lett. 33,911
3. 4. 5. 6.
REFERENCES
2.
Si
Ge x x x x x
10” 1O-35 1o-3 1o-4 10-6
3.20 3.17 6.01 87.4 0.53 1.1
x x x x x
10” 1o-37 lo-’ 1o-4 lo+
(1974).
Rev. B17, 2655 (1978).
Acknowledgements - We thank Professor N.H. March for interesting discussions. We would also like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. One of us (K.S.S.) wishes to acknowledge a grant (DMR 77-09937) from the National Science Foundation.
1.
Vol. 34, No. 3
liquids
Si[lOO] x x x x x
LIQUIDS
J. Frenkel, Kinetic Theory of Liquids. University Press, Oxford (1942). P.A. Egelstaff & B. Widom,J. Chem. Phys. 53,
7. 8.
2667 (1970). P. Vashishta, S.G. Das & K.S. Singwi, Phys. Rev. Lett. 33,911 (1974). R.K. Kalia & P. Vashishta, Phys. Rev. B17,2655 (1978). T.M. Rice, Solid State Phys. 32, 1. Academic Press, New York (1977). J.W. Cahn & J.E. Hilliard,J. C’hem. Phys. 28,258 (1958); B. Widom, J. Chem. Phys. 43,3892 (1965). R.C. Brown & N.H. March,J. Phys. C6, L363 (1973). G. Beni & T.M. Rice, Phys. Rev. B18,768 (1978).
AND SURFACE ENERGY OF ELECTRON-HOLE
LIQUIDS
K.S. Singwi* and M.P. Tosi International
Centre for Theoretical
Physics, Trieste, Italy
(Received 1 September 1979 by R. Fieschi) Attention is drawn to the existence of an empirical relation xu/alf - 1 between the compressibility, the surface energy and the excitonic radius in electron-hole liquids.
the various EHL’s, one does indeed see that the surface thickness is of the order of the excitonic Bohr radius in all these cases. A simple microscopic derivation of the relation (1) for an electron gas, which leads to an estimation of L, has been given by Brown and March [ 7 ] on the assumption that the electron gas near the surface can be treated by retaining only the lowest order term in the gradient expansion. A similar argument has already been given by Rice [5] for the EHL. The above simplifying assumption, coupled with the assumption that the energy density function E@) has the form
IT HAS BEEN KNOWN [ 1,2] for a long time that the product of the isothermal compressibility x and the surface tension 0 is roughly a constant for a variety of classical liquids ranging from liquid metals to insulating liquids and molten salts. This constant is of the order of 1 A for all these liquids near their triple point. The purpose of this communication is to stress that a similar relation also exists for quantum electron-hole liquids. However, the product ~a in the latter is, at low temperatures, of the order of the effective excitonic Bohr radius, i.e. of the order of 100 A in semiconductors like Ge and Si. In the table below we report those theoretical estimates [3,4] of x and u which have been obtained in a selfconsistent manner. One should bear in mind that these values are vastly different from those [2] of liquid metals or other atomic liquids. We also give the values of the product xa and of xa/ai , aI*,being the excitonic Bohr radius. It should be noted that while the ranges of variation of x and u span three orders of magnitude, the product xo/a$ is, within a factor of two, a constant close to unity. If we assume local charge neutrality everywhere in the EHL including the surface region (an approximation which has been found to be good [5] ), the general macroscopic considerations of Cahn and Hilliard [6] can be taken over to the present case. These authors consider the excess free energy associated with the surface as composed of a local elastic term and a nonlocal term expressed through the density gradient. They are then able to show that x0-L
[(f-J’- 2(g3]
e(P)= IEBIE,P
where ,oo is the bulk equilibrium density and EB is the binding energy of the liquid in units of the effective Rydberg E,, allows an analytic determination of the surface profile. This then leads to xc = &l;/@BI.
(1)
where L is the “surface thickness” loosely defined as the distance over which the density varies in going from its bulk value to its value in the vapour. If one examines the surface density profiles that have been reported [4] for * Permanent address: Department of Physics and Astronomy, Northwestern University, Evanston, 60201, U.S.A.
IL
209
(3)
Although the above expression still displays a dependence of xu/ai on the system through EB, it turns out in practice that the former is a constant of the order of 0.3-0.4 for the systems considered in Table 1. For the EHL in polar semiconductors considered by Beni and Rice [8], equation (3) leads to a value of xo/ai between 0.5 and 1 when one uses the experimental (or the theoretical) values of the binding energy. Although equation (3) has been derived on the assumption of a gradient expansion for the kinetic energy functional, we feel on the basis of the results displayed in Table 1 that the approximate constancy of the quantity xu/a; is of more general validity. This constant is of order unity, within a factor of two which is also typical of the range of variation of xu in liquid metals [2]. It would be interesting to have a more general proof of this conjecture. As it stands now, this relation may already prove to be of empirical usefulness.
210
COMPRESSIBILITY
AND SURFACE ENERGY OF ELECTRON-HOLE
Table 1. Bulk and surface properties of electron-hole Ge[lll] 00(cmm3) f$(meV cm6)a g(cm* dyn-‘) o(dyn cm-r)b ox(cm) axlaB
1.11 3.37 13.5 0.2 2.7 1.5
1o16 1o-33 10-a 1o-4 IO+
4.47 1.17 5.98 11.4 0.68 1.4
a P. Vashishta, S.G. Das & K.S. Singwi,Phys. b A.K. Kalia & P. Vashishta,Phys.
x x x x x
IO” 1o-35 lo+ 1o-4 10-6
2.20 1.88 3.14 3.7 1.2 0.7
Rev. Lett. 33,911
3. 4. 5. 6.
REFERENCES
2.
Si
Ge x x x x x
10” 1O-35 1o-3 1o-4 10-6
3.20 3.17 6.01 87.4 0.53 1.1
x x x x x
10” 1o-37 lo-’ 1o-4 lo+
(1974).
Rev. B17, 2655 (1978).
Acknowledgements - We thank Professor N.H. March for interesting discussions. We would also like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. One of us (K.S.S.) wishes to acknowledge a grant (DMR 77-09937) from the National Science Foundation.
1.
Vol. 34, No. 3
liquids
Si[lOO] x x x x x
LIQUIDS
J. Frenkel, Kinetic Theory of Liquids. University Press, Oxford (1942). P.A. Egelstaff & B. Widom,J. Chem. Phys. 53,
7. 8.
2667 (1970). P. Vashishta, S.G. Das & K.S. Singwi, Phys. Rev. Lett. 33,911 (1974). R.K. Kalia & P. Vashishta, Phys. Rev. B17,2655 (1978). T.M. Rice, Solid State Phys. 32, 1. Academic Press, New York (1977). J.W. Cahn & J.E. Hilliard,J. C’hem. Phys. 28,258 (1958); B. Widom, J. Chem. Phys. 43,3892 (1965). R.C. Brown & N.H. March,J. Phys. C6, L363 (1973). G. Beni & T.M. Rice, Phys. Rev. B18,768 (1978).