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Electron crystallization and bifurcation theory
Volume 85A, number 1
PHYSICS LETI’ERS
7 September 1981
ELECTRON CRYSTALLIZATION AND BIFURCATION THEORY~ Dan H. CONSTANTINESCU Institut für Theoretische Physik, Universitdt Erlangen-Nurnberg, 8520 Erlangen, Fed. Rep. Germany and Institut für Astrophysik, Max-Planck-Institut für Physik und Astrophysik, 8046 Garching, Fed. Rep. Germany Received 26 March 1981
The bifurcation ofperiodic solutions is a necessary, but not a sufficient, condition for electron crystallization. Consequently, the crystaffization domain in phase space may be considerably smaller than the bifurcation domain.
Electron crystallization has long been frustrating the efforts of both theorists and experimentalists. Theoretically, calculations of the fluid—crystal phase boundary have produced a disconcerting variety of results [1]. In particular, recent evaluations based on the Lindemann criterion [2] and on bifurcation theory [3,4] fail to agree. Experimentally, electron crystallization has proved very difficult to detect [5]. There are indications that it might indeed occur in inversion layers [6], in doped semiconductors [71and in magnetite [8]. However, the least controversial evidence to date comes from the experiment performed by Grimes and Adams [9] on a two-dimensional gas of electrons [10] trapped on the surface of liquid helium [11]. The purpose of the present letter is to show how this latter result fits into the bifurcation-theory picture. This analysis will lead to an important conclusion: the crystallization domain in phase space is considerably smaller than the bifurcation domain [3,4]. This applies also to the strong magnetic field case [12], of interest both in astrophysics and solid-state physics [13]. A few changes are required if the description of refs. [3,4] is to be applied to the Grimes—Adams experiment. First, the theoretical calculation refers to the three-dimensional case, whereas in the experiment the electrons are trapped on the helium surface. Second, refs. [3,4] are concerned with electron crystallization at relatively high density, when exchange and ~ Work partially supported by the Deutsche Forschungsgemeinschaft. 0
correlations dominate (Wigner transition); the experiment reveals the phenomenon in the extreme lOw-density region, where the electron gas behaves classically. Third, stability questions must be tackled before concluding on the nature of the phase transition associated with the bifurcation [4]. Finally, one must estimate to what extent the phase diagram differs from the bifurcation diagram [12]. These points will be dealt with below. I consider a slight generalization of the simple model of ref. [4]: a system of nondegenerate charged fermions in a neutralizing jellium background, in D dimensions (D = 3, 2, 1) and at temperature T. A selfconsistent field picture is supposed to exist. The number density of particles is given by ~ ~ ‘2 n [g/(27r11) ] ~2(2m,
where g is the multiplicity of the single-particle states and ~ = 4ir, 2n~2 is the solid angle in D = 3, 2, 1 dimensions, respectively; ~ is a yet unknown function, to be determined from the requirement of self-consistency at equilibrium, and I,, denotes a Fermi—Dirac function. The free energy per particle is 21D)ID/2(17)/ID/2_10i)] + u + eØ. (2) ( Here the first term represents the free energy for noninteracting fermions, and the interaction term has been split up: v is the interaction energy for a uniformly smeared-out fermion charge, and eçb is the additional Coulomb energy arising from the actual equilibrium distribution. The electrostatic potential ~ satisfies the
~
—
031—9163/81/0000-—0000/$ 02.50 © North-Holland Publishing Company
45
Volume 85A, number 1
PHYSICS LETTERS
Poisson equation —4ire(n
=
—
(3)
p)/l,
where p is the uniform jellium density and I is the thickness in three-dimensional space of the D-dimensional layer of fermions. At equilibrium the free energy F[n] = f nfdDr is stationary for variations ~n that leave N[n] = f n dDr constant; hence Km1+a(nv)/an+ect=p.,
(4)
with ji a constant. I am interested here in the existence of periodic solutions of eqs. (1), (3) and (4) at low densities, where quantum effects are negligible. For a classical gas 2/d, n~ —1,d and Ia(0) [‘(a + 1)e’~moreover, v = —eby where is the mean interelec.tron spacing, defined n = D/~2dD.Introducing the dimensionless quantities
x = r/b,
8 D a~—ll 2 b =
~
1~n(r), ~,
N(x)
=
(~Z/D)a
1~p,r = 2h2KT/me4 R
(5)
(~L/D)a
=
7 September 1981
F = ui/c, where e = ~-DKTis the kinetic energy particle, this condition reads r~’F0=2D/(1+D).
per
(11)
It is straightforward to compute the second variation of the 2F free~ energy (7), and to show 0 if andabout only the if Fsolution ~F that 6 0: at the bifurcation the trivial solution becomes unstable. ForD = 2, eq. (11) yields F1~= 4/3, whereas the experiment indicates crystallization at F~= 137 ±15 [91, in excellent agreement with recent theoretical calculations [141.Bifurcation thus appears to be a necessary, but not a sufficient, condition for crystallization. An explanation of this fact in of an instability the bifurcating solution at terms intermediate values of of F is highly implausible, both mathematically and physically. Though the instability of the trivial solution at F 0 does furcating solution, it isa difficult to stability see how the pennot imply of to Fthe biodicnecessarily solution could starttransfer by being unstable at 0 and then, at Fe. as Moreover, F increases physics monotonically, guaranteesrecover the existence stability of a ground state which, if secondary instabilities are
(a 0 is the Bohr radius), one obtains from eqs. (1), (3) and (4) e2’~ (6) N = (2ir)D ~ F(D/2)r~~/ ~ [ro 2(1 +D~)Nh/D} N— R. ‘
—
Eqs. (6) depend on the two parameters R and T. Bifurcations from the trivial solution
N0
=
—p---
_~~_F(D/2)i~DI2 e~0= R
(21r)D 2D
(7)
lations. However, the correlated phase associated with the bifurcating solution can be a macroscopic crystal only if X becomes of the order of d/b. It is convenient to define, inside the bifurcation domain, a function
(8)
O(d/a0, r) of the phase variables by the relation
may exist, if the linearized equation
ft
—
2(1 ~i~D_l)D_lN~/D]
~
=N0~’
has nonzero solutions. In wave-vector space one then has the dispersion relation 2[r—2(l +D1)D1N~j/1~1 N —1k1
0.
(9)
Hence, periodic solutions may bifurcate only if the bifurcation condition r ~ 2(1 +D_1)D1N~jI’)
(10)
is satisfied. In terms of the dimensionless parameter 46
ruled out, must be associated, for any F> F0, with the periodic bifurcating solution. tionThe may be expressed in terms of two characteristic difference between bifurcation and crystallizalengths: the mean interelectron spacing (in dimensionless units)d/b and the wavelength X 2ir/ikl of the bifurcating solution. At the onset of bifurcation the latter is zero and it increases as one penetrates deeper into the bifurcation domain F> F 0. Physically, this corresponds to the onset and enhancement of corre-
X = Od/b Then, crystallization is expected to occur if
(12)
~
(13)
.
,
where O~(d/ao,r) is an unknown function which, however, must be weakly dependent on its arguments and have values close to one. Using eqs. (9) and (12), the crystallization condition, eq. (13), may be written in the form
Volume 85A, number 1
F ~ F~= ~ D\
+
PHYSICS LE1’TERS
2D 7T&~
—~
T
(14)
aol_1d2_D).
Is this picture consistent with the experimental result [9]? For D = 2 and taking I 10—6 cm, T 0.44 K and [‘c 137, one gets O~ 0.6; given the uncertainty of at least a factor two in 1, the answer is yes. One sees that, in spite of the small value of O~,the leading contribution to Fe comes from the second term in eq. (14): the bifurcation threshold is a poor approximation for the crystallization threshold. The situation, illustrated here in the classical limit, remains qualitatively the same when quantum effects are dominant (Wigner transition). In particular, for D = 3 and at zero temperature the periodic solution appears at
7 September 1981
be answered by bifurcation theory. The astrophysical implications of this work (e.g. for white dwarf interiors and pulsar surfaces) ~ll be discussed elsewhere. I am indebted to Professors B. Jancovici, N.H. March, Sir N.F. Mott and E.P. Wigner for stimulating correspondence. This work was begun while I was a visitor at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette; I thank Professor N.H. Kuiper for his kind hospitality and Professor L. Michel for helpful discussions. The final stage of this research was sup. ported by the Deutsche Forschungsgemeinschaft.
References
d/a
0 5 [3], whereas crystallization is predicted to occur only at d/a0 65 [2,15], corresponding to
1.7. Summarizing, one has obtained the following pieture about the relationship and difference between bifurcation and crystallization. The bifurcation of a periodic solution marks the appearance of a correlated phase; microscopically, clustering sets in, making the new phase “lumpy” presumably a liquid. Macroscopic order typical of a solid crystal can appear only if the stronger crystallization condition is met. Simple considerations show that the critical 0 is of the order of one, but its exact evaluation requires a detailed dynamical calculation that falls outside the range of bifurcation theory. Moreover, the location of the phase boundary is very sensitive to changes in 0~. This sounds familiar: indeed, the crystallization condition is only a different expression of the Lindemann criterion [2,16]. The extrapolation from 00 = 0 to ~ 1 uses a tacit smoothness assumption which turns out to be inaccurate; therefore the hope of replacing the Lindemann criterion by the bifurcation condition [3] must be abandoned. This does not affect the validity of the dispersion relation which, supplemented with the bifurcation (0 = = 0) or the crystallization (0 = °c 1) condition determines the boundary between the uncorrelated gas and the correlated fluid phases or the approximate boundary between fluid and solid, respectively. Other questions, such as the exact evaluation of °c’ the properties of the correlated fluid phase or the crystalline structure of the solid, require more dynamical input [14,15,17] and cannot —
—
—
[1] C.M. Care and N.H. March, Adv. Phys. 24(1975)101.
[21 R. Mochkovitch
and J.P. Hansen, Phys. Lett. 73A (1979) 35. [31 D.H. Constantinescu, Phys. Lett. 73A (1979) 84. [4] D.H. Constantinescu, Phys. Lett. 75A (1980) 205. [5J N.H. March, in: Proc. 19th Scottish universities summer school in physics (St. Andrews, 1978) (SUSSP Publications, Edinburgh, 1979). [6] S. Kawaji and J. Wakabayashi, Solid State Commun. 22 (1977) 87; B.A. Wilson, S.J. Allen Jr. and D.C. Tsui, Phys. Rev. Lett. 44 (1980) 479. [7] G. Nimtz, B. Schlicht, E. Tyssen, R. Dornhaus and L.D. Haas, Solid State Commun. 32 (1979) 669. [8] SJ’. Ionov, G.V. lonova, V.S. Lubimov and E.F. Makarov, Phys. Stat. So!. B71 (1975) 11. [9] C.C. Grimes and G. Adams, Phys. Rev. Lett. 42 (1979)
eds., Electronic properties of two-dimensional systems, Surf. Sd. 73 (1978).
[10] G. Dorda and P.J. Stiles,
[11] R.S. Crandall and R. Williams, Phys. Lett. 34A (1971) 404; C.C. Grimes, in: Electronic properties of two-dimensional systems, eds. G. Dorda and P.J. Stiles, Surf. Sci. 73 (1978) 379. [12] D.H. Constantinescu, Phys. Rev. Lett. 43 (1979) 1267. [13] J.I. Kaplan and M.L. Glasser, Phys. Rev. Lett. 28 (1972) 1077; W.G. Kl~ppmannand R.J. Elliott, J. Phys. C8 (197~) 2729. [14] R.C. Gann, S. Chakravarty and G.V. Chester, Phys. Rev. B20 (1979) 326; R.H. Morf, Phys. Rev. Lett. 43 (1979) 931. [15] D. Ceperley, Phys. Rev. B18 (1978) 3126. [16] R.A. Coldwell-Honsfall and A.A. Maradudin, J. Math. Phys. 1(1960) 395. [17] D.M. Cepenley and B.J. Alder, Phys. Rev. Lett. 45 (1980) 566.
47
PHYSICS LETI’ERS
7 September 1981
ELECTRON CRYSTALLIZATION AND BIFURCATION THEORY~ Dan H. CONSTANTINESCU Institut für Theoretische Physik, Universitdt Erlangen-Nurnberg, 8520 Erlangen, Fed. Rep. Germany and Institut für Astrophysik, Max-Planck-Institut für Physik und Astrophysik, 8046 Garching, Fed. Rep. Germany Received 26 March 1981
The bifurcation ofperiodic solutions is a necessary, but not a sufficient, condition for electron crystallization. Consequently, the crystaffization domain in phase space may be considerably smaller than the bifurcation domain.
Electron crystallization has long been frustrating the efforts of both theorists and experimentalists. Theoretically, calculations of the fluid—crystal phase boundary have produced a disconcerting variety of results [1]. In particular, recent evaluations based on the Lindemann criterion [2] and on bifurcation theory [3,4] fail to agree. Experimentally, electron crystallization has proved very difficult to detect [5]. There are indications that it might indeed occur in inversion layers [6], in doped semiconductors [71and in magnetite [8]. However, the least controversial evidence to date comes from the experiment performed by Grimes and Adams [9] on a two-dimensional gas of electrons [10] trapped on the surface of liquid helium [11]. The purpose of the present letter is to show how this latter result fits into the bifurcation-theory picture. This analysis will lead to an important conclusion: the crystallization domain in phase space is considerably smaller than the bifurcation domain [3,4]. This applies also to the strong magnetic field case [12], of interest both in astrophysics and solid-state physics [13]. A few changes are required if the description of refs. [3,4] is to be applied to the Grimes—Adams experiment. First, the theoretical calculation refers to the three-dimensional case, whereas in the experiment the electrons are trapped on the helium surface. Second, refs. [3,4] are concerned with electron crystallization at relatively high density, when exchange and ~ Work partially supported by the Deutsche Forschungsgemeinschaft. 0
correlations dominate (Wigner transition); the experiment reveals the phenomenon in the extreme lOw-density region, where the electron gas behaves classically. Third, stability questions must be tackled before concluding on the nature of the phase transition associated with the bifurcation [4]. Finally, one must estimate to what extent the phase diagram differs from the bifurcation diagram [12]. These points will be dealt with below. I consider a slight generalization of the simple model of ref. [4]: a system of nondegenerate charged fermions in a neutralizing jellium background, in D dimensions (D = 3, 2, 1) and at temperature T. A selfconsistent field picture is supposed to exist. The number density of particles is given by ~ ~ ‘2 n [g/(27r11) ] ~2(2m,
where g is the multiplicity of the single-particle states and ~ = 4ir, 2n~2 is the solid angle in D = 3, 2, 1 dimensions, respectively; ~ is a yet unknown function, to be determined from the requirement of self-consistency at equilibrium, and I,, denotes a Fermi—Dirac function. The free energy per particle is 21D)ID/2(17)/ID/2_10i)] + u + eØ. (2) ( Here the first term represents the free energy for noninteracting fermions, and the interaction term has been split up: v is the interaction energy for a uniformly smeared-out fermion charge, and eçb is the additional Coulomb energy arising from the actual equilibrium distribution. The electrostatic potential ~ satisfies the
~
—
031—9163/81/0000-—0000/$ 02.50 © North-Holland Publishing Company
45
Volume 85A, number 1
PHYSICS LETTERS
Poisson equation —4ire(n
=
—
(3)
p)/l,
where p is the uniform jellium density and I is the thickness in three-dimensional space of the D-dimensional layer of fermions. At equilibrium the free energy F[n] = f nfdDr is stationary for variations ~n that leave N[n] = f n dDr constant; hence Km1+a(nv)/an+ect=p.,
(4)
with ji a constant. I am interested here in the existence of periodic solutions of eqs. (1), (3) and (4) at low densities, where quantum effects are negligible. For a classical gas 2/d, n~ —1,d and Ia(0) [‘(a + 1)e’~moreover, v = —eby where is the mean interelec.tron spacing, defined n = D/~2dD.Introducing the dimensionless quantities
x = r/b,
8 D a~—ll 2 b =
~
1~n(r), ~,
N(x)
=
(~Z/D)a
1~p,r = 2h2KT/me4 R
(5)
(~L/D)a
=
7 September 1981
F = ui/c, where e = ~-DKTis the kinetic energy particle, this condition reads r~’F0=2D/(1+D).
per
(11)
It is straightforward to compute the second variation of the 2F free~ energy (7), and to show 0 if andabout only the if Fsolution ~F that 6 0: at the bifurcation the trivial solution becomes unstable. ForD = 2, eq. (11) yields F1~= 4/3, whereas the experiment indicates crystallization at F~= 137 ±15 [91, in excellent agreement with recent theoretical calculations [141.Bifurcation thus appears to be a necessary, but not a sufficient, condition for crystallization. An explanation of this fact in of an instability the bifurcating solution at terms intermediate values of of F is highly implausible, both mathematically and physically. Though the instability of the trivial solution at F 0 does furcating solution, it isa difficult to stability see how the pennot imply of to Fthe biodicnecessarily solution could starttransfer by being unstable at 0 and then, at Fe. as Moreover, F increases physics monotonically, guaranteesrecover the existence stability of a ground state which, if secondary instabilities are
(a 0 is the Bohr radius), one obtains from eqs. (1), (3) and (4) e2’~ (6) N = (2ir)D ~ F(D/2)r~~/ ~ [ro 2(1 +D~)Nh/D} N— R. ‘
—
Eqs. (6) depend on the two parameters R and T. Bifurcations from the trivial solution
N0
=
—p---
_~~_F(D/2)i~DI2 e~0= R
(21r)D 2D
(7)
lations. However, the correlated phase associated with the bifurcating solution can be a macroscopic crystal only if X becomes of the order of d/b. It is convenient to define, inside the bifurcation domain, a function
(8)
O(d/a0, r) of the phase variables by the relation
may exist, if the linearized equation
ft
—
2(1 ~i~D_l)D_lN~/D]
~
=N0~’
has nonzero solutions. In wave-vector space one then has the dispersion relation 2[r—2(l +D1)D1N~j/1~1 N —1k1
0.
(9)
Hence, periodic solutions may bifurcate only if the bifurcation condition r ~ 2(1 +D_1)D1N~jI’)
(10)
is satisfied. In terms of the dimensionless parameter 46
ruled out, must be associated, for any F> F0, with the periodic bifurcating solution. tionThe may be expressed in terms of two characteristic difference between bifurcation and crystallizalengths: the mean interelectron spacing (in dimensionless units)d/b and the wavelength X 2ir/ikl of the bifurcating solution. At the onset of bifurcation the latter is zero and it increases as one penetrates deeper into the bifurcation domain F> F 0. Physically, this corresponds to the onset and enhancement of corre-
X = Od/b Then, crystallization is expected to occur if
(12)
~
(13)
.
,
where O~(d/ao,r) is an unknown function which, however, must be weakly dependent on its arguments and have values close to one. Using eqs. (9) and (12), the crystallization condition, eq. (13), may be written in the form
Volume 85A, number 1
F ~ F~= ~ D\
+
PHYSICS LE1’TERS
2D 7T&~
—~
T
(14)
aol_1d2_D).
Is this picture consistent with the experimental result [9]? For D = 2 and taking I 10—6 cm, T 0.44 K and [‘c 137, one gets O~ 0.6; given the uncertainty of at least a factor two in 1, the answer is yes. One sees that, in spite of the small value of O~,the leading contribution to Fe comes from the second term in eq. (14): the bifurcation threshold is a poor approximation for the crystallization threshold. The situation, illustrated here in the classical limit, remains qualitatively the same when quantum effects are dominant (Wigner transition). In particular, for D = 3 and at zero temperature the periodic solution appears at
7 September 1981
be answered by bifurcation theory. The astrophysical implications of this work (e.g. for white dwarf interiors and pulsar surfaces) ~ll be discussed elsewhere. I am indebted to Professors B. Jancovici, N.H. March, Sir N.F. Mott and E.P. Wigner for stimulating correspondence. This work was begun while I was a visitor at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette; I thank Professor N.H. Kuiper for his kind hospitality and Professor L. Michel for helpful discussions. The final stage of this research was sup. ported by the Deutsche Forschungsgemeinschaft.
References
d/a
0 5 [3], whereas crystallization is predicted to occur only at d/a0 65 [2,15], corresponding to
1.7. Summarizing, one has obtained the following pieture about the relationship and difference between bifurcation and crystallization. The bifurcation of a periodic solution marks the appearance of a correlated phase; microscopically, clustering sets in, making the new phase “lumpy” presumably a liquid. Macroscopic order typical of a solid crystal can appear only if the stronger crystallization condition is met. Simple considerations show that the critical 0 is of the order of one, but its exact evaluation requires a detailed dynamical calculation that falls outside the range of bifurcation theory. Moreover, the location of the phase boundary is very sensitive to changes in 0~. This sounds familiar: indeed, the crystallization condition is only a different expression of the Lindemann criterion [2,16]. The extrapolation from 00 = 0 to ~ 1 uses a tacit smoothness assumption which turns out to be inaccurate; therefore the hope of replacing the Lindemann criterion by the bifurcation condition [3] must be abandoned. This does not affect the validity of the dispersion relation which, supplemented with the bifurcation (0 = = 0) or the crystallization (0 = °c 1) condition determines the boundary between the uncorrelated gas and the correlated fluid phases or the approximate boundary between fluid and solid, respectively. Other questions, such as the exact evaluation of °c’ the properties of the correlated fluid phase or the crystalline structure of the solid, require more dynamical input [14,15,17] and cannot —
—
—
[1] C.M. Care and N.H. March, Adv. Phys. 24(1975)101.
[21 R. Mochkovitch
and J.P. Hansen, Phys. Lett. 73A (1979) 35. [31 D.H. Constantinescu, Phys. Lett. 73A (1979) 84. [4] D.H. Constantinescu, Phys. Lett. 75A (1980) 205. [5J N.H. March, in: Proc. 19th Scottish universities summer school in physics (St. Andrews, 1978) (SUSSP Publications, Edinburgh, 1979). [6] S. Kawaji and J. Wakabayashi, Solid State Commun. 22 (1977) 87; B.A. Wilson, S.J. Allen Jr. and D.C. Tsui, Phys. Rev. Lett. 44 (1980) 479. [7] G. Nimtz, B. Schlicht, E. Tyssen, R. Dornhaus and L.D. Haas, Solid State Commun. 32 (1979) 669. [8] SJ’. Ionov, G.V. lonova, V.S. Lubimov and E.F. Makarov, Phys. Stat. So!. B71 (1975) 11. [9] C.C. Grimes and G. Adams, Phys. Rev. Lett. 42 (1979)
eds., Electronic properties of two-dimensional systems, Surf. Sd. 73 (1978).
[10] G. Dorda and P.J. Stiles,
[11] R.S. Crandall and R. Williams, Phys. Lett. 34A (1971) 404; C.C. Grimes, in: Electronic properties of two-dimensional systems, eds. G. Dorda and P.J. Stiles, Surf. Sci. 73 (1978) 379. [12] D.H. Constantinescu, Phys. Rev. Lett. 43 (1979) 1267. [13] J.I. Kaplan and M.L. Glasser, Phys. Rev. Lett. 28 (1972) 1077; W.G. Kl~ppmannand R.J. Elliott, J. Phys. C8 (197~) 2729. [14] R.C. Gann, S. Chakravarty and G.V. Chester, Phys. Rev. B20 (1979) 326; R.H. Morf, Phys. Rev. Lett. 43 (1979) 931. [15] D. Ceperley, Phys. Rev. B18 (1978) 3126. [16] R.A. Coldwell-Honsfall and A.A. Maradudin, J. Math. Phys. 1(1960) 395. [17] D.M. Cepenley and B.J. Alder, Phys. Rev. Lett. 45 (1980) 566.
47