- Email: [email protected]
Behavior of electrons at soft edges in the fractional quantum Hall and Wigner crystal states
::,,,... ‘::
:. ,: ,.
Surface Science 263 (1992) 60-64
Behavior of electrons at soft edges in the fractional and Wigner crystal states Fertig
Center for
Theoretical
College Park.
Received
and
Program
fbr Adl~cmceti
for publication
the state of electrons
26 August
an open conducting
bulk, then the edge channel
Electronic
Muterids,
‘;.:I
‘:.:.
..)
.:.;
:,.
..
.‘.:.‘.
.’
.I.
.’
quantum Hall
Department
of
fIly.sic~,\. Utlil,er.sit,,
of ,$!uI).[N~I(I,
1991
near the edge of a two-dimensional
varies slowly on the scale of the magnetic
consequences
.....
USA
31 May 1991: accepted
.striped phase, forming
7.
Sarma
Phpsics and Joint
MD 20742-4111,
We investigate potential
S. Das
:.,’j:j?, :...
kurface science : y. >,.:. jj’.‘:..: ..::. ::.:.,,, ,,,:,:>::: ,,,..:., :,y..: .,.... ..:: ,,.,. .:... ,:>:.: ,.:.y.. ,,:. .. .. ...
North-Holland
H.A.
,...
edge channel.
may be destabilized
electron
gas in a strong magnetic
length. We find that under certain circumstances,
field when the edge
these electrons
We find that if the filling is low enough to form a Wigner
due to interaction
between
bulk and edge electrons.
A number
may favor a crystal in the
of experimental
are discussed
Over the past year, there has been a great deal of interest in the possibility of edge channels existing in systems exhibiting the fractional quantum Hall effect (FQHE) [l-5]. In this work, we analyze a model of a two-dimensional electron gas in a strong magnetic field whose electrochemical potential falls off slowly on the scale of the magnetic length (“soft edge”). This model is probably appropriate for edges created by gated geometries. It has been shown recently by Beenakker [l] that if one assumes that transport at the edges in this model is ballistic, then one may understand the unusual quantizations seen in the experiments [4,5], and indeed understand transport in the FQHE in terms of sample edges. An important inconsistency in the application of the ballistic transport approximation to the soft edge model arises when one considers what happens near the very edge of the sample, where the electron density will be quite low. For bulk electrons at such low densities, one expects the system to form a Wigner crystal [6--S], and hence become pinned in the presence of impurities. In the case of low density electrons at the sample edge, if we follow the prescription of ref. [l], and simply subtract out the portion of current associated with electrons that have crystallized (since 0039.6028/92/$05.00
c
1YY2
Elsevier
Science Publishers
one naively expects them to be pinned), we find that the calculated Hall conductance urV deviates slightly from quantization. This effect has never been observed experimentally, and clearly prcsents a problem with the approach of ref. [ 11. One is thus led to investigate in more detail what state the electrons at the low density end of a soft edge would take. Our calculations described below indicate that the confinement potential at the edge, under certain circumstances, has a very dramatic effect on the low density electrons: the Wigner crystal state becomes unstable against the formation of a striped phase. The striped phase may be described as a charge density wave, with a density profile that oscillates perpendicular to the edge, and is uniform along the edge. When the striped phase is stabilized, one expects the approach of ref. [I] to give the correct quantization in the FQHE, since in the absence of transverse density oscillations there will be no pinning. To understand how a striped phase arises in this system, consider what would happen to a Wigner crystal if its edge is subjected to a confining potential. If we focus on the lines of charge parallel to the sample edge, then for potentials varying faster than linearly, the natural drift vclocity associated with the external potential for
B.V. and Yamada
Science Foundation.
All rights reserved
H.A. Fertig, S. Das Sarma / Behauior of electrons at soft edges in the FQH and WC states
each line increases as one approaches the sample edge. To understand how the system overcomes this tendency to shear, we transform to a frame of reference co-moving with one of these lines. One may show that, in this frame, the edge potential pushes the neighboring lines of charge towards the line with whose drift velocity we are moving. The lines of charge are thus compressed together, until the Coulomb repulsion between them balances the force due to the external potential. The local electric field on each line then vanishes, and in the laboratory frame, all the charges would move together. However, if the system favors a compression of the electrons in the P-direction, it necessarily follows that the single-particle wavefunctions comprising the states we consider here become more extended in the y-direction, so long as we are unwilling to admix states from higher Landau levels (which would be inappropriate in high fields.) This latter result follows from the commutation relations for the guiding center coordination of single particle wavefunctions in a magnetic field [9], [X, Y] = iii. In terms of real coordinates, an electron whose real coordinate (say, JZ:)is confined to within a magnetic length has its guiding center coordinate confined to an infinitesimal interval [9], so that both the guiding center and real coordinates in the f-direction become unconfined. The striped phase is essentially the most compressed state one can get in a particular direction without exciting to higher Landau levels. We thus see that the stabilization of a striped phase is in essence a unique manifestation of the uncertainty principle in a magnetic field: electrons that favor confinement in a particular direction will necessarily become unconfined orthogonal to that direction. The most dramatic consequences of the existence of a striped edge phase occur when the overall density of a sample is low enough for the bulk to crystallize [6-81. We find that if the edge potential rises quickly enough, then the striped phase will still be stable, even in the presence of a periodic potential due to the bulk electrons, which tends to reinforce crystallization. Not surprisingly, the confinement potential necessary to stabilize the striped phase is stronger here than in the case of a bulk liquid, as in the FQHE.
61
Thus, we see that interactions among bulk and edge electrons have a non-trivial effect, and that the state of the electrons at the edge tell us something about the state in the bulk (note “B” phase in fig. 1). Physically, one may understand the behavior of the crystal edge in terms of edge melting: for weaker confinement potentials, the edge electrons are locked into the periodicity of the bulk crystal. As one turns up the confinement, the electrons become more localized perpendicular to the edge; they then necessarily become more extended along the edge. For strong enough confinement, the electrons must become completely delocalized along the edge; i.e., the crystal edge effectively melts. Thus, if one can control the confinement potential at the edge (say, by a sufficiently clever gate geometry), then one could in principle force the edge through its melting transition. If current contacts are placed along the edge as well, the transition would be observed as a unique metal-insulator transition, since edge electrons locked into the bulk crystal will not conduct, whereas the striped phase will. It is also a distinct possibility that a moderate voltage bias across a sample in a Wigner crystal state will be sufficient to melt the edge. Thus, if the electrons in samples such as those investigated in ref. [7] truly are crystallized, then the threshold conduction observed in these samples may be due to the opening of an edge channel, rather than a depinning of the entire bulk crystal. Our results are summarized in fig. 1. Here, we have considered a model in which the electro-chemical potential varies as p(x) = and present a phase diagram for (e’/lJ(x/xJ”, the edge state as a function of x0 and cr, both with and without a periodic potential (modeling crystallized bulk electrons) applied at the edge. We note that this phase diagram is presented for the explicit situation in which the spatial extent of the edge is only large enough to contain a single stripe. For large xc,, the situation in which the edge potential varies slowly, the striped phase is always unstable against formation of a Wigner crystal (region C). In region B, the striped phase is stable in the absence of a periodic potential but unstable when a potential of the form Vext(~, y) = V, e-“’ cos(Qy) is applied at the edge, where
A
4c
7
2i’.-, 2
4
6
8
10
u Fig. I. Phase diagram for electrons at edge in presence of a potential p(x) = (r’/l,,Xx/~,,Y. A = striped phase. C = Wigner crystal, B = Wigner crystal in presence of V”‘. striped phase otherwise. (See text.) (0) points at which !IE(~)= 0 for some Q in presence of periodic potential; (X) same. in absence of periodic potential. Lines drawn in are guides to the eye.
we have chosen V,, = 2e”/l,, and A = 1,;’ for concreteness. In region A, the striped phase is stable even in the presence of V”“‘. (The form of V”“’ is chosen for calculational convenience; however, any V”“’ that falls off as x increases and is periodic in y gives a qualitatively similar phase diagram.) We expect that the samples investigated in refs. [6,7] should fall in region B, and it seems likely that for an edge potential of the form p(x), they should lie rather close to the N = 1 line. Our model is as follows: We explicitly consider only electrons at the very edge of the sample, in the low density region. We work in Landau gauge, and project the Hamiltonian into the lowest Landau level (LLL), as is appropriate in the strong field limit. The resulting Hamiltonian is
H=
c ~(k)a:a, x>o +;Jd’r
d’+(r)
x [iV’)
-&(“)I~
-&(+(r-r’)
where e(k) = (k I p(k) I k) = (e2/lo)(kl~/x,,>” the strong field limit, u; creates an electron the state
(1) in in
p(r) = c,,,2~~1(r)~k’(r)a:,a,~ is the density operator projected into the LLL, and l’(r - r’) = e’/ I r - r’ I is the Coulomb potential. The quantity P,,(r) is an explicit background charge density, which must be included to avoid divergences in the energy. For simplicity, we have set p,(r) = z: A_,,, I +k(r) I ‘$f. We have tested models of the form I,!J: = V[1 - k/k,,]fl(k,, - k 1 and I/I:’ = vH(k,, - k 1; the results of both are qualitatively the same. Roughly, one may think of the external potential b(x) as being due to the presence of an external gate, and the neutralizing background represents the remote donors supplying electrons to the sample. Note that WC explicitly exclude states with k < 0 from the background, since WC consider these to lie in the bulk rather than at the edge. So long as the electron charge density is not large at x = 0, this is a consistent approach. If we find, however, that p(x = 0) is large, then one can simply move the origin to a position at which p is small, and recalculate the ground state for the new edge. Our model does not allow us to treat any complicated correlations of the bulk electrons with the edge; however, within Beenakker’s model, in which one is optimizing energy density rather than total energy, this is a consistent approach. For further calculational convenience, we explicitly introduce a second layer, whose only role is to act as a reservoir for electrons at the edge. The operator hi creates electrons in the state &/, in the reservoir. We then consider trial wavefunctions of the form [lo] $0 = n
(+a:
+ ‘.I,@)
IO>,
(2)
h>0
with of + 11: = 1. The energy (+ I H I 4) 3 E,, for the trial wavefunction (2) may be evaluated explicitly; using this form, we have numerically minimized E,, with respect to a finite set of ux’s. A typical example is illustrated in fig. 2, where we have chosen +j = 0.20(k - 8.5/I,,) and p(x) = (e’/1,,)(.~/10)~. Note that to obtain a physically allowable result, we have added a term of the form UI)C k U!~ with U,, and N both very large; X the minimization routine then excludes configurations for which of > 1. We see that within the space of functions defined by cq. (2), the lowest
H.A. Fertig, S. Das Sarma / Behacior of electrons at soft edges in the FQH and WC states I
0.8 0.6 2
'k
Fig. 2. uz versus
1
/
”
k for $i = 0.28C8.5 - kl,,): example striped phase.
I
of a
energy state has sharp stripes in k-space; if one calculates the charge density in real space, it is clear that the result will be uniform in the 9-direction, and a charge density wave in P-direction. From a calculational point of view, it is helpful to note that the striped phase may be characterized by a small number of parameters: one need only know the locations and widths of the stripes in k-space. With this parameterization, one can optimize the energy (for wavefunctions of the form in eq. (2)) in the thermodynamic limit. This will be especially helpful when we wish to check the stability of this phase against Wigner crystallization. To test this stability, we need to go beyond the trial wavefunction in eq. (2), since non-uniformities of the density in the j-direction may only be obtained by admixing different single particle kstates. Towards this end, we look at states of the form tia = p,+cj, where Pa = i(pQ + p_p) and Pa = Ckalak _o. One can show that the effect of PQ on ILo is to introduce periodic, transverse fluctuations in the density along the j-direction. Thus, ifwe find that E(Q)=(I)~IHII+G~)
63
stripe at the edge. In curve (a), we show AE(Q) when p(k) = 0; one can see A E < 0 over a large range of Q. This indicates that the striped phase is unstable against formation of a crystal in the absence of an edge potential, as we know must be if this theory is to make sense. In curve (b), we have recalculated the ground state for E(k) = (k1,/7)*. One can see that AE(Q> > 0, indicating that the striped phase is stable when a strong enough edge potential is present. Clearly, as p(x) is weakened, AE(Q) will eventually touch down near Ql, = 1.5, indicating that the edge electrons undergo a phase transition to a crystalline state. It is interesting to note that one may show for the striped phase that AE(Q) - Q in the long-wavelength limit. The presence of an acoustic mode of this sort has been predicted for the edge excitations in a considerably different model [3]. Finally, we need to consider how the edge phase behaves when the bulk electrons are crystallized. To a first order approximation, we may think of these as applying a periodic potential to the electrons at the edge. For simplicity, we have chosen a potential of the form Vext(x, y) = V,, eehx cos Qy. A potential of this form has no effect on the energy of a state within the set of trial wavefunctions given by eq. (2); however, a contribution is made to the states P,tJ,. Fig. 3c shows an example of AE(Q) in the presence of V ext; one can see that the presence of the periodic potential may destabilize the striped phase.
l.O~
-1.0 1 0.0
0.5
1.0
1.5
QQO
2.0
2.5
3.0
Fig. 3. Excitation energies away from striped phase: (a) k(x) = 0; (b) p(x) = (e2/l,Xx/x,j2 with x,, = 7.01,; (c) as in (b), except with a periodic external potential applied at the edge.
64
H.A. Fertig S. Das Sarma / Behavior of’electrons at soft &es
Fig. 1 illustrates, for several choices of cy, the value of x0 at which the excitation mode becomes soft, both in the presence and absence of I’““‘. In summary, we have developed a theory of the last edge branch in the FQHE in a sample with a slowly varying edge potential. We find that for certain sample parameters, a striped phase may exist, allowing for conduction at the edge. A bulk Wigner crystal can destabilize this phase. The possibility of having an open edge branch while the bulk is crystallized is shown to exist, leading to some interesting experimental implications. We believe that the issue of existence of edge channels for FQHE (and, Wigner crystal) states is a fundamentally important one, and hope that the predictions of this work will motivate experiments of the sort suggested here. The authors would like Donald, A.M. Chang, and discussions. This work was ARO, US-ONR, and the was provided by University
to thank A.H. MacJ.K. Jain for helpful supported by the USNSF. Computer time of Maryland. One of
in the FQH and WC states
the authors (H.A.F.) pitality of the Aspen
also acknowledges the hosCenter for Physics.
References [I] C.W.J. Beenakker, [2] [3] [4] [S]
[6]
[7] [8] [9]
[lo] [ 111
Phys. Rev. Lett. 64 (IYYO) 216. A.11. MacDonald, Phys. Rev. Lett. 64 (1990) 220. X.G. Wen, Phys. Rev. Lett. 64 (1900) 2206. A.M. Chang, Solid State Commun. 74 (1990) X71. A.M. Chang and J.E. Cunningham. Solid State Commun. 72 (lY89) 651: L.P. Kouwenhoven. B.J. van Wees, N.C. van der Vaart. C.J.P.M. Harmans. C.E. Timmering and C.T. Foxon. Phys. Rev. Lett. 64 (l9YO) 65X. P. Levesque, J.J. Weis and A.H. MacDonald, Phys. Rev. B 30 (1984) 1056; P.K. Lam and S.M. Girvin, Phys. Rev. B 30 (19X4) 473. H.W. Jiang, R.L. Willett, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 65 (lY90) 633. V.J. Goldman, M. Santos, M. Shayegan and J.E. Cunningham. Phys. Rev. Lett. 65 (1990) 2189. R. Kubo et al.. Quantum Theory of Galvanometric Effect at Extremely Strong Magnetic Fields, Vol. I7 of Solid State Physics 17, Eds. F. Seitz and D. Turmbull (Academic Press, New York, 1965). H.A. Fertig, Phys. Rev. B 40 (1989) 1087. S.M. Girvin, A.H. MacDonald and P.M. Platzman, Phys. Rev. B 33 (1986) 2481.
:. ,: ,.
Surface Science 263 (1992) 60-64
Behavior of electrons at soft edges in the fractional and Wigner crystal states Fertig
Center for
Theoretical
College Park.
Received
and
Program
fbr Adl~cmceti
for publication
the state of electrons
26 August
an open conducting
bulk, then the edge channel
Electronic
Muterids,
‘;.:I
‘:.:.
..)
.:.;
:,.
..
.‘.:.‘.
.’
.I.
.’
quantum Hall
Department
of
fIly.sic~,\. Utlil,er.sit,,
of ,$!uI).[N~I(I,
1991
near the edge of a two-dimensional
varies slowly on the scale of the magnetic
consequences
.....
USA
31 May 1991: accepted
.striped phase, forming
7.
Sarma
Phpsics and Joint
MD 20742-4111,
We investigate potential
S. Das
:.,’j:j?, :...
kurface science : y. >,.:. jj’.‘:..: ..::. ::.:.,,, ,,,:,:>::: ,,,..:., :,y..: .,.... ..:: ,,.,. .:... ,:>:.: ,.:.y.. ,,:. .. .. ...
North-Holland
H.A.
,...
edge channel.
may be destabilized
electron
gas in a strong magnetic
length. We find that under certain circumstances,
field when the edge
these electrons
We find that if the filling is low enough to form a Wigner
due to interaction
between
bulk and edge electrons.
A number
may favor a crystal in the
of experimental
are discussed
Over the past year, there has been a great deal of interest in the possibility of edge channels existing in systems exhibiting the fractional quantum Hall effect (FQHE) [l-5]. In this work, we analyze a model of a two-dimensional electron gas in a strong magnetic field whose electrochemical potential falls off slowly on the scale of the magnetic length (“soft edge”). This model is probably appropriate for edges created by gated geometries. It has been shown recently by Beenakker [l] that if one assumes that transport at the edges in this model is ballistic, then one may understand the unusual quantizations seen in the experiments [4,5], and indeed understand transport in the FQHE in terms of sample edges. An important inconsistency in the application of the ballistic transport approximation to the soft edge model arises when one considers what happens near the very edge of the sample, where the electron density will be quite low. For bulk electrons at such low densities, one expects the system to form a Wigner crystal [6--S], and hence become pinned in the presence of impurities. In the case of low density electrons at the sample edge, if we follow the prescription of ref. [l], and simply subtract out the portion of current associated with electrons that have crystallized (since 0039.6028/92/$05.00
c
1YY2
Elsevier
Science Publishers
one naively expects them to be pinned), we find that the calculated Hall conductance urV deviates slightly from quantization. This effect has never been observed experimentally, and clearly prcsents a problem with the approach of ref. [ 11. One is thus led to investigate in more detail what state the electrons at the low density end of a soft edge would take. Our calculations described below indicate that the confinement potential at the edge, under certain circumstances, has a very dramatic effect on the low density electrons: the Wigner crystal state becomes unstable against the formation of a striped phase. The striped phase may be described as a charge density wave, with a density profile that oscillates perpendicular to the edge, and is uniform along the edge. When the striped phase is stabilized, one expects the approach of ref. [I] to give the correct quantization in the FQHE, since in the absence of transverse density oscillations there will be no pinning. To understand how a striped phase arises in this system, consider what would happen to a Wigner crystal if its edge is subjected to a confining potential. If we focus on the lines of charge parallel to the sample edge, then for potentials varying faster than linearly, the natural drift vclocity associated with the external potential for
B.V. and Yamada
Science Foundation.
All rights reserved
H.A. Fertig, S. Das Sarma / Behauior of electrons at soft edges in the FQH and WC states
each line increases as one approaches the sample edge. To understand how the system overcomes this tendency to shear, we transform to a frame of reference co-moving with one of these lines. One may show that, in this frame, the edge potential pushes the neighboring lines of charge towards the line with whose drift velocity we are moving. The lines of charge are thus compressed together, until the Coulomb repulsion between them balances the force due to the external potential. The local electric field on each line then vanishes, and in the laboratory frame, all the charges would move together. However, if the system favors a compression of the electrons in the P-direction, it necessarily follows that the single-particle wavefunctions comprising the states we consider here become more extended in the y-direction, so long as we are unwilling to admix states from higher Landau levels (which would be inappropriate in high fields.) This latter result follows from the commutation relations for the guiding center coordination of single particle wavefunctions in a magnetic field [9], [X, Y] = iii. In terms of real coordinates, an electron whose real coordinate (say, JZ:)is confined to within a magnetic length has its guiding center coordinate confined to an infinitesimal interval [9], so that both the guiding center and real coordinates in the f-direction become unconfined. The striped phase is essentially the most compressed state one can get in a particular direction without exciting to higher Landau levels. We thus see that the stabilization of a striped phase is in essence a unique manifestation of the uncertainty principle in a magnetic field: electrons that favor confinement in a particular direction will necessarily become unconfined orthogonal to that direction. The most dramatic consequences of the existence of a striped edge phase occur when the overall density of a sample is low enough for the bulk to crystallize [6-81. We find that if the edge potential rises quickly enough, then the striped phase will still be stable, even in the presence of a periodic potential due to the bulk electrons, which tends to reinforce crystallization. Not surprisingly, the confinement potential necessary to stabilize the striped phase is stronger here than in the case of a bulk liquid, as in the FQHE.
61
Thus, we see that interactions among bulk and edge electrons have a non-trivial effect, and that the state of the electrons at the edge tell us something about the state in the bulk (note “B” phase in fig. 1). Physically, one may understand the behavior of the crystal edge in terms of edge melting: for weaker confinement potentials, the edge electrons are locked into the periodicity of the bulk crystal. As one turns up the confinement, the electrons become more localized perpendicular to the edge; they then necessarily become more extended along the edge. For strong enough confinement, the electrons must become completely delocalized along the edge; i.e., the crystal edge effectively melts. Thus, if one can control the confinement potential at the edge (say, by a sufficiently clever gate geometry), then one could in principle force the edge through its melting transition. If current contacts are placed along the edge as well, the transition would be observed as a unique metal-insulator transition, since edge electrons locked into the bulk crystal will not conduct, whereas the striped phase will. It is also a distinct possibility that a moderate voltage bias across a sample in a Wigner crystal state will be sufficient to melt the edge. Thus, if the electrons in samples such as those investigated in ref. [7] truly are crystallized, then the threshold conduction observed in these samples may be due to the opening of an edge channel, rather than a depinning of the entire bulk crystal. Our results are summarized in fig. 1. Here, we have considered a model in which the electro-chemical potential varies as p(x) = and present a phase diagram for (e’/lJ(x/xJ”, the edge state as a function of x0 and cr, both with and without a periodic potential (modeling crystallized bulk electrons) applied at the edge. We note that this phase diagram is presented for the explicit situation in which the spatial extent of the edge is only large enough to contain a single stripe. For large xc,, the situation in which the edge potential varies slowly, the striped phase is always unstable against formation of a Wigner crystal (region C). In region B, the striped phase is stable in the absence of a periodic potential but unstable when a potential of the form Vext(~, y) = V, e-“’ cos(Qy) is applied at the edge, where
A
4c
7
2i’.-, 2
4
6
8
10
u Fig. I. Phase diagram for electrons at edge in presence of a potential p(x) = (r’/l,,Xx/~,,Y. A = striped phase. C = Wigner crystal, B = Wigner crystal in presence of V”‘. striped phase otherwise. (See text.) (0) points at which !IE(~)= 0 for some Q in presence of periodic potential; (X) same. in absence of periodic potential. Lines drawn in are guides to the eye.
we have chosen V,, = 2e”/l,, and A = 1,;’ for concreteness. In region A, the striped phase is stable even in the presence of V”“‘. (The form of V”“’ is chosen for calculational convenience; however, any V”“’ that falls off as x increases and is periodic in y gives a qualitatively similar phase diagram.) We expect that the samples investigated in refs. [6,7] should fall in region B, and it seems likely that for an edge potential of the form p(x), they should lie rather close to the N = 1 line. Our model is as follows: We explicitly consider only electrons at the very edge of the sample, in the low density region. We work in Landau gauge, and project the Hamiltonian into the lowest Landau level (LLL), as is appropriate in the strong field limit. The resulting Hamiltonian is
H=
c ~(k)a:a, x>o +;Jd’r
d’+(r)
x [iV’)
-&(“)I~
-&(+(r-r’)
where e(k) = (k I p(k) I k) = (e2/lo)(kl~/x,,>” the strong field limit, u; creates an electron the state
(1) in in
p(r) = c,,,2~~1(r)~k’(r)a:,a,~ is the density operator projected into the LLL, and l’(r - r’) = e’/ I r - r’ I is the Coulomb potential. The quantity P,,(r) is an explicit background charge density, which must be included to avoid divergences in the energy. For simplicity, we have set p,(r) = z: A_,,, I +k(r) I ‘$f. We have tested models of the form I,!J: = V[1 - k/k,,]fl(k,, - k 1 and I/I:’ = vH(k,, - k 1; the results of both are qualitatively the same. Roughly, one may think of the external potential b(x) as being due to the presence of an external gate, and the neutralizing background represents the remote donors supplying electrons to the sample. Note that WC explicitly exclude states with k < 0 from the background, since WC consider these to lie in the bulk rather than at the edge. So long as the electron charge density is not large at x = 0, this is a consistent approach. If we find, however, that p(x = 0) is large, then one can simply move the origin to a position at which p is small, and recalculate the ground state for the new edge. Our model does not allow us to treat any complicated correlations of the bulk electrons with the edge; however, within Beenakker’s model, in which one is optimizing energy density rather than total energy, this is a consistent approach. For further calculational convenience, we explicitly introduce a second layer, whose only role is to act as a reservoir for electrons at the edge. The operator hi creates electrons in the state &/, in the reservoir. We then consider trial wavefunctions of the form [lo] $0 = n
(+a:
+ ‘.I,@)
IO>,
(2)
h>0
with of + 11: = 1. The energy (+ I H I 4) 3 E,, for the trial wavefunction (2) may be evaluated explicitly; using this form, we have numerically minimized E,, with respect to a finite set of ux’s. A typical example is illustrated in fig. 2, where we have chosen +j = 0.20(k - 8.5/I,,) and p(x) = (e’/1,,)(.~/10)~. Note that to obtain a physically allowable result, we have added a term of the form UI)C k U!~ with U,, and N both very large; X the minimization routine then excludes configurations for which of > 1. We see that within the space of functions defined by cq. (2), the lowest
H.A. Fertig, S. Das Sarma / Behacior of electrons at soft edges in the FQH and WC states I
0.8 0.6 2
'k
Fig. 2. uz versus
1
/
”
k for $i = 0.28C8.5 - kl,,): example striped phase.
I
of a
energy state has sharp stripes in k-space; if one calculates the charge density in real space, it is clear that the result will be uniform in the 9-direction, and a charge density wave in P-direction. From a calculational point of view, it is helpful to note that the striped phase may be characterized by a small number of parameters: one need only know the locations and widths of the stripes in k-space. With this parameterization, one can optimize the energy (for wavefunctions of the form in eq. (2)) in the thermodynamic limit. This will be especially helpful when we wish to check the stability of this phase against Wigner crystallization. To test this stability, we need to go beyond the trial wavefunction in eq. (2), since non-uniformities of the density in the j-direction may only be obtained by admixing different single particle kstates. Towards this end, we look at states of the form tia = p,+cj, where Pa = i(pQ + p_p) and Pa = Ckalak _o. One can show that the effect of PQ on ILo is to introduce periodic, transverse fluctuations in the density along the j-direction. Thus, ifwe find that E(Q)=(I)~IHII+G~)
63
stripe at the edge. In curve (a), we show AE(Q) when p(k) = 0; one can see A E < 0 over a large range of Q. This indicates that the striped phase is unstable against formation of a crystal in the absence of an edge potential, as we know must be if this theory is to make sense. In curve (b), we have recalculated the ground state for E(k) = (k1,/7)*. One can see that AE(Q> > 0, indicating that the striped phase is stable when a strong enough edge potential is present. Clearly, as p(x) is weakened, AE(Q) will eventually touch down near Ql, = 1.5, indicating that the edge electrons undergo a phase transition to a crystalline state. It is interesting to note that one may show for the striped phase that AE(Q) - Q in the long-wavelength limit. The presence of an acoustic mode of this sort has been predicted for the edge excitations in a considerably different model [3]. Finally, we need to consider how the edge phase behaves when the bulk electrons are crystallized. To a first order approximation, we may think of these as applying a periodic potential to the electrons at the edge. For simplicity, we have chosen a potential of the form Vext(x, y) = V,, eehx cos Qy. A potential of this form has no effect on the energy of a state within the set of trial wavefunctions given by eq. (2); however, a contribution is made to the states P,tJ,. Fig. 3c shows an example of AE(Q) in the presence of V ext; one can see that the presence of the periodic potential may destabilize the striped phase.
l.O~
-1.0 1 0.0
0.5
1.0
1.5
QQO
2.0
2.5
3.0
Fig. 3. Excitation energies away from striped phase: (a) k(x) = 0; (b) p(x) = (e2/l,Xx/x,j2 with x,, = 7.01,; (c) as in (b), except with a periodic external potential applied at the edge.
64
H.A. Fertig S. Das Sarma / Behavior of’electrons at soft &es
Fig. 1 illustrates, for several choices of cy, the value of x0 at which the excitation mode becomes soft, both in the presence and absence of I’““‘. In summary, we have developed a theory of the last edge branch in the FQHE in a sample with a slowly varying edge potential. We find that for certain sample parameters, a striped phase may exist, allowing for conduction at the edge. A bulk Wigner crystal can destabilize this phase. The possibility of having an open edge branch while the bulk is crystallized is shown to exist, leading to some interesting experimental implications. We believe that the issue of existence of edge channels for FQHE (and, Wigner crystal) states is a fundamentally important one, and hope that the predictions of this work will motivate experiments of the sort suggested here. The authors would like Donald, A.M. Chang, and discussions. This work was ARO, US-ONR, and the was provided by University
to thank A.H. MacJ.K. Jain for helpful supported by the USNSF. Computer time of Maryland. One of
in the FQH and WC states
the authors (H.A.F.) pitality of the Aspen
also acknowledges the hosCenter for Physics.
References [I] C.W.J. Beenakker, [2] [3] [4] [S]
[6]
[7] [8] [9]
[lo] [ 111
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