Coulomb Ordering in Anderson-Localized Electron Systems

4 Pages·0.13 MB·

by  A. A. Slutskin

Coulomb Ordering in Anderson-Localized Electron Systems.
A. A. Slutskin1, M. Pepper2, and H. A. Kovtun1.
1Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkov, Ukraine.
2Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
WeconsideranelectronsystemunderconditionsofstrongAndersonlocalization,takingintoaccount
2 interelectronlong-rangeCoulombrepulsion. Wehaveestablishedthatwiththeelectrondensitygoing
0 to zero the Coulomb interaction brings the arrangement of the Anderson localized electrons closer
0 and closer to an ideal (Wigner) crystal lattice, provided thetemperatureis sufficiently low and the
2 dimension of the system is >1. The ordering occurs despite the fact that a random spread of the

n energy levels ofthelocalized one-electron states, exceedingthemean Coulomb energy perelectron,
a renders it impossible the electrons to be self-localized due to their mutual Coulomb repulsion This
J differsprincipallytheCoulomborderedAndersonlocalizedelectronsystem(COALES)fromWigner

1 crystal,Wignerglass, andanyotherordered electronorholesystemthatresultsfromtheCoulomb
2 self-localization of electrons/holes. The residual disorder inherent to COALES is found to bring
aboutamulti-valleyground-statedegenerationakintothatinspinglass. Withtheelectrondensity
] increasing, COALES is revealed to turn into Wigner glass or a glassy state of a Fermi-glass type
n
n dependingon thewidth of the random spread of theelectron levels.
-
s PACS number(s): 71.45.-d, 72.15.Rn, 73.20.-r, 73.40.-c
i
d
a. Introduction. As was shown by Wigner long type of Coulomb ordering that occurs under conditions
.
at ago1, slow decrease in the Coulomb electron-electronin- of a strong Anderson localization. Specifically, we deal
m teraction potential v(r) = e2/κr (e is the free electron withAndersonlocalizedelectronsystemswhereintheex-
charge,r is a distance between the interacting electrons, ternal random field localizing the electrons is a random
-
d κ is the permittivity) with an increase in the interelec- setofpotentialwells that aresufficiently deep forthe lo-
n tron distance inevitably causes the Coulomb energy of calization length rL to be <∼ the mean wells separation,
co free electron gas to exceed its kinetic energy at suffi- r0 ∼n−01/d (n0 isthedensityofthewells,disthedimen-
[ cientlylowelectrondensitieswiththeresultingtransition sion of the system). We show further that at sufficiently
3 othfethweakgeasoifnWtoigannere’lsecptrroednicctriyosntaal n(Watuigrnaelrqcureyssttioanl).wIans ltouwrese,leTct,rtohnedmenustiutaielsC, noue,loamnbdrseupffiuclsiieonntlyoflothwetleomcapliezread-
v
raised whether long-range (weakly screened) Coulomb electrons inevitably forces them to be arranged close to
1
9 forces can lead to ordering of charge carrier ensembles the WCL sites, provided d > 1. Such Coulomb ordered
0 in conductors. However, a strong evidence of ”Wigner- Anderson localized electron system (COALES) differs
0 crystal-in-crystal”existencehasnotbeenfoundyet. This qualitativelyfrombothWignercrystalandOSLCCLSfor
1 suggests that the Wigner crystallization, at least in the the following reason. An electron/hole self-localization
1 conventialconductors, is very difficult (if at all possible) underlyingformationofWignercrystalorOSLCCLSoc-
0 to observe in pure form. Therefore, seeking mechanisms cursonly iftheCoulomb energy of mutual charge carrier
/
t of charge carrier Coulomb ordering that are beyond the repulsion is the prevailing one. In the electron systems
a
m above Wigner’s scenario is of great interest. undergoing strong Anderson localization this is not the
At present much attention is being given to an elec- case,asthe width ofrandom spread of the electron levels
d- tron/hole Coulomb self-localization in lattice systems in the wells, ∆ (the disorder energy) is much more than
n with so small overlapintergalt that tunnelling of charge the mean Coulombenergyper electronε ∼v(r¯). Under
c
o carriers is supressed by their mutual Coulomb interac- theseconditionsaCoulombself-localizationisimpossible
v:c dtieornin.gTihnetwseolfc-laosceasl:izi/atthioenhborsitnlgasttaicbeoiustreelgeucltarron(g/ehnoelreaollry- scihnacnegaesshtihfteoefleacntreolnecternoenrgoyvemruacshmmalolrdeistthaannceε∼.rB0e≪lowr¯
i anincommensurateelectron/holestructureisformed2,3), we describe a new mechanism by which Coulomcb order-
X
ii/the hostlattice is disordered,butthe meanseparation ing occurs despite the inequality ε ≪∆.
c
ar of its sites is much less than that of the charge carri- The smallness of εc as compared with ∆ is believed
ers r¯(the so-called Wigner glass is formed, whose space to be a good reasonto describe an influence of Coulomb
structure, though disordered, is close to a Wigner crys- electron-electron interaction on the systems with strong
tal lattice (WCL)4 in a sense5). Ordered self-localized Anderson localization in terms of Fermi glass,6 a ran-
charge carrier lattice systems (OSLCCLS) of both types dom system of the electrons occupying all wells with the
are obviously not the same as Wigner crystal. In ad- energies ≤ the Fermi energy ε . Up to now, it is the
F
dition, unlike the Wigner crystallization, the higher is Fermi-glass approximation modified with regard to ex-
the electron/hole density the weaker are dynamic effects istence of the so-called Coulomb gap (Efros-Shklovskii
caused by the charge carrier tunnelling. gap)7 at the Fermi energy has been the basic approach
The present paper aims to draw attention to a new to the problem. COALES existence at low n is beyond
e
1
the scopeofthe Fermi-glassconcept. This indicatesthat Weaimtofindthestructureoftheelectronconfiguration
the convential line of reasoning should be revised. R that minimizes E(R).
g
COALES posseses a peculiar “duality”: similar to c. Electron Ordering and Fermi-Glass Instabil-
Wignercrystal,itarisesasn →0,while itsspacestruc- ity. First let us discuss how mutual electron repulsion
e
ture similar to that of Wigner glass. As will be seen affects the ground state of the Fermi-glass in the limit
from the following, such “duality” not only differs CO- n → 0. As follows from Eq.(1), n is linear in ε at
e e F
ALES from both Wigner crystal and Wigner glass but ε ≪∆:
F
imparts it a number of new features that make COA-
LESanappealingsubjectofinvestigation. We alsoshow ne =n0εF/∆,
that with an increase in n COALES turns into Fermi
glass or OSLCCLS dependieng on a value of the param- On the other hand, εc is proportional to n1e/d. Hence,
d−1
eter γ = ∆/v(r0), governing an interplay between the the ratio εF/εc ∝ ned tends to zero (for d > 1) with
random spread of the electron levels and the Coulomb a decrease in n . What this means is that Fermi glass
e
interelectron repulsion. The residual disorder inherent withsufficientlylown isunstablewithrespecttomutual
e
to COALES is described here in terms of dipole glass, Coulomb repulsion of electrons.
a random system of interacting dipoles that issue from If the electrons were free to move and ε(R~) = 0, the
theWCLsites. Thedipolerepresentationofthedisorder ground-state configuration R would be a WCL. Since
g
proves to be helpful for revealing a multi-valley degener- the electrons of Fermi glass are randomly arranged, the
ation of the COALES ground state reminiscent of that Coulomb energy per electron of the Fermi-glass exceeds
in the spin glasses. that of the WCL, ε0 ∼ εc, significantly (i.e. by a value
b. The BasicAssumptions. Owingtotheinequal- ∼ ε ). This suggests that for sufficiently low n the
c e
ity rL <∼ r0 the radius-vectors R~ of the centers of the configuration Rg falls into a class of R that meet the
random-potential wells can be considered as quantum folowing conditions: i/ for each WCL site there is an
numbersoftheone-electronlocalizedstates,|R~ >. Their electron located in a small neighborhood of the site,
energies ε(R~) are random values, the structure of the R~ ii/theupperboundεb oftheelectronenergiesinthewells
set is assumed to be arbitrary, in particular, it can be ε(R~1),...,ε(R~N) satisfies the inequalities
regular. Without essential loss of generality, the density
ν of the number of ε(R~) with a given value ε can be εF ≪εb ≪εc, (4)
presented in the form
The energyper electronε =E(R)/N ofsuchRis near
R
ν =n0f(ε)/∆, (1) ε0. Itcannotbe lessthanε0 asthemeanelectronenergy
in the wells ε =E /N ≥ε .
w w F
where f(ε) is ∼ 1 within the interval [εmin, εmin + ∆] Anyconfigurationoftheaboveclassbelongstotheset
and equals zero outside it; εmin is the least of the ε(R~). of points R~ for which ε(R~) ≤ ε . The density of these
b
Further we put εmin =0, f(0)=1. points equals n0(εb/∆), and their mean separation
Here it is assumed that r¯ ≫ r0. In particular, this
allows to neglect perturbations produced in the eigen- ρ(εb)∼r0(∆/εb)1/d, (5)
states of the system by the electron-electron interaction
(withaccuracytoadditions∼(r0/r¯)2exp(−r0/rL)≪1). is much less than the WCL spacing a0 ∼ r¯owing to the
In this approximation, the eigenstates can be identified first of the inequalities (4). Therefore, for each WCL
with site m~ there are inevitably severalR~ of the setsuch that
|R>=|R~1 >|R~2 >...|R~N > . (2) |(Ro~n−e me~le|c∼troρn. Ppeorpusliatet)inygietlhdesseju“sptrothxiemcaotnefi”gsutraatteison|R~s o>f
whereR=R~1,...,R~N denotesasetofthewellsoccupied the class we are interested in. For such a configuration
by the electrons, N is the number of the electrons. The the energy εR is the sum
complete set of the eigenstates comprises all possible R
with different R~1...R~N. With the same accuracy, the εR =ε0+δε, (6a)
eigenvalues E(R) corresponding to the states |R> take
where δε is expected to be a small correction to ε0. It
the form
consists of two terms:
E(R)=Ec(R)+Ew(R), (3a) δε=a (ρ/r¯)2ε0+b(r0/ρ)d∆ (6b)
where E and E are the energy of the mutual electron
c w
The first term is the deformation energy produced by
repulsionandtheenergyofthenon-interactingAnderson
electrondisplacementsoverdistances∼ρfromtheWCL
localized electrons in the wells, respectively:
sites. The second term is ε expressed in terms of ρ
w
Ec = 1 XN v(|R~i−R~k|), Ew =XN ε(R~i); (3b) ain=viae(wR)o,fbE=qb.((5R))adnedpetnhde ofanctdetthaailtsεowf R∼, bεub.t tFheayctaorres
2
i,k=1 i=1 both ∼1 for any of the considered configurations.
i6=k
2
The ground-state correction, δεg, to ε0 is the least of Heff = X Jm~m~ ′sm~sm~ ′ + Xh(sm~), (10)
theδεvalues. Puttinga,b=1intheexpression(6b)and m~6=m~ ′ m~
finding its minimum in ρ, we obtain the estimate
where s are the independent variables, the “exchange
m~
δεg ∼γd+22 (r0/r¯)2(dd+−21) ε0, (7) integral” Jm~m~ ′ = d20Λαα′(m~ − m~ ′)emα~emα~′′ is a random
matrix, and h(sm~)=ε˜(d0sm~~em~) plays the role of an ex-
the minimum being reached at ternal random field. The system with the Hamiltonian
(10), being a special case of spin glass8, shares with the
1 d−1 1
ρ=ρ¯∼γd+2 (r0/r¯)d+2 a0 ∼(∆/δEc)d+2 r0, (8) spinglassestheir knowngeneralproperty: amulti-valley
ground-state degeneration. It can be described in terms
where γ is the parameter defined in item a; the physical
of “N-excitations” that are s configurations differing
sense of the energy δEc = (r0/r¯)2εc will be explained from the ground-state one bym~a big number N of spin
below (item e). The quantity ρ¯ characterizes deviation
flips. The multi-valley degeneration implies that separa-
of R from WCL.
g tion of the low bound of the N-excitations energy spec-
Expressions (7) and (8) show that δεg/ε0 and ρ¯/a0 trumfromthe ground-stateenergyis∼thetypicalsepa-
both go to zero when r¯/r0 → ∞. Hence, for suffi- rationbetweentheneighboringenergiesofthespectrum,
ciently low n the ground-state electron configuration
e δE ∝ N/Z (Z is the total number of N-excitations
N
is WCL slightly perturbed by random electron displace-
with a given N, lnZ ∝ N), and in consequence of this,
ments from the WCL sites. This is just the COALES
is exponentially small in N.
mentionedintheitema. ThespacestructureofCOALES
ThetruedipoleHamiltonian(9)differsfromthemodel
differsfromthatofWignerglass5 atanimportantpoint:
one(10)onlyinthateachdipoled~ runsthroughagiven
the typical displacement of the electrons/holes from the m~
finiterandomsetcontainingmorethantwovectors. (The
WCL sites in Wigner glass is of order of the geometri-
vectorsofthesetaresuchthattheirmoduliarecompara-
cal constant of the system, the mean host-lattice sites
blewithρ¯). Therefore,theabovepropertiesoftheenergy
separation, while that in COALES, ρ¯, depends not only
spectrum of the N-excitations (they are dipole configu-
on the geometrical parameter r0 but also on both the
rations with N dipoles other than in the ground state)
disorder energy ∆ and the electron density.
holdinthe generalcase. Inotherwords,themulti-valley
d. DipoleGlass. InordertodojusticetotheCOA-
degeneration does take place in the dipole system, and
LES ground state it is necessary to describe the residual
hence, in COALES.
disorder inherentto this system, extending the consider-
e. The Region of COALES Existence. Expres-
ation to cover all R that are close to R . To this end, it
is convenientto introduce the effective Hg amiltonian Heff sion(8) shows that with an increase in ne the ratio ρ¯/a0
increases, while ρ¯ itself decreases. This brings about,
by the formula E(R) = Nε0 +Heff, expressing Heff in
depending on a value of the parameter γ, two different
terms of dipoles d~ = R~ −m~ as independent variables
m~ scenariosofwhathappenswithCOALESasn increases.
e
(m~ are the radius-vectorsof the WCL sites) whose mod-
uli are ≪ a0. Expanding Ec in powers of dmα~ (index α Fwoitrhsunffi,cireenatclhyesbiigneγvitthaeblryatsioomρ¯e/ac0r,itiinccarlevaasliunegβtog<et1h/e2r
e
enumeratesthecomponentsofvectors)andrestrictingto
at which the length of the space correlations in the sys-
quadratic in d~m~ terms, it is easy to obtain temis∼r¯,andCOALESturns(supposedly,byasecond
order transition) into a glassy state. This occurs at
1
Heff =Xε˜(d~m~) + 2 X Λαα′(m~ −m~ ′)dmα~dmα~′′, (9)
m~ m~,m~ ′ ne =ne1 ∼βcd(dd−+12)γ−d−d1n0.
where matrix Λ (m~) = |m~|−3(δ − 3m m /|m~|2)
αα′ αα′ α α′ At this point the ratio ε /ε is ∼ βd+2, i.e. it is signif-
determines the interaction of two dipoles issuing from F c
sites0,m~;thefunctionε˜(d~m~)=ε(m~ +d~m~)+Λ¯αα′dmα~dmα~′, aicanntlryanlegses tohvaenr w1.hiTchhetrheeforgel,asfsoyr nsteat≥e ncae1nnthoterbeeexaidstes-
the matrix Λ¯ = − Λ (m~); here and further on e
αα′ Pm~ αα′ quatelydescribedintermsofthe notionofFermienergy.
summation over repeated indexes α,α′ is implied, the
A further decrease in n leads to transformation of such
sums ofm~,m~ ′ are takenoverallWCL sites. The second e
“non-Fermi”glass into conventional Fermi glass (modi-
terminε˜(d~ )istheenergyofinteractionofagivendipole
m~ fied by the Coulomb electron-electron interactions) only
withWCL;thesecondterminEq.(9)istheenergyofthe
d−1
dipole-dipole interaction. if γ ≥1. Otherwise, εF/εc ∼ γ(ne/n0) d is less than 1
The distinctive feature of Heff is that the dipoles are for any ne, and non-Fermi glass exists for all ne ≥ne1.
If γ is sufficiently small, an increase in n reduces ρ¯
variables taking on their values on a given random set. e
Thissuggeststhattheconsidereddipolesystemhasmuch down to its least possible value, r0, the ratio ρ¯/a0 re-
in common with the spin glass8. The analogy becomes maining ≪ 1. This takes place at ∆ ∼ δEc, or in terms
clear for the simplest model in which d~m~ = d0sm~~em~, of ne, at ne = ne2 ∼ γd3n0. The parameter δEc is the
typical change in the Coulomb energy of the system as
asm~gi=ven±1ra,nd0doismasceotn,sHtaenfftt,aaknidngunthitevfeocrtmors~em~ constitute an electron is shifted over distance ∼ r0. Therefore, for
3
δE > ∆ the mutual electron repulsion dominates the — n-type GaAs — p-type GaAs with charge transfer
c
randomspreadoftheelectronlevels,causinglocalization in an impurity band.9 It is distinguished by pronounced
of the electrons by itself. In other words, in the con- regular-typeoscillationsinn ,whichcannotbeexplained
e
sidered case the state into which COALES turns as n in conventional one-electron terms and most likely are
e
increases is nothing but the above-mentionedOSLCCLS a manifestation of two-dimensional COALES or OSLC-
whose host lattice is a regular or disordered R~ set. CLS. We intend to provide new evidence to confirm the
It follows from the aforesaid that the region of COA- suggestionandtospecifytheelectronstructureinnearest
LES existence on the n ,γ-plane is restricted by γ-axis future.
e
and two curves, ne = ne1(γ), ne = nc1(γ) (ne1(γ) → 0 The main outlines of COALES mentioned here are to
as γ → ∞; ne2(0) = 0). They intersect at point be issues of our further detailed publications.
ne = n¯e ∼ βdn0,γ = γ¯ ∼ β3, the value n¯e being the We gratefuly acknowledge interesting and useful dis-
maximal n at which COALES exists. cussions with I.Lernerand I.Yurkevich. We are sincerely
e
Heating affects COALES if T exceeds δε , both terms thankful to B.Shklovskii for helpful remarks.
g
in Eq.(6b) (ρ = ρ¯) being ∼ T. This gives the dipole The visit of A. Slutskin to the Cavendish Laboratory
thermal-fluctuations amplitude d ∼ r¯(ε /T)1/2, the was supported by the Royal Society. AS would like to
T c
fluctuating dipole vector taking ∼ (T/δε )1+d/2 values. express his gratitude.
g
As T increases, dT becomes ∼ a0, and at some critical
T ∼ε COALES turns into a glassy state.
c
f. Some General Features of COALES. Macro-
scopically, COALES manifests both electron crystal and
glassy-state features.
Due to the proximity of the COALES space struc-
ture to WCL the low bound of the energy spectrum of 1E. Wigner, Trans. Farad. Soc., 34, 678 (1938)
one-electron excitations produced by displacements of 2HubbardJ., Phys.Rev.B 17, 494 (1978).
an electron over distances ≫ r¯ is separated from the 3A.A.Slutskin,V.V.Slavin,andH.E.Kovtun,Phys.Rev.B
ground state energy by a big gap ∼ ε . This renders 61, 14184 (2000)
c 4To avoid confusion, it should be noted that here and fur-
one-electron variable-range hopping over COALES im-
ther on we use this term, for the sake of shortness, only to
possible. Low-temperature conduction in COALES is
designate the electron/hole ideal crystal lattice that would
by multi-electron exchange processes of a creep type (a
realizetheabsoluteminimumoftheCoulombelectron/hole
power dependence of the COALES conductivity σ on T
repulsion energy if theelectrons/holes were free tomove.
isexpected)orbytransferofchargedpointdefects(posi- 5M.S.Bello, E.I.Levin,B.I.Shklovskii,A.L.EfrosJETP53,
tivelychargedWCL vacanciesandinterstitialelectrons),
822(1981).WerefertothepaperinwhichWignerglasswas
σ being proportional to their concentration that in its
predicted, though the term itself appeared, to our knowl-
turn is ∝ exp((ε −ε )/T) (ε and ε > ε are the en-
v i v i v edge, later.
ergies of vacancy and interstitial-electron formation re- 6N.F. Mott, E.A. Davis, Electron Processes in Non-
spectively; εi−εv <εc). Thus,σ byno means obeysthe crystalline Materials (Clarendon Press, Oxford,1979)
known Mott’s low6 for Fermi glass. 7A.L.Efros, B.I. Shklovskii, J. Phys. C8, L49 (1975)
The multi-valley degenerationofthe dipole glass,sim- 8K.Binder, A.P. Young,Rev. Mod. Phys. 58, 801 (1986)
ilarly to that in spin glass8, is bound to cause an infinite 9M. Pepper, J. Phys. C 12, L617 (1979)
spectrumofrelaxationtimes,and,inconsequenceofthis,
anon-ergodicbehaviorofCOALES.Thiscanberevealed
by observation of different relaxation processes in COA-
LESproceedingforanomalouslylongtimes. Anexample
is relaxation of a non-equilibrium COALES polarization
createdinonewayoranother. Thishasmuchincommon
withrelaxationofanon-equilibriummagneticmomentin
spin glass.8
Asfollowsfromtheaforesaid,undermoderate-disorder
conditions (∆<∼v(r0)) COALEScan exist up to ne that
areonlyseveraltimeslessthann0. Thepropermaterials
toobserveCOALESarevariousamorphousnarrow-band
conductors, superlattices, and inversion layers wherein
n can be varied within wide limits without affecting
e
the disorder. Favorable conditions for two-dimensional
COALES existence are expected to be realized in semi-
conductor superlattices with the so-called δ-layers. Of
special interest is a conductive sheet in a system metal
4