PHYSICAL REVIEW B VOLUME 32, NUMBER 12 15 DECEMBER 1985

Two-dimensional polaron in a magnetic field

Wu Xiaoguang, F. M. Peeters, and J. T. Devreese*University ofAntwerp (Universitaire Instelling Antwerpen), Department ofPhysics, Universiteitsplein j,

B 26IO-Antwerpen (Wilrij k), Belgium(Received 24 April 1985)

The ground-state energy of a Frohlich optical polaron confined to two dimensions, placed in aperpendicular magnetic field is calculated within the Feynman path-integral approach. TheFeynman-model mass, the magnetization and the susceptibility are calculated as a function of themagnetic field strength for different values of the electron-phonon coupling. We find that withinthe generalized Feynman approximation the polaron exhibits a discontinuous transition from adressed state to a stripped state if the electron-phonon constant a is larger than 1.60. For a & 1.60,the transition occurs continuously with increasing magnetic field.

I. INTRODUCTION

In the present paper we study the Frohlich optical pola-ron in a constant magnetic field which is perpendicular tothe plane in which the motion of the electron is confined.This ideal two-dimensional (2D) system has been studiedrecently by several authors' in the limit for smallelectron-phonon interaction. In the present paper we willconcentrate on the ground-state property of the 2D pola-ron system for arbitrary electron-phonon coupling and ar-bitrary magnetic field strength. We will use Feynman'spath-integral approach of the polaron to calculate theground-state energy, the mass of the Feynman polaronmodel, the magnetization, and the susceptibility as a func-tion of both the electron-phonon coupling and the mag-netic field strength. Results for the Gaussian approxima-tion will also be given. The motivation for using theFeynman approximation lies in the fact that for thethree-dimensional (3D) optical polaron in the presence ofa magnetic field it was found by two of the present au-thors that the Feynman path-integral method is superiorto all the other existing approaches. For nonzero magnet-ic field the Feynman approach leads to lower values forthe polaron ground-state energy than obtained from othertheories. Unfortunately there exists no mathematicalproof that, in the case of a nonzero magnetic field, theground-state energy in the Feynman approximation is anupper bound to the exact ground-state energy. But in Ap-pendix A of Ref. 6 strong arguments were given, based onphysical intuition, in favor of the upper-bound nature ofthe polaron ground-state energy calculated in the Feyn-man approximation when a magnetic field is present.

For zero magnetic field the Feynman approximationhas been applied to the 2D polaron problem. The re-sults show that both ground-state energy and effectivemass are continuous functions of the electron-polaroncoupling constant o.. The same conclusion is true for the3D optical polaron. However, in Ref. 6 the influence ofa magnetic field was examined: the ground-state proper-ties of a 3D polaron were studied and it was found thatthere exists a transition of the Feynman polaron from a

dressed polaron state to a stripped polaron state with in-creasing magnetic field strength. In the present work wewill investigate whether or not a similar magnetic-field-induced transition can be found for the 2D polaron.

The organization of the present paper is as follows. InSec. II the problem is formulated, the approximations areoutlined, and the ground-state energy is calculated. Ana-lytic results for limiting values of a and co, are presentedand discussed in Sec. III together with our numericaldata. The conclusions are presented in Sec. IV.

II. FORMULATION AND APPROXIMATION

The Hamiltonian describing the 2D polaron in a mag-netic field is given by'

'2

H= p+ —A. + gmakak1 e2&l C

s=~, +swhere

(2)

S,= f du {r (u)+ico, [y'(u)x(u) —x(u)y(u)] I2m

(3)

+ g( Ir&akeik r+ y&+akte—ik r)

k

where Vk =fico, (v 2trct/Sk)'~ (fi/mco, )'~ . p and r arethe electron momentum and positron operators, respec-tively, ak (ak) is the creation (annihilation) operator of anoptical phonon with wave vector k and energy Ace„a isthe electron-phonon coupling constant, S is the volume ofthe 2D crystal, and A is the vector potential which is tak-en in the symmetrical gauge: A= —,B(—y, x,o).

In the well-known Feynman path-integral representa-tion of the partition function the phonon variables can beeliminated exactly. After this elimination each electronpath contributes e '"~ to the path integral, with the ac-tion

32 7964

32 TWO-DIMENSIONAL POLARON IN A MAGNETIC FIELD 7965

is the action of a free electron in a magnetic field withco, =eB/mc the cyclotron frequency of the noninteractingelectron, P= 1/ktt T is the inverse temperature, and

P PSt ———g i Vk i f du f du'G„(u —u')

)& expI ik [r(u) —r(u')] J

(4)

The variational ground state is thenP

3 3 21 2 2 sndnE=—, gsn —w —(v —w )

n=1 , (w+s„)

where' 1/2

1 00

a dt2 2 o VD(t)

(10)

is the action which contains a memory effect as a conse-quence of the elimination of the phonons. G„(u )= —,'n(co)(e" ~" ~+e"'~ ~" ~') is the Green's function ofthe free phonon system with n(co) =(e"~ 1—) ' the num-ber of phonons with energy %co. In the following, for con-venience, we will use units such that A'=m =co, = 1.

Since the path integral with action (2) cannot be solvedexactly, we will follow Feynman' in order to get an ap-proximation to the exact ground-state energy. For the 3Dpolaron and in the absence of a magnetic field Feynmanintroduced a trial action So to derive a variational upperbound to the exact ground-state energy ( Es ), i.e.,Ee &E =ED A —B, whe—re Eo is the ground-state energycorresponding to the trial action while A and B are func-tions which depend on the variational parameters of thetrial action So. We will use a similar approximation inthe present paper. In the case when a magnetic field ispresent the action is complex and we no longer have aproof that the approximate ground-state energy is anupper bound to the exact polaron ground-state energy. InRef. 6 intuitive arguments were given which suggest thatfor a particular choice of the trial action, namely for atrial action derivable from a Hermitian Hamiltonian, theFeynman inequality is still valid.

The trial action we choose is

So=s.+Swhere

w v2 — 2

S~t = f du f du G~(u —u ')

with

3

D(t)= g d„(1—e ")n=1

and

1 S~ —W2 2

3s„+2( —1)"s„co,—v

For co, —+0 Eq. (10) reduced to the familiar result given inRefs. 7 and 8. For given electron-phonon coupling con-stant a and magnetic field strength co„ the parameters v

and w are determined by a minimalization of E withrespect to u and w.

III. RESULTS AND DISCUSSION

It seems impossible to derived from Eq. (10) an expres-sion in a closed form for the ground-state energy. Onlyfor some limiting values of a and co, analytic results canbe obtained.

-In the small-coupling limit ix«1 we have u=w andthe ground-state energy becomes

an ~co, I ( 1+1/co, )(12)

I ( —'+1/co, )

which was recently obtained in Refs. 3 and 5 withsecond-order perturbation theory. For small magneticfields co, «1, the following series expansion results fromEq. (12):

X [r(u) —r(u')]' (6)

2ma ~c ~c2 8 128

(13)

and S, is given by Eq. (3). The action So can be obtainedfrom the Hermitian Hamiltonian (see Ref. 6)

'2

Ho= p+ —A +, + —,'K(r —r')1 e (p')(7)

2m ' c 2fPl

after eliminating the variables of the fictitious particle(r', p'). Here we have defined the usual Feynman varia-tional parameters u and w. The Hamiltonian Ho can bediagonalized exactly,

1HO= g Sn(CnCn+ 2 )n=1

(8)

with the eigenfrequencies s„ to be determined from thesolution of the third-order algebraic equation in s„(fordetails we refer to Ref. 6)

COc

2

a+n coc ann' ln22 ~co&

Note that the electron-phonon correction to the ground-state energy is proportional to a~co, which is muchlarger than for the 3D case where the correction goes as

c.For large electron-phonon coupling a&~1 and weak

magnetic fields co, «1 we have v »w and the ground-state energy, after minimalization with respect to v and w,can be reduced to

1 c2u' '

In the strong-magnetic-field case we have (v —w)/v «1, and consequently we may again start from v =win Eq. (10). Performing an expansion of Eq. (12) for largechic values we obtain the ground-state energy

7966 WU XIAOGUANG, F. M. PEETERS, AND J. T. DEVREESE 32

3.0 =«~ (X=1.65

K=1.60

0.

GaussianFeynman

C)~ 0.26-

0.25 '=

2P — K=1.55 g

f ~&a ~I

1P I I i

0.27-I

I

I

I

II

llI

I

~gy ~ II

~lg~ I--+(x=1.65~ a=i.m;I I

/a. i.ss ,

'

1 I i

7

P 06

0.5

I

II I

I.Ig

I

I

K=1,55'

r%J

mmmm+

I

0.2—CQ

CL0.1—LJ

1.9 2.0 2.1 2.2 2.3

~/~FIG. 1. The mass of the Feynman polaron model, the mag-

netization, and the susceptibility are shown as functions of co„near the critical point a =1.60,

u= 3.5

where the variational parameters are given by w =1 andu =(ir/4)a —(4 ln2 —1). The terms —(m. /8)a—2 ln2 ——,

' are the ground-state energy of a 2D polaron inthe strong-coupling limit in the absence of a magneticfield. This result can be obtained directly from the corre-sponding 3D result by using a scaling argument intro-duced in Ref. 8. co, /2u is the zero-point energy of a par-ticle with mass u =(na/4) place. d in a magnetic field.

For arbitrary strength of the electron-phonon couplingconstant and the magnetic field the minimalization of Ewith respect to u and w has been performed numerically.We have calculated numerically the mass of the Feynmanpolaron model M = (u/w), the magnetizationp= —BE/Bco„and the susceptibility X= BE/Bco, fo—rseveral values of the coupling constant a and the magnet-

0.03;

0.02-',II

I

001 -'I1

i

QI

II

II'ai -0.01

CL

-0.02t/l

IIIIIIII

IIIIIIIIIIIIIIIIIIIIllI II II II

II

GaussianFeynman

()3 I I I I I I l

0 1 2 3 0 5 6 7 8 9 10

(dc/4)s

FIG. 3. Magnetization as a function of co, for a=3.5 and 4.0in the case of the generalized Feynman approximation (thicksolid curve) and for the Gaussian approximation {thin solidcurve). Behavior of the metastable state is indicated by thedashed curves.

4

4

~ +s'ear JA':: 715

I I i%I Ir I I

-0.03

-Q.05—

l

//

l

/// /

///

0 1 2 3 4 5 6 7 8 9w liu

FIG. 2. Feynman polaron mass as a function of co, fora=3.5 and 4.0.

I I I I I I I I

0 1 2 3 4 5 6 7 8 9 10

(dc ]u)s

FIG. 4. Same as Fig. 3, but now for the susceptibility.

TWO-DIMENSIONAL POLARON IN A MAGNETIC FIELD 7967

0.1 0.2 0.3 0./+

l

0.5 0.6

refer to Refs. 6 and 13 for further details on the physicsof this "polaron stripping transition. "

The discontinuity in the polaron state is shown muchmore clearly in the behavior of the magnetization IM andthe susceptibility X. At the point where the discontinuoustransition starts to occur, i.e., a = 1.60+0.01,co,, =2.10+0.01 (see Fig. 1), p is still continuous, but Xdiverges which is reminiscent of a critical point where thetransition is of second order. For Iz & 1.60, both IM and Xare discontinuous at a certain value of the magnetic field(see Figs. 1—4). The transition behavior of the polaron issummarized in Fig. 5 where the magnetic field at whichthe transition occurs is plotted versus the inverse of theelectron-phonon coupling constant.

The effect of the electron-phonon interaction on theproperties of the polaron are enhanced in two dimensions

W/wc

FIG. 5. Phase diagram for the polaron in the generalizedFeynman approximation plotted vs the inverse of the electron-phonon coupling strength (1/a) and the inverse of the magneticfield strength (1/co, ).

ic field strength co, . In Figs. 1—6, the magnetizationp, =p —p, and the susceptibility X=X—X, are referred tothe free-electron values p, = —0.5 and +, =0. Conse-quently /T, and X are purely a consequence of the electron-phonon interaction.

For a &1.60 (see Fig. 1) the ground-state, the model-mass M =(u/w), the magnetization, and the susceptibili-ty of the 2D polaron are continuous functions of theelectron-phonon constant a and the magnetic field co, .However, for a & 1.60 the polaron undergoes a transitionfrom a "dressed state" [M =(U/w) »1] to a "strippedstate" [M=(U/w) 1]. The transition is similar to thatfound for the 3D-polaron (see Ref. 6) case and for theproblem of an electron on a liquid helium film. ' We

( U/w)EGaussian

a=0. 1

0.00.1

0.20.40.60.81.01.52.04.0

10.0

—0.157 54—0.109 51—0.061 49

0.034 510.130490.226470.322 470.562 610.803 001.767 594.682 67

—0.15708—0.10905—0.061 05

0.034 920.130860.226 800.322 750.562 800.803 131.767 634.682 67

1.0361.0391.0421.0461.0451.0431.0381.0221.0141.0041.000

a=1.0

TABLE I. Polaron ground-state energy for a=0. 1, 1.0, and4 for different values of the magnetic field strength within thegeneralized Feynman approximation (E„, ,„,) and within theGaussian approximation (EG,„„;,„). The mass of the Feynmanpolaron model M = (u /w) is also given.

a, /co, Epresent

0.00.1

0.20.40.60.81.01.52.04.0

10.0

—1.623 22—1.592 02—1.560 80—1.498 15—1.435 09—1.371 52—1.307 39—1.145 16—0.983 30—0.327 48

1.826 27

—1.570 80—1.540 55—1.51049—1.450 81—1.39140—1.33201—1.27245—1.122 04—0.968 70—0.323 74

1.826 71

1.5301.5651.6041.6741.7371.7831.7951.5111.2211.0391.005

p I I M I I I I I I

0 1 2 3 4 5 6 7 8 9

FIG. 6. Phase diagram for the polaron as obtained in thegeneralized Feynman approximation and in the Gaussian ap-proximation.

0.00.1

0.20.40.60.8

'

1.01.52.04.0

10.0

—8.2074—8.2067—8.2057—8.2031—8.1996—8.1952—8.1899—8.1728—8.1502—8.0090—7.7004

a=4.0—7.6945—7.6944—7.6940—7.6927—7.6904—7.6872—7.6832—7.6691—7.6496—7.5194—6.7286

76.7876.8877.0277.3377.6978.1078.5779.9381.5990.77

1.021

WU XIAOGUANG, F. M. PEETERS, AND J. T. DEVREESE

as compared to the 3D case (see Refs. 2—5). As a conse-quence the critical electron-phonon coupling constant andcritical magnetic field strength, at which the transitionstarts to become discontinuous, are smaller in 2D than inthe case of the 3D polaron, where it was found thata =4.20 and co, =2.24. However, in terms of thebehavior of the 2D polaron and the 3D polaron at thetransition point, there are no qualitative differences.

For comparison, the ground-state energy, the magneti-zation (see Fig. 3), and the susceptibility (see Fig. 4) arealso calculated for the Gaussian approximation which isobtained from Eq. (10) by putting w =0, in which case weare left with only one variational parameter. In Table Iwe compare the resulting values for the ground-state ener-

gy with the results from the generalized Feynman approx-imation presented here. It is obvious that our generalizedFeynman approximation gives lower values for theground-state energy than the Gaussian approximation. Itis interesting to note that for a & 2.48 the ground-state en-ergy obtained from the Gaussian approximation is, for all

identical to the result given by second-orderRayleigh-Schrodinger perturbation theory [this result isgiven by Eq. (12)]. The reason is that for a&2.48 thevariational parameter U in the Gaussian approximation iszero for all values of the magnetic field.

In Fig. 6 we compare the discontinuous behavior ofboth approximations with each other. Note that theGaussian approximation gives, as a function of a, a first-order transition for all values of co„even for co, =0. Asdiscussed in Ref. 8 (see also Ref. 9) this behavior is an ar-tifact of the Gaussian approximation.

IV. CONCLUSION

We have recently found in Ref. 8 that the Feynman ap-proximation to the polaron ground-state energy in 2D(EzD ) can be obtained from the 3D result ( E» ) by usingthe scaling relation EzD(a) = —,E3D(3na/4). This scalingrelation breaks down when a magnetic field is applied, thereason being that the scaling relation results from the factthat in the Feynman approximation for co, =0 the com-ponents of the motion of the electron in the differentspace directions do not couple with each other and aretreated within the same approximation. But a magneticfield introduces a special direction into the problem anddestroys this spherical symmetry which as a consequenceimplies that E2D can no longer be obtained from E3D bythe scaling relation of Ref. 8.

The "stripping transition" of the polaron found in thepresent paper is analogous to the one found for the 3D po-laron by two of the authors in Ref. 8. Recently a similartransition was found for an electron on a liquid-heliumfilm. ' There the electrons interact with the surface exci-tations of the liquid-helium film which are called ripplons(this problems can be mapped into a 2D acoustic polaronproblem). For such a system it was found in Ref. 13 thatthe effect of the transition is much more dramatic and

could lead to a change in the model mass M =(v/w) of5—6 orders of magnitude.

The discontinuity of the polaron state found in 2D fora&1.60 and which occurs at a well-defined magneticfield strength is obtained within a generalization of theFeynman approximation as presented in Ref. 6 for the 3Dpolaron. One may argue that this discontinuity can be anartifact of the present approximation and may not be aproperty of the Frohlich Hamiltonian. Concerning thispoint one may reflect on the discussion on the absence ofa discontinuous self-trapping transition of the Frohlichpolaron as was presented in Ref. 9 (see also Ref. 14).Note also that within a variation scheme it is impossibleto prove the existence of a phase transition.

In conc1usion, on physical grounds, it is intuitively clearthat as a function of the magnetic field there should betwo regions: (i) a relatively small magnetic field region inwhich the polaron as an effective particle moves as a rigidentity, and (ii) a high magnetic field region in which theelectron oscillates so rapidly that the phonon cloud is nolonger able to follow the electron. Whether or not thetransition from the dressed polaron state to the strippedpolaron state is continuous or discontinuous is open todiscussion. We found that within the generalized Feyn-man approximation the transition is continuous for ma-terials with a & 1.60 and discontinuous when a & 1.60.

The fact that confinement of the electron motion to twospace dimensions enhances the polaron effects in compar-ison with polaron motion in 3D implies that the criticalcoupling in 2D, a=1.60, is reduced in comparison withthe critical coupling in 3D where a=4.20. As a conse-quence a larger number of materials are available inwhich, from the present study, we expect that the polaronundergoes a stripping transition as a function of the mag-netic field if the electron motion is confined to 2D. Forexample, the present ca1culation predicts that for a hetero-structure of AgC1 (a=1.84) a discontinuity occurs forco /coLQ 2.6 or at a magnetic field of about B—140 T.In this calculation we neglected the finite width of theelectron layer and temperature effects (which is expectedto be valid at helium temperatures). The transition shouldbe visible in a cyclotron resonance experiment in whichwe expect a sudden narrowing of the cyclotron resonanceline due to the sudden decrease in the effective electron-phonon interaction. At the transition point, the polaroncyclotron resonance spectrum transforms to the spectrumof a quasifree electron in a magnetic field.

ACKNOWLEDGMENTS

One of the authors (F.M.P.) acknowledges support fromthe National Fund for Scientific Research (Belgium).%'.X. wishes to thank the Ministry of Education of Chinaand the International Culture Co-operation of Belgiumfor financial support. This work is partially sponsored byF.K.F.O. (Fonds voor Kollektief Fundamenteel Onder-zoek, Belgium), under Project No. 2.0072.80.

32 TWO-DIMENSIONAL POLARON IN A MAGNETIC FIELD 7969

'Also at University of Antwerp (Rijksuniversitair CentrumAntwerpen), B-2020 Antwerpen (Wilrijk), Belgium, andDepartment of Physics; Eindhoven University of Technology,NL-5600 MB Eindhoven, The Netherlands.

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