2D electron gas in the magnetic field

8/13/2019 2D electron gas in the magnetic field

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ANNALS OF PHYSICS 97, 216-241 (1976)

Quantum Theory of Longitudinal Dielectric ResponseProperties of a Two-Dimensional Plasma in a

Magnetic FieldNORMAN J. MORGETNSTERN HORING AND MUSA M. YILDIZ*

Department of Physics and Cryogenics Center, Stevens Institute of Technology,Hoboken, New Jersey 07030Received September 19, 1974

An analysis of dynamic and nonlocal longitudinal dielectric response properties of atwo-dimensional Landau-quantized plasma is carried out, using a thermodynamicGreens function formulation of the RPA with a two-dimensional thermal Greensfunction for electron propagation in a magnetic field developed in closed form. Thelongitudinal-electrostatic plasmon dispersion relation is discussed in the low wave-number regime with nonlocal corrections, and Bernstein mode structure is studied forarbitrary wavenumber. All regimes of magnetic field strength and statistics are in-vestigated. The class of integrals treated here should have broad applicability in othertwo-dimensional and finite slab plasma studies.The two-dimensional static shielding law in a magnetic field is analyzed for lowwavenumber, and for large distances we find k(f) - Q/kOV. The inverse screeninglength k, = 2aezap/Q (p = density, [ = chemical potential) is evaluated in all regimesof magnetic field strength and all statistical regimes. k, exhibits violent DHVA oscillatorybehavior in the degenerate zero-temperature case at higher field strengths, and theshielding is complete when [ = rhw, but there is no shielding when [ # rfi iw, , Acareful analysis confirms that there is no shielding at large distances in the degeneratequantum strong field limit fiw, > 1. Since shielding does persist in the nondegeneratequantum strong field limit hw, > KT, there should be a pronounced change in physicalproperties that depend on shielding if the system is driven through a high field statisticaltransition. (It should be noted that the static shielding law of semiclassical and classicalmodels has no dependence on magnetic field in two dimensions, as in three dimensions.)Finally, we find that the zero field two-dimensional Friedel-Kohn wiggle staticshielding phenomenon is destroyed by the dispersal of the zero field continuum of electronstates into the discrete set of Landau-quantized orbitals due to the imposition of themagnetic field.

* Present address: Department of Mechanical Engineering, University of New Hampshire,Durham, New Hampshire 03824.216Copyright 0 1976 by Academ ic Press, Inc.

All rights of reproduction in any for m reserved.


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2-D LANDAU QUANTIZED PLASMA 2171. INTRODUCTION

The theory of two-dimensional systems has become a subject of considerableinterest over the past few years, particularly in connection with the physicalproperties of inversion layers and thin films [l-5]. Several nvestigations concerningthe effects of a magnetic field on a two-dimensional plasma have been carried out,and the structure of the dynamic nonlocal RPA magnetoconductivity tensor hasbeen derived [6-81. The work reported here s concerned with a detailed analysis ofthe longitudinal dielectric responseproperties of a two-dimensional electron plasmasubject to Landau quantization, and it has two specific purposes. The first is toexplore the physical ramifications of the two-dimensional dielectric responsefunction in terms of the longitudinal-electrostatic plasmon dispersion relation(including nonlocality, Bernstein modesand natural damping) and also n terms ofstatic shielding as modified by Landau quantization due to the magnetic field. Thisstudy of longitudinal dielectric phenomena is undertaken using a thermodynamicGreens function formulation of the RPA [9], and the two-dimensional Greensfunction for electron propagation in a magnetic field is developed here in closedform [lo]. The second purpose is to facilitate the extraction of information des-cribing the low/intermediate field case, as well as the high field case, andthe various statistical regimes (degenerate, nondegenerate, etc.) in a relativelysimple form. The classof integrals analyzed for this purpose will also be useful inthe detailed exploration of phenomena nvolving the full two-dimensional magneto-conductivity tensor [l I], and will have further application in the theory of a finiteslab of quantum plasma n a magnetic field [lo].

NOTATION: ? = (x, 7) and T; = (x, v) are two-dimensional position vectorsalong plane; E = (k, , k,) = two-dimensional wavevector along plane; r = / r 1k = I E I; v(i - i) = Coulomb potential, Q = frequency; w, = cyclotron fre-quency = eH/mc; H = magnetic field perpendicular to plane; m = effectivemass; p = two-dimensional equilibrium density (per unit area); fO(w) = Fermi-Dirac distribution function = (1 + exp(w - 5) /3)-l; 4 = chemical potential;/3 = (thermal energy)-l = (KT)-I; p,, = eii/2mc (g = anomalous g-factor= 1); 7+(X) = unit step function = 1 for X > 0, + for X = 0, and 0 for X ( 0;a3 = Pauli spin matrix = (t -3. (Note that we employ an integration variable T,which should not be confused with temperature T, which occurs n p = (KT)-l.)II. LONGITUDINAL DIELECTRIC PROPERTIES OF A TWO-DIMENSIONAL QUANTUM

PLASMA IN MAGNETIC FIELDThe analysis of the longitudinal dielectric properties of a two-dimensionalquantum plasma n a magnetic field will be carried out here using a thermodynamic


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218 HORING AND YILDIZGreens function formulation of the random phase approximation (RPA). Theprocedures and techniques employed will closely follow those of the correspondinganalysis of the three-dimensional quantum plasma carried out by Horing [9] (wewill use the same notation, as explained in Section I). For electron motion confinedto the two-dimensional x-y plane (z = 0), the effective potential V(1) at a space-time point on the plane (1 = (x, y; t)) is linearly related to the impressed potentialU(2) at other space-time points on the plane (2 = (x, y; t)) as V(1) =J 43 m 2) UC% or K(1,2) = 6 V(l)/SU(2). This serves to define the two-dimensional longitudinal dynamic nonlocal inverse dielectric function K(1, 2). Asa consequence of thermal averaging, K( 1, 2) possesses al l of the spectral representa-tions and the Fourier series representation normally associated with a bosonthermodynamic Greens function, which are given explicitly in [9, p. lo]. (Thewavevector variables E that appear in the spectral representations, must be under-stood in our present context as being two-dimensional wavevector variablesconjugate to the two-dimensional position difference variables through two-dimensional spatial Fourier transforms. We shall understand this to be the case,henceforth, and shall adjust all spatial Fourier transforms to two-dimensionalitywithout explicit comment in the remainder of this work). Employing the randomphase approximation (RPA) for K(1,2) with space arguments restricted to the two-dimensional plane, we have spatial translational invariance along the plane, andmay solve the RPA integral equation with a single spatial Fourier transform7 - F -- f R: The result for the Fourier series coefficient X(E, V) is obtained fol-lowing the procedures of [9], and noting that the two-dimensional Fourier trans-form of the Coulomb potential v is 2vre2/k, we find [lo]

where G refers to the uncorrelated two-dimensional thermodynamic Greensfunction for one electron propagation subject to Landau quantization due to themagnetic field (or its component) perpendicular to the plane. Further, followingthe development of [9], the physical inverse dielectric function given byX(k, Sz + ic) may be expressed as[X(k i-2 + k)]-1 = E(E, 52 + k)

= 1 - y Im Z (k; $ [a + ic])

(2)


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2-D LANDAU QUAN TIZED PLASMA 219where E(E, Sz + ic) is the direct physical dielectric function, and the RPA free-electron polarizability may be identified as

with4%-o@, Q) = l (~,A-2 k) - 1 = (i27+/k) I(/%; (+r)[SZ + ic]), (3)

andRe 4~r~y(Iz,Q) = -(2re2/k) Im Z(g; (T/T)& + in]), (44

Im Lhra(E, L?) = (2ne2/k) Re I@; (+)[J2 + in]). (4b)Here, I(E; (T/v)[.~ + in]) denotes a two-dimensional analog of the correspondingthree-dimensional ring diagram integral in [9], and it is given by

I(k; v) = IT dt e-invt/T j (&7/(277)2) c&j; -t) G>(ij - E; t),0

andwhere

r(E; (T/rr)[SZ + k]) = --p;* +JT< , (6)

-a,= cos

dt e-i(a-ie)t j (d2q/(27r)2) G&; -r) C,(S - E; t), (740

-a;= OI dt e-i(R-4t s (&j/(2+) G,(q; 4) G


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220 HORING AND YILD IZand G is effectively spatially translationally invariant and is independent of thearbitrary gauge function 4. This corresponds to the fact that the magnetic fielddeprives the system of spatial translational invariance because the momentumvector is not conserved in the circular motion of the electrons. However, the can-cellation of the factors C(F, i) C(F, F) = 1 in the formulas above reflects the factthat the dielectric properties of the system are indeed spatially translationallyinvariant for a homogeneous magnetic field, as well as being independent of gauge.This is to say that it is G that should be used in place of G in the formulas above.Since the constraint to planar two-dimensional motion eliminates the degree offreedom for motion perpendicular to the plane, the equation for G takes the form(see [lo, Paragraph I.31 and [9, Eq. II, 1 a; ti - 11)

whereand

x = x - x; y = y - y'; R zz.z _ f; T = t _ t

and Lz=f(X7&Y& 1The spectral weight A (R, T) of e satisfies the corresponding homogeneousequation subject to the condition 2 (8, T = 0) = 6(R), and the solution for thetwo-dimensional Greens function is given by [lo, Paragraph I.31

(10)With this result, we may now construct the integrals 2$ in the form

3, = Lw dT 1 -$& j g e-i(S2+w-w-ir)T *f,(w)(l - fO(w)) Q(ww; li), (lla)$< = s_, dT j 2 1 f$ e-i(S2+w-w+ir)T +fO(o)(l - Jo(w)) Q(ow; I;), (llb)

whereQ(wo; k) = jfm dT lm dT .@T&wTp(TT; r;>, (12)-a --m


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2-D LANDAU QUANTIZED PLASMA 221andP(TT; k) = Tr Js (&%/(277)) e-ioHa3(T+r) . sec(w,ir/2) sec(o,T/2)

. exp[(--ik2/mw,) tan(w,ir/2)] exp[-i(E - E)2 tan(w,r/2)/mw,],(13)

which can be shown to result in [9, lo]

Introducing the integration variables x = T + T; 4 = T - T, we then obtainQ(ww'; k) = it" dx jim dy e~(w+~)r12 . ei(w-w)v12 . 471i ;; x,2)u-02 --iD c

I- ii;2- exp ~. cos(w,y/2) - cos(w,x/2) )mu, 2 sin(w,x/2) ) (15)

which is real. Forming I$, (~/rr)[Q + k]) = -Y>* + &, we note that the termsin the integrands involving double distributions fo(w)fo(w) cancel out, andobtain

= jam dT j g j f&& eiG+w-w+ic)T . [fo(w)which may be recast in the form

= ; Lrn & j & j g (@--w+M - ei(R+~+i4T) .fo(w) . R(ww; j& (17)

whereR(wo; I;> = Q(w, o + w; k) - Q(w, w - w; E) = real

= j-r dx 1-T dy eiwZ 2i sin (q) e+Ju/z Eli tziCx,2)

* exp I-ik2 cos(w,y/2) - cos(w,x/2) {~ mw, 2 sin(w,x/2) ) (18)


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2-D LANDAU QUANTIZED PLASMAmining the long wavelength behavior of the two-dimensionalConsidering Eq. (20), we have

.s+= dy eiUZ g sin ( fi;x) mrh,-02 e- I * 47ri tan(fiw,x/2)

c - ilE%-.. exp 2mw, cos(w, y/2) - cos(?h,x/2) 1in(fiw,x/2) (23)

223plasmon spectrum.

and expansion of the integrand in powers of (E2fi/2mw,) along with a subsequentbinomial expansion in powers of exp[&iw,y/2] yields the y- integral as a row ofS(w - [integer] WJ functions, so that the w- integral is immediate and we obtainthe result-Re 47r(~(i&) = (F) f & (s) i (r,) + i [i)

n=0 7=0 rl=o4[2q - r] w,

* 322 - ([2q - r] co,)2 *mfiw,

(1 n

4rri tan@o,x/2) i sin(#iw,x/2) 1. (-,,, ( fiyx ))+r . i sin(Pq -;I fi~G9 .

For low wavenumber, the first few terms are given by

where

and

-Re 47&, Q) = (= - 277e21E20 Rm2wc2 522 - WC2 1+ 27re2k2a (

Em2wc2 J-22 - 4cLJ,2 1

(24)

(25)

(26)

(27)It is readily verified that p is the two-dimensional density (per unit area) in accor-dance with N = p(Area) = --i Tr sd? G&r, F; T = 0). It is at once evident from595/97/l-15


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224 HORING AND YILDIZthe structure of Re &a(& Q) that the low wavenumber plasmon spectrum fortwo dimensions will include a local principal mode [13] given by

i-2,2 = co,2 + 2rre2p k/m,and nonlocal corrections that modify the principal mode and introduce anotherbranch of the Bernstein mode type into the plasmon spectrum in the vicinity ofD - 20, . A more careful analysis of the wavenumber dependence of

Re 47x@, a),which will be discussed below, reveals structure that will induce a Bernstein modebranch into the plasmon spectrum near each multiple of wc, 8 N nw, . (Suchstructure is already in evidence in the full wavenumber power series given above.)The effects of Landau quantization are embodied in integrals such as p and u.Considered as a general class, we are dealing with integrals of the form

where j(s) has isolated singularities (including the possibility of isolated essentialsingularities as well as poles of finite order) at S, = ii2m/&o, and is periodicwith period s1 such thatj(s) =j(s + s,) =j(~ + s,). Deforming the contour of thes-integral to encircle the isolated singularities for w > 0,

where we have used the translation s = z + s, , and have invoked the periodicityof j(s). (q+(w) is the unit step function.) The n-sum may be recognized as aFourier series that is periodic under the shift w --f w - rfiwe, and since itrepresents S(o/fiwJ in the fundamental interval, we haver]+(o) y ems = q+(w) F ei2nnwfioC = hw,~+(w) F S(w - rfmc). (30)

n=-m 12=--m p=--m

Evaluating the w-integral for J, we obtain(31)


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2-D LANDAU QUAN TIZED PLASMA 225This structure is readily interpreted in terms of Landau quantization, since it isclearly composed of discrete contributions arising from the discrete set of Landaueigenstages. It is particularly useful at high fields fiw, N 5 when only a few Landaustates are occupied since only a few corresponding terms of the r-sum yieldnontrivial contributions. It should be noted that the discrete nature of the contribu-tions is uniquely associated with the fact that we have a completely quantizedsituation in the two-dimensional plasma in magnetic field, since there is no con-tinuum of states associated with kinetic energy of motion parallel to the fie ld (thereis no such motion), and hence, the individual electron states describing circularmotion on the plane have a finite energy separation of hw, . The spin-splitting hasbeen taken equal to the Landau level separation, and this results in a doubledegeneracy for states corresponding to r 3 1, while the ground state r = 0 isnondegenerate, which is reflected in the fact that q+(rfiw,) = 1 for r 3 1, whereasq+(O) = 4 for r = 0. (Of course there are no contributions for r < 0.) Al l of thisis evident in the density of states D(w) which may be identified from the s-integralof the density p as

cc= 2h.0, C 7j+(r iwJ S(w - r fiwc) fdz

pieTf iWCZ . m f i w ,

V = O 45~ tanh(fiw,z/2)= *) ,r,j+(rrbJ S(w - rh,). (32)It is interesting to note that the general integral J may be expressed in terms ofD(w) as

J = (n/m) lrn dwf,(w) D(w) jco (dz/2&) ewj(z)).

The structure of D(w) indicates that there are no electron states between thediscrete Landau-quantized states at r hwc , so there is no continuum aspect to thespectrum of uncorrelated electron states. For r > 1, y+(rh,) = 1 weights thedoubly degenerate states twice as heavily as does v+(O) = 4 for the nondegenerateground state. The corresponding expression for the density p is given by

(33)


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226 HORING AND YILD IZSimilar considerations yield the integral u as

which is clearly proportional to the average Landau orbital energy since theindividual Landau state energy is ~9%~ .The integration technique presented above is exact and most useful at highfield AU, N 5, but it is inconvenient at low/intermediate fields fiio, < [ when manyLandau levels are occupied, so we will develop another procedure for low field [ 141.Considering J and integrating by parts with respect to w we have

(35)We shall restrict our considerations to the zero temperature degenerate case forwhichf,(w) = -S(o - 5) (but it should be noted that finite temperature correc-tions could be evaluated using this procedure if the restriction were lifted), whence,

where we have deformed the contour to encircle the isolated singularities, andseparated of f the integral about the origin c,, since the nature of the n = 0 singular-ity is different from the others n # 0 because of the factor (l/s). Introducing thetranslation s = z + s, and invoking the periodicity ofj(s), the integrals about c,becomeC - 1 eEsn $

nfo ?a#0 -ST eZj(z) = C fP $co(or,gln) 5 &.Wd,c,(origin) hi z + sn n#O (37)where we have introduced the definition j(z/+) = j(z) noting that the periodicityof j (period sl) implies that it depends on z (or s) through z/s1 (or s/s& alone.Setting x = tz, we have

(38)


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2-D LANDAU QUAN TIZED PLASMA 227For low-intermediate fields & - t/fiwC > 1, so we may approximate

x -I- & - fsnin the denominator as indicated above, and evaluate the remaining x-integral(which is independent of n) using just the leading term of the Laurent expansion ofj(x/&) about the origin . Now, the n-sum has characteristic de Haas-van Alphen(DHVA) oscillatory behavior, and can be summed in closed form as follows[15, TISP, p. 38, No. 1.441.1]

(39)

where Kr - JlPlp er is (Z - Y)/2 in the fundamental interval 0 < Y -C 2~ and isperiodically repeated outside this interval because of its Fourier sine series represen-tation above. This yields the DHVA terms in closed form as a periodic saw-toothfunction of .$/iiw,

Turning our attention to the n = 0 term, (x = &r here),

which may be evaluated for low-intermediate fields & - [/fiw, > 1 using just theleading term of the Laurent expansion ofj(x/&). It is to be expected that thisleading contribution should correspond to a semiclassical model in which quantummagnetic field effects are ignored and two-dimensional dynamics are governed byclassical dynamics subject to initial averaging with a Fermi distribution (so thatthere will be no Landau quantization effects, although classical magnetic fieldeffects will arise through the circular nature of the classical electron trajectories).It should be noted that in the low field limit, the DHVA terms (corresponding ton # 0 above) oscillate very rapidly, averaging to naught. In summary, we havefor low-intermediate field strengths,


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228 HORING AND YILD IZFor convenience, we note that the two final integrals here are related by

Applying this to the evaluation of p at low-intermediate fie ld strengths, we have(sl = 2+&J,)

andj(s) = (2/h2)(m~w,/4n)[tanh(irrs/s3]-l = j(~/sJ,

(43)Considering the integral u,

j(s) = (mwc2/4-rr>[tanh(iz-.s/S1)]-2 =j(s/s,),we obtain the result

(44)Finally, we note that in the nondegenerate case,&(w) -+ en?eVwp,so that the w-

and s-integrals are Laplace transform and inverse. The evaluation of J is immediate

and the corresponding nondegenerate result for p is2ecR

(m?iw,P=- )i2 4x tanh(fiw$/2) *

The nondegenerate evaluation of o is

For the classical lim it, #k~J3/2 < 1, and we have

(45)

(47)


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2-D LANDAU QUANTIZED PLASMA 229IV. ANALYSIS OF BERNSTEIN MODE PLASMON RESONANCE STRUCTURE FOR

ARBITRARY WAVENUMBERThe Q-frequency poles of Re 49~cu(E,Q) at all integral multiples of the cyclotronfrequency, which are manifested in the low wavenumber power expansion (Eq. (24),induce additional roots of the plasmon dispersion relation in the nature of Bernsteinmode plasmon resonances. There is one such Bernstein mode branch of the plasmonspectrum near each multiple of wg, and the low wavenumber behavior of theBernstein mode near 2w, is readily obtained from Eq. (25). We shall carry out theanalysis of such structure for arbitrary wavenumber here, taking full account of

Landau quantization effects. Considering N as given by Eq. (22) and expanding theT-integrand in a series of modified Bessel functions In(x) according to the identityexp(x cos wJ) = CT=-m e~*lWJlJ~), one finds that the T-integral is elementary sothat Eq. (21) leads to the result

where the prime on C means that the n = 0 term should be divided by 2 to avoiddouble-counting, and .Fn) is defined by

(Proper handling of the y1= 0 term is particularly important in the static limit(0 + 0) discusssd in Section VII.) The calculation of Z(@ for low-intermediatemagnetic field strength in the zero-temperature degenerate case may be undertakenusing Eq. (42). This calls for an evaluation of the integral

and since tsl N f/Awe > I, we may employ the leading term of the Laurentexpansion ofj(x/&) with the semiclassical result


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230 HORING AND YILD IZ[17, BIT I, p. 280, No. 151. Differentiation of this integral with respect to ayields the integral coefficient of the DHVA oscillatory part of Eq. (42), andthen setting a = 1, we obtain the result

z(n)= f (J, [(2x)12])z + &Y$ = - ylfiw,lper. & (s)12 J, [(S)]. p-1 [(S, ] - J,+1(ggI/. (51)

For higher field strengths, the calculation of .Zcn) may be undertaken usingEq. (31), which calls for an evaluation of the integral (LWJJ = x).

It is not possible to make approximations of this integral with respect to fiw,/f, andit is too difficult to carry out exactly. However, one may introduce a low wave-number approximation for Z, when that is appropriate, with the result,

The exact evaluation of these integrals yields agreement with the low wavenumberterms of Section III for n = 1,2, and further development of this type can provideinformation concerning low wavenumber behavior for arbitrary n.To lift the restriction to low wavenumber, we shall introduce a generally usefulalternative procedure [lo, 181 for analyzing Ztn) at higher field strengths andarbitrary wavenumber. Considering the definition of Ztn) above (Eq. 50), weemploy the identitye*p [

#iis n= 2 (,,,I exp [- &I sinh(fiw,s/2)


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232 HORING AND YILDIZIn the zero temperature degenerate case, the w-integral is elementary, resulting in(note that p = rnf/rr for the zero fie ld, zero temperature degenerate case)

-Re ~TK@, Q) = - $?$ /g + p-2 - k2/2m]. 71+ (-$ [$ - q - f)- [I - 2k2f/m 1g - Q IL]1 + (SZ +

The static limi t of this is given by (Q -+ 0)-Re 4nol(k, J2 = 0) = y (v+ (1 - A$-)(1 - $)l

-

-

8). (54a)

11 (54b)Eq. (54), in agreement with the zero-field two-dimensional studies in the literature[l-5]. The corresponding nondegenerate zero field result is readily obtained usingEq. (53) withy,(w) --f e@e- WB,which leads to a complementary error function.It is also of interest to determine Im 4~a(E, Q) in the zero fie ld limi t. EmployingEq. (4b), (19) and R(wSZ; E) as determined above with wC ---f 0 (and A + 1 here)we obtainIm 477ol(l;, Q

(554= 271 ( ;z2 )12 5 (kl) f dwh(w) . v+ (w - (m/2k2)P F k2/2m12) .(w - (m/2k2)[Q T kz/2m]2)1/2

Wb)In the nondegenerate case the first form yields (Eq. 45)

e2m Srrm Ii2Im 47x@, Sz) = et0 2k (F) 1pip c ktl) esc~ca/2,-cr2/sm,- (~~~/z~~)) (564+which represents two-dimensional Landau damping [5] (with quantumcorrections). In the degenrate case of zero temperature, the second form yields

Im 4~-a(lT, S) = y ($)l 2 (Al) 77+ (5 - +$ [Q F $12)f. ( 5 - $ [Q =F &]2y2, (56b)

which represents two-dimensional plasmon decay into electron-hole pairs. The


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2-D LANDAU QUANTIZED PLASMA 233reality of the square root is explicitly ensured here by the unit step function,y+(f - e..), whose significance is that plasmon decay cannot occur unless theplasmon-photon momentum is large enough to satisfy the requirements of energy-momentum conservation.

VI. PLASMON DAMPINGThe evaluation of plasmon damping devolves upon the analysis of F(li, Q),

T(k, Q) = 2 Im &ol(K ) = 2(27re2/k) Re ,I(& (7/r)[Q + ie]),Using Eqs. (18) and (19). Expanding the y-integrand of R(w, Q; E) in a series ofmodified Bessel functions I,(x) according to

exp(x cos(w,y/2)) = f einwcrizl~(x),n=-mit is readily seen that the y-integral yields 6(Q f nw,)-functions, and that theremaining integrals define Z@), which appeared above in Section IV. Thus, weobtain

T(E, 52) = (237e2/k)(2mw,) jJ Zcn)[S(Q - n4 - S(Q + w)l. (57)?L=lIn view of the fact that -Re 4~@, A?) has poles as a function of Sz at +nw, , noplasmon root of 1 = -Re 47rc@, Q) can lie on any multiple of w, . Hence,r(E, L?) --f 0 for such plasmon roots, and there is no plasmon damping for finitemagnetic field. However, it is important to note that in the zero field limit the manyundamped Bernstein mode plasmon roots cluster together and a phase averagingtakes place, in which they lose their identity as distinct plasmon resonances, andonly the principal plasmon mode survives intact. In this same limit, the phaseaveraging process dictates that the sum over the row of&O & nw,)-functions abovein r(K, C?) be replaced by an integral, which yields a finite result for T(I%, 8) andcorresponding nontrival damping for the principal plasmon mode in the zero fieldlimit. Explicit results for zero field plasmon damping may be obtained fromEqs. (55) and (56) using F(E, J2) = 2 Im 47rar(E,Sz).

VII. STATIC SHIELDINGThe study of static shielding (Eq. (8)) calls for detailed information concerningthe zero frequency limit of

Re ~~cx(E, L? = 0) = -(2re2/k) Im 1(E, (~/rr)[O + in]),


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234 HORING AND YILDIZand it is convenient to employ Eqs. (21) and (22). In the zero frequency limit, it isuseful to change the integration variable of K, in accordance with T + 2T/is F 1,with the result s1H, = --s dT exp (& cosh(fiw,sT/2) - cosh(&,s/2)sinh(fiw,s/Z) 1. (W-1 cFor low wavenumber, tt, -- f -2.5 and correspondingly,

foG4 ia+ ds miiw-Re 47&, D = 0) = y I* dw 7 /wim+a z es0 --A (-2s)0 4~ tanh@w,s/2)2ne2 ap

k -z$as one can readily verify from Eq. (26) (note that a/@ -+ --a/&, and integrate byparts). This result, which is exact with respect to field strength at low wavenumber,corresponds to a quasiclassical model [19-211, which correctly accounts for theeffects of Landau quantization in the density of states and averages with respectto the initial Fermi distribution f. , but otherwise treats the electron dynamicsclassically. The corresponding static shielded potential in position space is given by

V(F)= j $$ . &-r. 2nQk + 2rre2 ap/a.$ .The quantity k, = 2pe2ap/a& may be interpreted as an inverse screening length.It is clear from Eqs. (33) and (43) that ap/at (and hence, the inverse screeninglength) will have delta function peaks whenever [ = rhwc in the degenerate caseof zero temperature. This singular behavior would certainly be moderated byinclusion of electron-electron collisional effects in the analysis, but it is reasonableto expect that the qualitative feature of large inverse screening length fort = rhwc would persist. (Of course, finite temperature effects also moderate suchsingular behavior.) The positional dependence of V(P) may be evaluated in closedform. The angular integral in k-space yields ZnJ,(kr) [ 16, BHTF II, p. 81,No. 21 whence, (exponentiating the denominator)

V(f) = Q joa dk m = Q 1: ds ePako 1: dk eeSkkJo(kr)(61)

z -Q jam ds ePSkO$ jow dk e@J,,(kr).


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2-D LANDAU QUAN TIZED PLASMA 235Noting that the Laplace transform of a Bessel function is an elementary function[17, BIT II p. 9, No. 181, we have

W) = Q 6 ds eeskor2 +ss2j3,2= -Q & Jam s (,.2 _&,. . (62)Putting s = tr, the t-integral may be identified in terms of the Struve function HP, ,and the Bessel function Y-, [17, BIT I, p. 138. No. IO], with the result

V(f)= $5 & {@orMH-,&or) Y-,Uv)lI.0

For k,r > 1, we employ the asymptotic form of [H-, - Y_,] [16, BHTF II p. 39,No. 631, obtaining a l/r3 fall-off V(r) - Q/ko2r3.The various forms assumed by the inverse screening length k, = 2ne2(8p/3f) aretabulated below:(A) Nondegenerate Case.

k, = 2rre2&. (644(B) Degenerate Case. Low/intermediate fields (zero temperature)

k. _ 2;f2 1 2m;% if [ 7F ;f/%] .per

(C) Degenerate case. Arbitrary field (arbitrary temperature)(f-1

k 0 = q f ~+(r?iw,) ~f,(rliw,).7=0

(64~)For zero temperature, delta function peaks occur in k. whenever t = rfiwe, asone can see by evaluating the &derivatives of the sharply cut-off Fermi function in(C) and the discontinuous sawtooth periodic function in (B). On the other hand, if

f # rfiw, , it is clear from (C) that k, vanishes at zero temperature when only afew Landau levels are populated at higher field strengths.This is to say that there is no shielding at zero temperature and high fields if[ # rfzwe within the framework of the quasiclassical model (which correspondsto K, + -2s). Therefore, it is appropriate to explore further higher-order terms inN, , and expanding it to order k2, we havex0 -2s i 1 + ii/P 2= - 1___ 1-2mw, fiw,s tanh(fiw,s/2) I* (65)


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236 HORING AND YILD IZCorresponding to this, we find that

-Re 4~a(E, Q = 0) = y I-- $$ - & [p - %]I.mm, (66)For example, in the degenerate quantum strong field limit at zero temperature(&J, > 5) when only the lowest Landau state is populated, k, N (+/at) = 0, andwe have -Re 47To1(k,Q = 0) = -(2~e2/k)(kz/2&oJ. (67)Using this as a prototype for the general case in which k, N i?p/af = 0, we con-struct I(?) following the procedures above, with the result

V(F) = Q La dk 1 J&r) - Q Lrn ds cs . 1+ (e2/hwc>k (r2 + s2[e2/hoc]2)1/2 (68)and find that this may be expressed in terms of the Struve function H, and Besselfunction Y, as

V(F) = (Q/r)(7T~w,r/2e2)[H,(fiw,r/e2) - Yo(Ao,r/e2)1. (69)Asymptotically for r > e2/ko, , this exhibits unshielded behavior, althoughshielding is in evidence when I N e2/fiw, . It is possible to explore static shieldingfurther using the zero frequency limit of Eqs. (49) and (52), which give the fullwavenumber dependence of the degenerate zero temperature quantum strong fieldlimit as follows-Re 47rol(li, Q = 0) = - Llg .s . exp (s) 5 i . -& (Elm, (70)n=1but we will not pursue this.

Finally, it is appropriate to examine the Friedel-Kohn wiggle static shieldingphenomenon in the context of the two-dimensional quantum plasma under con-sideration. At zero field, the full wavenumber dependence of the degenerate zerotemperature limit of -Re ~ZT& Sz = 0) for zero frequency is given by Eq. (54b)-Re 4~cu(k, 52 = 0) = F (v+ (1 - $)(l - -$$-)I - 1),

and this exhibits wavenumber singularities at k2 = (2k,)2 = 8mf in the form of abranch point and a cutoff. Such singular behavior in the wavenumber dependenceof -Re 47To~(k,Q = 0) generates a distinct oscillatory contribution to the positionaldependence of the shielded potential of the Friedel-Kohn wiggle type for thetwo-dimensional plasma in the absence of a magnetic field [l]. However, for finite


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2-D LANDAU QUAN TIZED PLASMA 237magnetic field strength, there are no such wavenumber singularities, as one canreadily see from the static limit (Q --+ 0) of Eqs. (49) and (52) which give

-Re 4~&, 8 = 0)for arbitrary finite magnetic fie ld strength as

with

- 71+(2n + 2r + 1 F l)fO (&F [2n + 2r t 1 F I])[.The absence of high wavenumber singularities from the wavenumber dependence ofthe static polarizability for finite magnetic fie ld strength, means that there will beno Friedel-Kohn wiggle shielding phenomenon as long as the discrete nature of theLandau-quantized energy spectrum is discernable. It is only at very low fields,when the energy separation Aw, of adjacent Landau states is so small (tiw, < 8)that the states phase average and merge into the continuum of states charac-teristic of zero fie ld, that the Friedel-Kohn wiggle will occur. This is to say thatthe two-dimensional Friedel-Kohn wiggle static shielding phenomenon is destroyedby the dispersal of the zero field continuum of electron states into the discrete setof Landau-quantized orbitals due to the imposition of the magnetic field. It shouldbe noted that the same fate befalls the Friedel-Kohn wiggle in three dimensions [26-27] for static shielding in directions perpendicular to the magnetic field because ofthe dispersal by the magnetic field of the zero field continuum of electron statesassociated with motion perpendicular to the fie ld into the discrete set of Landau-quantized states. The continuum of electron states associated with motion parallelto the field (which is not present in the two-dimensional problem) does not preventthe demise of Friedel-Kohn wiggle in three dimensions for shielding in directionsperpendicular to the field.

Incidentally, it is interesting to note that the semiclassical static shielding lawmay be obtained using Zen) as given by Eq. (51) (first term)Z(n)= + (J [(s)*2])2,


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238 HORING AND YILDIZwith the result

-Re 47r& D = 0) = - g g; (Jn2 [(gy]).Employing the identity [16, BHTF II, p. 100, NO. 20; p. 102, No. 381

we find that1 = .7,2(z) + 2 i 5,2(z) = 2 $*5,2(z),

9Z=l n=0

-Re 47rc& Q = 0) = - -$$ ,which does not depend on the magnetic field, and is in agreement with the zerofield limit of Eq. (64b). The fact that the semiclassical static shielding law does notdepend on the magnetic field could also be readily predicted by putting A -- f 0 inEq. (58). The physical reason for this is that the classical description of dynamics,which is embedded in the semiclassical model, permits no work to be done by themagnetic field, and therefore, the magnetic field cannot supply any of the energythat would be needed for a redistribution of the charges producing the shieldedpotential, so that the energy, charge distribution, and shielded potential, are allindependent of the magnetic field [12]. Obviously, these considerations are appli-cable to the static shielding law of the classical model, as well as the semiclassicalmodel, and are true in three dimensions as well as in two dimensions [12]. Thestatic shielding law can depend on magnetic field only when energy does, and thisoccurs only with the onset of Landau quantization of the energy spectrum.Magnetic field effects on static shielding, therefore, only can occur as quantumeffects, and must be null in any classical or semiclassical description.

VIII. CONCLUSIONSThe results of our low wavenumber analysis of the two-dimensional longitudinalplasmon dispersion relation in the electrostatic limit (Eq. (25) may be expressed inthe form

2&Z = Qo2 + (242 f (p-202 - (20J~3~ + (2he2a/m2) P)l12. (71)This yields nonlocal corrections to the principal mode in magnetic field [13]Go2 = co,2 + 2re2pklm as follows


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2-D LANDAU QUANTIZED PLASMA 239and the role of Landau quantization effects in G and p has been evaluated for allregimes of magnetic field strength in Section III. The mode Q- is a two-dimensionalcounterpart of a Bernstein mode near Sz- - 2w,, and outside of the region of acrossover with the principal mode, Q, # 2w, , it is approximately given by

(73)In the crossover region, the two modes on, interact strongly, and their mixing ismanifested in a repulsion measured by

(J-2+2 - Q-2) = ((247&u/m) key, (74)where k, = 3wc2m/2ne2p is the wavenumber corresponding to the crossovercondition 52, = 2w, . Further analysis of higher wavenumber phenomena devolvesupon the exact two-dimensional RPA dispersion relation

(75)which has two-dimensional Bernstein mode roots near each multiple of w, . Zen)has been evaluated for all regimes of magnetic field strength in Section IV. Wehave shown that there is no natural damping for the principal mode and for thefull complement of Bernstein modes in a finite magnetic field. However, in the zerofield limit the principal mode is damped and the Bernstein modes are subject to aphase averaging which destroys their identity as distinct plasmon resonances. Thezero-field damping of the principal mode is given by Eq. (55). In the nondegeneratecase, it corresponds to two-dimensional Landau damping (Eq. (56a)) and in thezero-temperature degenerate case, it corresponds to two-dimensional plasmondecay into electron-hole pairs (Eq. (56b)).Our investigation of two-dimensional static shielding in a magnetic field showsthat at large distances (kg > I), the shielded potential falls off like 1 r3

V(J) - Q/k,,V3, (76)and the inverse screening length k, = 2rre2ap/a[ includes quantum magnetic fieldeffects in its structure. This result is in agreement with the zero-field zero-tempera-ture analysis of Cane1 et al, [22], save for the fact that our zero-field zero-tempera-ture limit yields k, = 2/a, (a, = Bohr radius = Tz2/me2),which differs from theirresult by a factor of 2 due to the inclusion of spin in our analysis. Our result is alsoin agreement with other zero field two dimensional static shielding calculations


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240 HORING AND YILDIZreported in [I, 51. The low/intermediate field DHVA-oscillatory structure of k,for zero temperature is given by Eq. (64b). In the nondegenerate case, we find thatk, = 2ne2& (with Landau quantization effects included).The most pronounced effects of Landau quantization occur at higher fieldstrengths (for which a few Landau levels are populated) and zero temperature. Inthis case, k, exhibits violent DHVA-oscillatory behavior, manifested in deltafunction peaks when ,$ = rfiw, , and vanishing otherwise for 8 # rhwc . Suchsingular behavior would be moderated by the inclusion of collisional effects, but itis reasonable to expect the qualitative features exhibited here to persist. Theshielding is essentially complete when 5 = rfzwc , and there is essentially no-shielding when 8 # r?icoe . The prototype of the latter case associated with thevanishing of k, is the degenerate quantum strong field limit, kw, > 5, when onlythe lowest Landau eigenstate is occupied. We have carefully examined the shieldinglaw in this case beyond the confines of the quasiclassical model, which is limited byits characterization of screening in terms of k, alone. This examination confirmsthe absence of shielding for r > e2/fiw, = rc2/ao (r, = [~/PuJJ,]~/~ = radius oflowest Landau orbit), while indicating shielding activity at shorter distancesr - rc2/ao .The striking absence of two-dimensional shielding in the degenerate quantumstrong field limit at zero temperature (associated with k, + 0) stands in sharpcontrast to the presence of shielding characterized by k, = 2rre2j3p in the non-degenerate quantum strong field limit (kw, > KT), in which a Maxwellian distribu-tion describes the population of the lowest Landau state [23-251. In view of this, onemay expect dramatic changes in physical properties that depend on shielding if thesystem is driven through a statistical transition [23-251 from degeneracy to nonde-generacy by sufficiently increasing the applied magnetic field. (The critical magneticfield strength for inducing such a high field statistical transition is readily obtainedfrom Eq. (33) by retaining only the r = 0 term for the quantum strong field limitand setting c = 0. This yields the critical field strength as w, = 47rtip/m.)The zero field two-dimensional Friedel-Kohn wiggle static shielding phenomenonis destroyed by the dispersal of the zero field continuum of electron states into thediscrete set of Landau-quantized orbitals due to the imposition of the magneticfield. Moreover, the static shielding law of semiclassical and classical models hasno dependence on magnetic field in two dimensions, as in three dimensions.Finally, we note that the thermodynamic Greens function developed here(Eq. (10)) for electrons in a quantizing magnetic field constrained to motion on aplane, and the low/intermediate and high field evaluations of the class of integralsJ (Eqs. (28), (31), (42), (45), (48)) should be of general utility in analyzing magneticfield effects on other phenomena of the two-dimensional quantum plasma. Suchintegrals are also important in the theory of a slab of quantum plasma in a magneticfield.


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2-D LANDAU QUANTIZED PLASMA 241ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful discussions with M. Orman, J. J. LaBonney, andS. Silverman.

REFERENCES]1. F. STERN, Phys. Rev. Lett. 18 (1967), 546; F. STE RN AND W. E. HOWARD, Phys. Rev. 163

(1967), 816.2. F. STERN, in Proc. 10th Int. Conf. Phys. Semicond, /p. 451. (S. P. Keller et a/., Eds.),

USAEC, 1970.3. P. B. VISSCH ER AND L. M. FALICOV, Phys. Rev. B3 (1971), 2541.4. T. R. BROWN AND C. C. GRIMES, Phys. Rev. Leti. 29 (1972), 1233.5. A. L. FETTER, Ann. Phys. (N. Y.) 81 (1973), 367; Phys. Rev. BlO (1974), 3739.6. N. J. M. HORING, M. ORMAN, AND M. YILDIZ, Phys. Lett. 48A (1974), 7.7. M. ORMAN AND N. J. M. HORING, Solid State Comm. 15 (1974), 1381.8. K. W. CHIU AND J. J. QUINN, Phys. Rev. B9 (1974), 4724.9. N. J. HORING, Ann. Phys. (N.Y.) 31 (1965), 1.10. M. YILDIZ, Ph.D. Thesis, Stevens Institute of Technology, 1973.11. M. ORMAN, Ph.D. Thesis, Stevens Institute of Technology, 1974.12. N. J. HORING, Ann. Phys. (N.Y.) 54 (1969), 405.

13. N. J. M. HORING AND M. YILDIZ , Phys. Lett. 44A (1973), 386.14. J. J. LABONNEY, JR., private communication, 1975.15. TISP refers to I. S. GRADSH TEYNAN D I. M. RYZHIK, Tables of Integrals, Series and Products,Academic Press, New York, 1965.16. BHTF refers to BATEMAN MANUSCRIPT PROJECT, Higher Transcendental Functions, (A.Erdelyi, Ed.), McGraw-Hill, New York, 1953.17. BIT refers to BATEMAN MANUSCRIPT PROJECT, Tables o f Integral Transforms, (A. Erdelyi,Ed.), McGraw-Hill, New York, 1954.18. N. J. HORING, Phys. Rev. 186 (1969), 434, Appendix.19. A. K. DAS AND E. DEALBA, J. Phys. C. 2 (1969), 852.20. N. J. HORING, in Electronic Structures in Solids, (E. D. Haidemenakis, Ed.), p. 223.

Plenum, 1969.21. N. J. M. HORING AND R. W. DANZ, J. Phys. C. 5 (1972), 3245.22. E. CANEL, M. P. MATTHEW S, AND R. K. P. ZIA, Phys. Kondens. Materie 15 (1972), 191.23. S. ASKENAZY, J. P. ULMET, AND J. LEOTIN, Solid State Comm. 7 (1969), 989.24. S. ASKENAZY, J. P. ULMET, AND J. LEOTIN, in High Magnetic Field Conference, PhysicalSociety, Nottingham, England, 1969.25. N. J. M. HORING, Znt. .Z. Quanr. Chem. 5 (1971), 763.26. N. J. M. HORING, Ann. Phys. (N.Y.) 68 (1971), 337.27. N. J. M. HORING, Phys. Rev. 186 (1969), 434.

1 This list is not intended to be complete and does not reflect credit due to many other authors.

2D electron gas in the magnetic field

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