Lifetime of a Quasiparticle in an Electron Liquid

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by University of Missouri: MOspace PHYSICAL REVIEW B 71, 075112 ͑2005͒

Lifetime of a quasiparticle in an electron liquid

Zhixin Qian Department of Physics and State Key Laboratory for Mesoscopic Physics, Peking University, Beijing 100871, China and Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA

Giovanni Vignale Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA ͑Received 31 May 2004; published 18 February 2005͒

We calculate the inelastic lifetime of an electron quasiparticle due to Coulomb interactions in an electron liquid at low ͑or zero͒ temperature in two and three spatial dimensions. The contribution of “exchange” processes is calculated analytically and is shown to be non-negligible even in the high-density limit in two dimensions. Exchange effects must therefore be taken into account in a quantitative comparison between theory and experiment. The derivation in the two-dimensional case is presented in detail in order to clarify the origin of the disagreements that exist among the results of previous calculations, even the ones that only took into account “direct” processes.

DOI: 10.1103/PhysRevB.71.075112 PACS number͑s͒: 71.10.Ay, 72.10.Ϫd

I. INTRODUCTION Refs. 7 and 8, for example, the quasiparticle lifetime was extracted directly from the width of the electronic spectral The calculation of the inelastic scattering lifetime of an function obtained from a measurement of the tunneling con- excited quasiparticle in an electron liquid, due to Coulomb ductance between two quantum wells. In the case of large interactions, is a fundamental problem in quantum many- wells separation, like the ones ͑175–340 Å͒ studied in Ref. body theory. According to the Landau theory of Fermi 8, the couplings between electrons in different well are weak liquids1 the inverse lifetime of an electron quasiparticle of ␰ ͑ ͒ and can be ignored. For such weakly coupled wells, the life- energy p relative to the Fermi energy EF at temperature T time is principally due to interactions among electrons in 2D, ͑ ͒ in a three-dimensional 3D electron liquid should scale as while the contribution of the impurities is relatively small. ␰ 2 In spite of these wonderful advances, a quantitative com- p ͩ ͪ , k T Ӷ ␰ Ӷ E , parison between theory and experiment remains very diffi- ប E B p F ϰ F ͑1͒ cult. There are several reasons for this to be so. First of all, ␶ 2 e Ά kBT · the 2D samples studied in the experiments are not yet suffi- ͩ ͪ ␰ Ӷ Ӷ ͑ ͒ , p kBT EF, 3D , ciently “ideal,” namely disorder and finite width effects still EF play a non-negligible role: as a result, the measured lifetimes where kB is the Boltzmann constant. In a two-dimensional are typically found to be considerably shorter than the theo- ͑2D͒ electron liquid the above dependencies are modified as retically calculated ones. Secondly, the electronic density in follows:2 these systems falls in a range in which the traditional high- density/weak-coupling approximations,1,9–12 are not really ␰ 2 E ͩ p ͪ F Ӷ ␰ Ӷ justified. Finally, there is still confusing disagreement among ln , kBT p EF, ␰ 2,13–20 ប EF p various theoretical results in 2D, even in the random ϰ ͕ ͑2͒ ␶ 2 phase approximation ͑RPA͒. e Ά kBT EF ͩ ͪ ␰ Ӷ Ӷ ͑ ͒ ln , p kBT EF, 2D . This paper is devoted to a critical analysis of the last E k T F B question, i.e., specifically, we calculate analytically the con- In addition to its obvious importance for the foundations stants of proportionality in the relations ͑1͒ and ͑2͒ in the of the Landau theory of Fermi liquids,1 the inelastic lifetime weak coupling regime, and try to clear up the differences that also plays a key role in our understanding of certain transport exist among the results of different published calculations. phenomena, such as weak localization in disordered metals. One particular aspect of the confusion is the widespread be- In this case, the distance an electron diffuses during its in- lief that the Fermi golden rule calculation of the lifetime, elastic lifetime provides the natural upper cutoff for the scal- based on the RPA screened interaction, is exact in the high- ing of the conductance, and thus determines the low- density/weak-coupling limit. In fact, this is only true in 3D, temperature behavior of the latter.3–5 but not in 2D. To our knowledge, this fact was first recog- During the past decade some newly developed experi- nized by Reizer and Wilkins,20 who introduced what they mental techniques, combined with the ability to produce called “non-golden-rule processes,” i.e., exchange processes high-purity 2D electron liquids in semiconductor quantum in which the quasiparticle is replaced in the final state by one wells have enabled experimentalists to attempt for the first of the particles of the liquid. In point of truth, these processes time a direct determination of the intrinsic quasiparticle life- are still described by the Fermi golden rule, provided one time, i.e., the lifetime that arises purely from Coulomb inter- recognizes that the initial and final states are Slater determi- actions in a low-temperature, clean electron liquid.6–8 In nants, rather than single plane wave states. In three dimen-

1 2 =2␲͚ ͚ W ͑q͒¯n ␴n ␴ ¯n ␴ ␶͑D͒ p+q k Ј k−q Ј ␴ k,q ␴Ј ϫ ␦͑␰ ␰ ␰ ␰ ͒͑͒ p + k␴Ј − k−q␴Ј − p+q␴ 4 and 1 ␲ ͑ ͒ ͑ ͒ ͑ex͒ =−2 ͚ W p − k + q W q ¯np+q␴¯nk−q␴nk␴ ␶␴ k,q ϫ ␦͑␰ ␰ ␰ ␰ ͒ ͑ ͒ p + k␴ − k−q␴ − p+q␴ , 5 where W͑q͒ is the effective interaction between two quasi- ␤␰ ͑ k ͒ particles nk␴ =1/ e +1 the Fermi-Dirac distribution func- ␤ ប ␦ FIG. 1. ͑a͒ A typical scattering process between electrons of the tion at temperature =1/kBT, and we have set =1. The same spin orientation near the Fermi surface has contributions from functions ensure the conservation of the energy in the colli- both a “direct” ͑solid line͒ and an “exchange” ͑dotted line͒ term. ͑b͒ sions. Obviously, from Eqs. ͑4͒ and ͑5͒, one can see that the A special class of low momentum transfer processes gives the contribution from the exchange process tends to cancel that leading-order contribution to the scattering amplitude in 2D at high from the direct process. density. As can be seen from Eq. ͑4͒, there are two types of collisions contributing to the direct term, the collisions with sions, such exchange contributions to the lifetime were cal- same-spin electrons ͑␴Ј=␴͒, and those with opposite-spin culated ͑numerically͒ in Refs. 11 and 12, but they are easily electrons ͑␴Ј=−␴͒. We denote the former 1/␶␴␴, and the shown to become irrelevant in the high-density limit. In 2D, latter 1/␶␴␴¯, where ¯␴=−␴. It can be easily shown that by contrast, the exchange contribution remains of the same 1 1 order as the direct contribution even in the high density limit. ജ ͑ ͒ ͑D͒ − ͑ex͒ . 6 Reizer and Wilkins found the exchange contribution to re- ␶␴␴ ␶␴ 1 ͑ ͒ duce to 2 of the direct one with the opposite sign in the 1 In the paramagnetic state, one evidently has high-density limit, while we ﬁnd it here to be only 4 of the direct contribution in the same limit. More generally, we give 1 1 = . ͑7͒ an analytical evaluation of both the “direct” and the “ex- ␶͑D͒ ␶͑D͒ Ӷ␰ ␰ ␴␴ ␴␴¯ change” contributions vs density, for boh kBT p and p Ӷ kBT. Therefore, The rest of this paper is organized as follows. In Sec. II, ប ␶ we provide the general formulas for / e including exchange 1 1 ͑ ͒ ജ − ͑ ͒ . ͑8͒ processes. We then devote Sec. III to the analytical calcula- 2␶ D ␶ ex ប ␶ ␴ ␴ tion of / e in 3D and Sec. IV to the same calculation in 2D. The 2D calculation is presented in greater detail in order to The effective interaction W͑q͒ between quasiparticles is explain the origin of the disagreements among the results of short-ranged compared to the bare Coulomb potential due to previous calculations. We explain the reason for the much the screening effects from the remaining electrons. Such stronger impact of exchange on the lifetime in 2D than in 3D screening effects are normally characterized by a screening at high density. Section V presents a comparison between the wave vector ks. Following this practice we approximate present theory and the experimental data of Ref. 7 and sum- 4␲e2 marizes the “state of the art.” ͑3D͒, q2 + k2 ͑ ͒ s ͕ ͑ ͒ W q = 2 9 II. GENERAL FORMULAS Ά 2␲e ͑2D͒, q + k We consider an excited quasiparticle with momentum p s and spin ␴. Its inverse inelastic lifetime due to the electron- where electron interaction is a sum of two terms, corresponding to the contributions from the “direct” and “exchange” pro- 4k ͱ F ͑3D͒, cesses, respectively, ␲ a0 ͕ ͑ ͒ ks = 10 1 1 1 Ά 2 = + , ͑3͒ ͑2D͒ ␶ ͑␰ ͒ ͑D͒ ͑ex͒ e p,T ␶␴ ␶␴ a0 ␰ ϵ 2 ␮ where p p /2m− is the free-particle energy measured and kF and a0 are the Fermi wave vector and the Bohr radius, from the chemical potential ␮ ͑see Fig. 1͒. We use D to respectively. At very low density, the screening wave vector denote the direct term and ex the exchange term. becomes much larger than the Fermi wave vector. It can be Making use of the Fermi golden rule, we get,1 shown that, in this limit,

075112-2 LIFETIME OF A QUASIPARTICLE IN AN ELECTRON LIQUID PHYSICAL REVIEW B 71, 075112 ͑2005͒

ϱ 1 1 1 1 ͑ ͒ ␲͵ ␻ ͑ ͒ ͑ ͒ =− ͑ ͒ , 11 ͑ ͒ =−2 d ␤͑␻ ␰ ͒ ͚ W q ¯nknq+k ␶ D ␶ ex ␶ ex − p ␴␴ ␴ −ϱ 1+e k,q ϫ ␦͑␻ ␰ ␰ ͒␦͑␻ ␰ ␰ ͒ ͑ ͒ or, in other words, by using Eq. ͑7͒ − k + q+k − p + q+p W p − k . ͑18͒ 1 1 =− . ͑12͒ ␶͑D͒ ␶͑ex͒ ␶͑D͒ ␶͑ex͒ The fact that 1/ and 1/ depend only of the mag- 2 ␴ ␴ nitude of p allows us to average over the unit vector of pˆ =p/p on the right-hand side of Eqs. ͑16͒ and ͑18͒. To this Equation ͑8͒ and the low density limit result of Eq. ͑12͒ are end, we deﬁne exact results, which, to the best of our knowledge, were not explicitly established before. The validity of the results in Eqs. ͑11͒ and ͑12͒ is of course questionable in the low den- pq q2 ⍀ ͑q͒ϵ ± − ͑19͒ sity limit of a real system since it is obtained from the per- ± m 2m turbative formula of Eqs. ͑4͒ and ͑5͒, which are supposed to be valid only in the high-density/weak-coupling regime. and use the fact that However, they help us understand the mathematical structure of the weak-coupling formulas. In what follows we will only consider the case of the 1 ͵ dpˆ ␦͑␻ − ␰ + ␰ ͒ paramagnetic electron liquid, which allows us to trivially dis- 2d−1␲ p q+p pose of the spin indices. Furthermore, by making the change ⌰͑ ͒␪͓⍀ ͑ ͒ ␻͔␪͓␻ ⍀ ͑ ͔͒ ͑ ͒ of variable k→k+q in the momentum summation in Eq. ͑5͒, = p,q + q − − − q , 20 and correspondingly, k→−k in Eq. ͑4͒, we rewrite Eqs. ͑4͒ and ͑5͒ as where ␪͑x͒=1 for xϾ0, ␪͑x͒=0 for xഛ0, and

1 =2␲͚ ͚ W2͑q͒¯n n ¯n ␦͑␰ + ␰ − ␰ − ␰ ͒ m ␶͑D͒ p+q k k+q p k k+q p+q ͑3D͒, k,q ␴Ј 2pq ⌰͑p,q͒ = ͕ ͑21͒ ͑13͒ Ά 2m ͑2D͒. ␲ͱ 2 2 ͑ ␻ 2͒2 and 4p q − 2m + q

1 Therefore 1/␶͑D͒ can be rewritten as =−2␲͚ W͑p − k͒W͑q͒¯n ¯n n ␶͑ex͒ p+q k k+q k,q ϱ 1 1 ϫ ␦͑␰ + ␰ − ␰ − ␰ ͒. ͑14͒ ͵ ␻ p k+q k p+q ͑ ͒ =−2 d ␤͑␻ ␰ ͒ ␤␻ ␶ D ͓ − p ͔͓ − ͔ −ϱ 1+e 1−e By using the identity ϫ 2͑ ͒ ␹ ͑ ␻͒⌰͑ ͒␪͓⍀ ͑ ͒ ␻͔ ͚ W q Im 0 q, p,q + q − Im ␹ ͑q,␻͒ q 0 =−2␲͚ n ¯n ␦͑␻ + ␰ − ␰ ͒, ͑15͒ −␤␻ k k+q k k+q ϫ␪͓␻ − ⍀ ͑q͔͒. ͑22͒ 1−e k −

␹ ͑ ␻͒ ͑ where 0 q, is the Lindhard function i.e., the density- We note that this equation is not restricted to the regime of ͒ Ӷ density response function of the noninteracting electron gas , kBT EF, but holds for arbitrary temperature. ␶͑D͒ ͑ ͒ we rewrite 1/ in Eq. 13 as In this paper, we are only interested in the case that kBT Ӷ EF, and therefore the Fermi energy EF is always well de- ϱ 1 1 fined and E Ӎ␮. To perform the average over pˆ in Eq. ͑18͒, ͵ ␻ F ͑ ͒ =−2 d ␤͑␻ ␰ ͒ ␤␻ ␰ Ӷ ␶ D ͓1+e − p ͔͓1−e− ͔ we use the fact that, for kBT, p EF, the contribution to −ϱ ␶͑ex͒ ␰ ͉␻͉Ӷ 1/ only arises from the region in which k, EF. ϫ ͚ W2͑q͒␦͑␻ − ␰ + ␰ ͒Im ␹ ͑q,␻͒. ͑16͒ Furthermore, the first ␦ function in Eq. ͑18͒ fixes the angle p p+q 0 ␰ q between k and q to be such as to satisfy the condition k ␰ ␻Ϸ − q+k = 0. With this in mind, one obtains In obtaining Eq. ͑16͒, we have also used the fact that 1 1 ͵ ˆ ␦͑␻ ␰ ␰ ͒ ͑ ͒ ␦͑␻ ␰ ␰ ͒ ␦͑␻ ␰ ␰ ͒ d−1 dp − p + q+p W p − k ¯np+q − p + p+q = ␤͑␻ ␰ ͒ − p + p+q . 2 ␲ 1+e − p ⌽͑ ͒␪͓⍀ ͑ ͒ ␻͔␪͓␻ ⍀ ͑ ͔͒ ͑ ͒ ͑17͒ = p,q + q − − − q , 23

Similarly, one has where

075112-3 Z. QIAN AND G. VIGNALE PHYSICAL REVIEW B 71, 075112 ͑2005͒

⌽͑p,q͒ m 4␲e2 ͑3D͒, ͱ 2 2 2 2pqks k +4k − q = s F Ά me2 1 1 · ͫ + ͬ ͑2D͒. ͱ 2 2 ͑ ␻ 2 ͒2 k ͱ 2 2 p q − m + q /2 s 4kF − q + ks ͑24͒ A detailed derivation of this key result is presented in the Appendix. Thus finally ϱ 1 1 ͵ ␻ ͑ ͒ = d ␤͑␻ ␰ ͒ ␤␻ ␶ ex ͓ − p ͔͓ − ͔ −ϱ 1+e 1−e ϫ ͑ ͒ ␹ ͑ ␻͒⌽͑ ͒␪͓⍀ ͑ ͒ ␻͔ FIG. 2. The ratio of 1/␶͑ex͒ over 1/␶͑D͒ via r in 3D. ͚ W q Im 0 q, p,q + q − s q ϱ ϫ␪͓␻ ⍀ ͑ ͔͒ ͑ ͒ 1 ␲e2m 2 − − q . 25 ͵ ␻ ͑ ͒ = d ␤͑␻ ␰ ͒ ␤␻ ␶ ex ͑ ␲͒3 ͓ − p ͔͓ − ͔ 2 pks −ϱ 1+e 1−e III. THE INVERSE LIFETIME IN 3D Im ␹ ͑q,␻͒ ϫ ͵ ͑ ͒ 0 ␪͓⍀ ͑ ͒ ␻͔ The theory of the electron inelastic lifetime in 3D is rather dqW q + q − qͱk2 +4k2 − q2 well established9,10 at zero temperature. However, no analyti- s F ϫ␪͓␻ ⍀ ͑ ͔͒ ͑ ͒ cal expression including the exchange has been presented so − − q . 30 far, even though Kleinman,11 and later Penn,12 have reported ͑ ͒ numerical calculations of the exchange contribution. This de- By using Eq. 27 , one has ϱ ficiency is remedied in the present section. Our calculation is 1 m3e2 2␻ ͵ ␻ done at nonzero temperature, with zero temperature as a spe- ͑ ͒ =− d ␤͑␻ ␰ ͒ ␤␻ ␶ ex ͑ ␲͒2 ͓ − p ͔͓ − ͔ cial case. 2 pks −ϱ 1+e 1−e ͑ ͒ In 3D, Eq. 22 becomes 2kF 1 ϫ ͵ ͑ ͒ ͑ ͒ ϱ dqW q . 31 1 m 2 ͱ 2 2 2 ͵ ␻ 0 ks +4kF − q ͑ ͒ =− d ␤͑␻ ␰ ͒ ␤␻ ␶ D 2͑2␲͒3p ͓1+e − p ͔͓1−e− ͔ −ϱ After carrying out the integrations, one obtains the final re- 1 sult ϫ ͵ 2͑ ͒ ␹ ͑ ␻͒␪͓⍀ ͑ ͒ ␻͔ dqW q Im 0 q, + q − q 3 4 ␲2 2 2 ␰2 1 m e kBT + p 1 ͑ ͒ =− ␤␰ ϫ␪͓␻ ⍀ ͑ ͔͒ ͑ ͒ ␶ ex ␲ 3 − p ͱ 2 − − q . 26 pks 1+e ␭ +2 ␰ Ӷ We are interested in the case that kBT, p EF. Therefore we ␲ 1 1 only need consider the region of ␻ӶE , in which, ϫͫ − tan−1ͩ ͱ ͪͬ. ͑32͒ F 2 ␭ ␭2 +2 2 m ␻ ͑ex͒ ͑D͒ Im ␹ ͑q,␻͒ =− ␪͑2k − q͒. ͑27͒ We plot the ratio of 1/␶ to 1/␶ vs the Wigner-Seitz 0 ␲ F 2 q radius rs in Fig. 2. Notice that at very high density, ͉ ␶͑ex͉͒Ӷ͉ ␶͑D͉͒ Substituting Eq. ͑27͒ into ͑26͒ leads to 1/ 1/ , and the direct-process-only theory is then relatively good. On the other hand, at low density, 1/␶͑ex͒ ϱ 1 m3 2␻ =−1/2␶͑D͒, which agrees with the general conclusion of Eq. ͵ ␻ ͑D͒ = 3 d ␤͑␻−␰ ͒ −␤␻ ͑12͒. The contribution from exchange processes therefore ␶ ͑2␲͒ p ϱ ͓1+e p ͔͓1−e ͔ − cannot be ignored in most density range. Once again, the 2kF validity of Eqs. ͑29͒ and ͑32͒ is limited to the weak-coupling ϫ 2͑ ͒ ͑ ͒ ͵ dqW q . 28 regime. They might well not hold in the low-density regime 0 of a real system, and should be regarded as mathematical The integrations over q and ␻ can be carried through, and properties of the weak-coupling equations. ␰ Ӷ one obtains In the limiting case of small excitation energy, p kBT Ӷ ͑ ͒ EF, Eq. 29 reduces to 1 m3e4 ␲2k2 T2 + ␰2 ␭ B p ͫ −1 ␭ͬ ͑ ͒ ͑ ͒ = ␤␰ + tan , 29 ␲ 3 4 ␭ ␶ D ␲ 3 − p ␭2 1 m e pks 1+e +1 = k2 T2ͫ + tan−1 ␭ͬ. ͑33͒ ␶͑D͒ 3 B ␭2 +1 ␭ 2pks where =2kF /ks. Ӷ␰ Ӷ Next we move to evaluate the contribution from the ex- In the opposite of very low temperature kBT p EF one change process. In 3D, Eq. ͑25͒ becomes has

075112-4 LIFETIME OF A QUASIPARTICLE IN AN ELECTRON LIQUID PHYSICAL REVIEW B 71, 075112 ͑2005͒

m Im ␹ ͑q,␻͒ = ͕␪͓k2 q2 − ͑m␻ + q2/2͒2͔ 0 ␲q2 F ϫͱ 2 2 ͑ ␻ 2 ͒2 ␪͓ 2 2 kFq − m + q /2 − kFq ͑ ␻ 2 ͒2͔ͱ 2 2 ͑ ␻ 2 ͒2͖ − m − q /2 kFq − m − q /2 . ͑37͒

Therefore,

2 ␰ ͱ 2 ͱ 2 1 m p k − k −2m␻ k + k +2m␻ = ͵ d␻ͫ͵ F F dq + ͵ F F dqͬ ͑D͒ 3 ␶ ␲ ͱ 2 ␻ ͱ 2 ␻ FIG. 3. The three regions of integration over q, at given ␻,in I+III 0 −kF+ kF+2m kF+ kF−2m Eq. ͑36͒ are labeled I, II, and III, respectively. Only region II con- k2 q2 − ͑m␻ − q2/2͒2 tributes to the leading order in 2D. ϫ q−1W2͑q͒ͱ F . ͑38͒ p2q2 − ͑m␻ + q2/2͒2 1 m3e4 ␭ = ␰2ͫ + tan−1 ␭ͬ. ͑34͒ It is straightforward to show that ␶͑D͒ ␲ 3 p ␭2 pks +1 2 2 ͑ ␻ 2 ͒2 Ͻ 2 2 ͑ ␻ 2 ͒2 ͑ ͒ kFq − m − q /2 p q − m + q /2 39 In the high density limit ␭→ϱ, Eq. ͑34͒ becomes in the above integral. Thus, we have 3 4 ␰ 1 m e 2 2 p ͱ 2 ␻ = ␰ , ͑35͒ 1 m kF− kF−2m ␶͑D͒ 3 p Ͻ ͵ d␻ͫ͵ dq 2pks ͑D͒ 3 ␶ ␲ ͱ 2 ␻ I+III 0 −kF+ kF+2m 9 a result obtained earlier by Quinn and Ferrell. k +ͱk2 +2m␻ + ͵ F F dqͬq−1W2͑q͒. ͑40͒ ͱ 2 ␻ kF+ kF−2m IV. THE INVERSE LIFETIME IN 2D Evidently the leading order of the two terms on the right- As mentioned in the Introduction, there is still some dis- hand side of the above inequality is both O͑␰2͒. Hence we ␶ p agreement among the results of previous calculations of 1/ e have shown that the first and third terms in Eq. ͑36͒ have no in 2D. The main purpose of this section is to exactly evaluate contributions to the leading order of O͑␰2 ln ␰ ͒. ͑ ͒ ͑ ͒ p p the prefactors of 1/␶ D and 1/␶ ex in 2D, and at the same Hereafter we therefore focus only on the calculation of time attempt to clarify the origin of those disagreements. We the second term. In region II, for small ␻, present our derivations in the two different regimes of kBT Ӷ␰ Ӷ ␰ Ӷ Ӷ 2 p EF and p kBT EF separately. For greater clarity, 2m ␻ Im ␹ ͑q,␻͒ =− . ͑41͒ we also show our derivations for the “direct” and “exchange” 0 ␲ ͱ 2 2 contributions in separate subsections. q 4kF − q Substituting the above equation into Eq. ͑36͒ leads to ™␰ A. kBT p: Direct process 3 ␰ ͱ 2 1 4m p k + k −2m␻ Ӷ␰ Ӷ ͑ ͒ ͵ ␻␻͵ F F 2͑ ͒ In 2D, for kBT p EF, Eq. 22 becomes ͑D͒ = 3 d dqW q ␶ ␲ ͱ 2 ␻ 0 kF− kF−2m ␰ p ͱ 2 ␻ ͱ 2 ␻ 1 2m kF− k −2m kF+ k −2m 1 =− ͵ d␻ͫ͵ F dq + ͵ F dq ϫ ͑ ͒ ͑D͒ 2 . 42 ␶ ␲ ͱ 2 ␻ ͱ 2 ␻ ͱ͓ 2 2͔͓ 2 2 ͑ ␻ 2͒2͔ 0 −kF+ kF+2m kF− kF−2m 4kF − q 4p q − 2m + q ͱ 2 k + k +2m␻ Im ␹ ͑q,␻͒ + ͵ F F dqͬqW2͑q͒ 0 . Thus we have ͱ 2 2 2 2 k +ͱk2 −2m␻ 4p q − ͑2m␻ + q ͒ F F ␰ 1 4m3 p ͑36͒ = ͵ d␻␻Q ͑␻͒, ͑43͒ ␶͑D͒ ␲3 1 0 The three regions of integration, at a given value of ␻ are shown in Fig. 3. It can be shown22 below that the first and where the third terms in the square bracket make no contributions ͱ 2 ␻ ͑␰2 ␰ ͒ kF+ kF−2m 1 to the leading order of O p ln p , which arises only from ͑␻͒ ͵ 2͑ ͒ ͑ ͒ Q1 = dqW q 2 2 44 the second term. We denote the contributions from the first ͱ 2 ␻ q͑4k − q ͒ kF− kF−2m F ␶͑D͒ ␶͑D͒ and the third terms to 1/ as 1/ I+III. We start with the ␹ ͑ ␻͒ expression for Im 0 q, in 2D, which is or, to leading order,

075112-5 Z. QIAN AND G. VIGNALE PHYSICAL REVIEW B 71, 075112 ͑2005͒

␻ 2kF−m /kF 1 lifetime with half the weight of W¯ ͑0͒2. Of course Eq. ͑52͒ is Q ͑␻͒ = ͵ dqW2͑q͒ . ͑45͒ 1 ͑ 2 2͒ only valid ͑within the RPA͒ at the very lowest energies and m␻/k q 4kF − q F temperatures, where the frequency dependence of the effec- Evidently, the integral in Eq. ͑45͒ has a logarithmic diver- tive interaction becomes irrelevant. Jungwirth and gence at both the upper and lower limits. We split it into two MacDonald17 have shown that at higher energies and/or tem- parts peratures the use of the “average” approximation for the ͑␻͒ ͑a͒͑␻͒ ͑b͒͑␻͒ ͑ ͒ wave-vector dependence of the interaction results in very Q1 = Q1 + Q1 , 46 good agreement with their full-ﬂedged numerical calcula- where tions. We have nothing to say about this: our aim here is simply to obtain the correct low-energy asymptotics for the kF ͑ ͒ 1 lifetime, of which Eq. ͑52͒ gives the direct part within the Q a ͑␻͒ = ͵ dqW2͑q͒ ͑47͒ 1 ͑ 2 2͒ m␻/k q 4kF − q RPA. F Except for the work by Reizer and Wilkins,20 all the cal- and culations cited above in 2D explicitly consider only the di- ␻ 2kF−m /kF rect process, without taking account of the exchange process, ͑ ͒ 1 Q b ͑␻͒ = ͵ dqW2͑q͒ . ͑48͒ which we deal with in the next subsection. 1 q͑4k2 − q2͒ kF F ͑ ͒ ͑ ͒ a ͑␻͒ b ͑␻͒ ™␰ To leading order, Q1 and Q1 can be evaluated as B. kBT p: Exchange process 1 m␻ In 2D, for k TӶ␰ ӶE , Eq. ͑25͒ becomes ͑a͒͑␻͒ 2͑ ͒ ͑ ͒ B p F Q1 =−W 0 2 ln 2 49 ␰ ͱ 2 ͱ 2 4kF kF 1 2m p k − k −2m␻ k + k −2m␻ = ͵ d␻ͫ͵ F F dq + ͵ F F dq ͑ex͒ 2 and ␶ ͑2␲͒ ͱ 2 ␻ ͱ 2 ␻ 0 −kF+ kF+2m kF− kF−2m ͱ 2 ͑ ͒ 1 m␻ k + k +2m␻ Q b ͑␻͒ =−W2͑2k ͒ ln . ͑50͒ ͵ F F ͑ ͓͒ ͑ ͒ ͑ͱ 2 2͔͒ 1 F + dqͬqW q W 0 + W 4k − q 2 2 F 8kF kF ͱ 2 ␻ kF+ kF−2m Therefore, in summary, Im ␹ ͑q,␻͒ ϫ 0 . ͑55͒ 1 m␻ ͱ 2 2 ͑ 2 ␻͒2 ͑␻͒ ͓ 2͑ ͒ 2͑ ͔͒ ͑ ͒ 4p q − q +2m Q1 =− 2 2W 0 + W 2kF ln 2 . 51 8kF kF Again only the second term in the square bracket contributes Substituting Eq. ͑51͒ into ͑42͒, and performing the integra- to the leading order. Thus, tion over ␻, we ﬁnally arrive at ␰ 1 4m3 p =− ͵ d␻␻Q ͑␻͒, ͑56͒ 1 ␰2 1 2E ␶͑ex͒ ␲͑2␲͒2 2 = p ͫW¯ 2͑0͒ + W¯ 2͑2k ͒ͬln F , ͑52͒ 0 ␶͑D͒ 4␲E 2 F ␰ F p where

where we have deﬁned the dimensionless quantity ͱ 2 k + k −2m␻ 1 ͑␻͒ ͵ F F ͑ ͓͒ ͑ ͒ m Q2 = dq 2 2 W q W 0 ¯ ͱ 2 ␻ q͑4k − q ͒ W͑q͒ϵ W͑q͒. ͑53͒ kF− k −2m F ␲ F + W͑ͱ4k2 − q2͔͒. ͑57͒ The quantity in the square brackets of Eq. ͑52͒ can be ex- F pressed in terms of the Wigner-Seitz radius rs as follows: To the leading order, 1 1 r 2 1 m␻ ¯ 2͑ ͒ ¯ 2͑ ͒ ͩ s ͪ ͑ ͒ ͑␻͒ ͑ ͓͒ ͑ ͒ ͑ ͔͒ ͑ ͒ W 0 + W 2kF =1+ ͱ . 54 Q2 =− 2 W 0 W 0 +2W 2kF ln 2 . 58 2 2 rs + 2 4kF kF The fact that Eq. ͑45͒ also has a logarithmic contribution Substituting Eq. ͑58͒ into Eq. ͑56͒ and performing the inte- Ӎ ␻ from the upper limit of integration at q 2kF was missed in gration over , we arrive at almost all previous analytical calculations. This is one of the ប ␰2 ␰ main reasons leading to errors in the numerical prefactor of = p W¯ ͑0͓͒W¯ ͑0͒ +2W¯ ͑2k ͔͒ln p . ͑59͒ ␶͑ex͒ ␲ F the lifetime. The second term in the square brackets of Eq. 16 EF 2EF ͑52͒ is absent in the works of Refs. 2, 18, and 20. Jungwirth ͑ ͒ ͑ ͒ In Fig. 4, we show the ratio of 1/␶ ex to 1/␶ D vs r . The and MacDonald17 were the ﬁrst to clearly recognize the ex- s remarkable fact is that, at variance with the 3D case, this istence of the 2k term: however, they made a further ap- F ratio does not vanish for r →0. The reason for this differ- proximation in replacing the square of the effective interac- s ence can be understood as follows. In 3D a typical scattering ¯ 2͑ ͒ ¯ ͑ ͒2 ¯ ͑ ͒2 tion W q by the average of W 0 and W 2kF . Equation process near the Fermi surface, such as the one shown in Fig. ͑ ͒ ␰2 ␰ 52 above shows that to leading order in p ln p this is not 1͑a͒, involves two particles that are well separated ͑by a ¯ ͑ ͒2 ͒ quite correct: W 2kF enters the expression for the inverse wave vector of the order of 2kF in momentum space. The

075112-6 LIFETIME OF A QUASIPARTICLE IN AN ELECTRON LIQUID PHYSICAL REVIEW B 71, 075112 ͑2005͒

͑␻͒ ␮ ␰ ͑␻͒ 1 m 0 q+ + p q+ =− ͫ͵ d␻͵ dq + ͵ d␻͵ dqͬ ͑D͒ 2 ␶ ␲ ϱ ͑␻͒ ͑␻͒ − −q− 0 q− 1 qW2͑q͒Im ␹ ͑q,␻͒ ϫ 0 , ͑62͒ sh␤␻ ͱ4p2q2 − ͑2m␻ + q2͒2 ͑␻͒ ⍀ ͑ ͒ ␻ where q± are the solutions of the equation + q = , ͑␻͒ ͓ ͱ 2 ␻͔ ͑ ͒ q± = p ± p −2m . 63 ␻Ӷ ␶͑D͒ Again, only the regime of EF contributes to 1/ p to the accuracy of the leading order. Thus, by using Eq. ͑41͒, one has

3 ϱ ͱ 2 1 2m ␻ k − k −2m͉␻͉ ͵ ␻ ͵ F F ͑ ͒ ͑ ͒ = d ͫ dq ␶ ex ␶ D ͑D͒ 3 FIG. 4. The ratio of 1/ over 1/ via rs in 2D. ␶ ␲ ϱ sh␤␻ ͱ 2 ͉␻͉ − −kF+ kF+2m

k +ͱk2 −2m͉␻͉ k +ͱk2 +2m͉␻͉ direct scattering amplitude for such a process is maximum + ͵ F F dq + ͵ F F dqͬ when the momentum transfer q is much smaller than k , and ͱ 2 ͉␻͉ ͱ 2 ͉␻͉ F kF− kF−2m kF+ kF−2m is thus typically proportional to W͑0͒. The exchange scatter- ͑ ͒ 1 1 ing amplitude, on the other hand, is of the order of W 2kF ϫ W2͑q͒ . ͑64͒ ␶͑ex͒ ␶͑D͒ ͱ 2 2 ͑ ␻ 2͒2 ͱ 2 2 for all values of q: hence the ratio of 1/ to 1/ goes as 4p q − 2m + q 4kF − q W͑0͒W͑2k ͒/W͑0͒2, which vanishes for r →0. The reason F s Once again only the second term in the bracket makes con- why this argument fails in 2D is that the logarithmic contri- tribution to the leading order, and we find bution to the inverse lifetime in the high density limit does ϱ not arise from typical scattering processes, but rather, from 1 2m3 ␻ ͵ ␻ ͑␻͒ ͑ ͒ special ones in which the two colliding particles are very ͑ ͒ = d Q1 , 65 ␶ D ␲3 sh␤␻ close in momentum space ͓see Fig. 1͑b͔͒: hence the direct −ϱ and the scattering amplitude are comparable, and give simi- ͑␻͒ ͑ ͒ ͑ ͒ where Q1 is defined in Eq. 45 and evaluated in Eq. 51 . lar contributions to the inverse lifetime. A careful analysis of Therefore the integrals involved shows that in the high density limit, ϱ the exchange contribution cancels 1 of the direct contribution 1 m3 ␻ m␻ 4 =− ͓2W2͑0͒ + W2͑2k ͔͒͵ d␻ ln , to the inverse lifetime. This result is at variance with that of ␶͑D͒ ␲3 2 F ␤␻ 2 4 kF −ϱ sh kF Ref. 20, according to which the exchange contribution can- 1 ␶͑ex͒ ͑66͒ cels 2 of the direct one. We find that the relation 1/ ͑D͒ =−1/2␶ holds only in the low density limit ͓see Eq. ͑12͒ which can be further evaluated leading to and Fig. 4͔, where the weak coupling theory is not reliable. ប ͑␲ ͒2 Combining direct and exchange contributions in a single kBT 1 2EF = ͫW¯ 2͑0͒ + W¯ 2͑2k ͒ͬln . ͑67͒ ␶͑D͒ ␲ F formula we finally find that 8 EF 2 kBT As in the low-temperature case, the second term in the ␰2 1 3 1 1 2EF square brackets of this equation was missed in almost all the = p ͫ W¯ ͑0͒2 + W¯ ͑2k ͒2 − W¯ ͑0͒W¯ ͑2k ͒ͬln , ␶ ␲ F F ␰ e 4 EF 4 2 2 p previous theories except the one by Jungwirth and MacDonald,17 which, however, overestimates it by a factor 2. ͑60͒ Without the second term in the square bracket, Eq. ͑67͒ would agree with the expression obtained by Zheng and Das where the quantity in the square brackets is given by Sarma18 and by Reizer and Wilkins,20 but it would be four times smaller than the result of Fukuyama and Abrahams,13 ␲2 2 3 r and /4 times larger than the result of Giuliani and Quinn. − s . ͑61͒ 4 ͱ ͑ ͱ ͒2 ␰ ™ 2 rs + 2 D. p kBT: Exchange process ӷ␰ ͑ ͒ In 2D, for kBT p, Eq. 25 becomes Thus in the high density limit the total inverse lifetime dif- 0 q ͑␻͒ ␮+␰ q ͑␻͒ fers by a factor 3 from the result of the direct-scattering-only 1 m + p + 4 = ͫ͵ d␻͵ dq + ͵ d␻͵ dqͬ 3 ͑ex͒ 2 calculation, and by a factor from the result of Ref. 20. ␶ 4␲ ϱ ͑␻͒ ͑␻͒ 2 − −q− 0 q− 1 Im ␹ ͑q,␻͒ ϫ 0 ͑ ͒ ͓ ͑ ͒ ␰ ™ W q q W 0 C. p kBT: Direct process sh␤␻ ͱ4p2q2 − ͑2m␻ + q2͒2 For ␰ Ӷk TӶE , Eq. ͑22͒ becomes ͑ͱ 2 2͔͒ ͑ ͒ p B F + W 4kF − q . 68

075112-7 Z. QIAN AND G. VIGNALE PHYSICAL REVIEW B 71, 075112 ͑2005͒

FIG. 5. Momentum- and energy-conserving tunneling between two identical free-electron bands separated by a potential difference FIG. 6. Electron relaxation rate ⌫ in 2D. Experimental data are eV is possible only if the spectral width of the single particle states ͑ ͒ from Ref. 7, calculated ones are from Eq. 72 . Here TF =EF /kB. in each band ͑indicated by the shaded regions͒ is at least as large as eV. The situation changes profoundly if electron-electron interactions are allowed. Now momentum is still conserved ͑if Proceeding as in the previous section we rewrite this as impurity scattering and surface roughness are negligible͒ but ϱ 1 m3 ␻ the energy of the electron quasiparticle is no longer a well =− ͵ d␻ Q ͑␻͒, ͑69͒ ␶͑ex͒ ␲3 ␤␻ 2 defined quantity, due to the possibility of inelastic scattering 2 −ϱ sh processes involving other electrons in each quantum well. As where Q ͑␻͒ is defined in Eq. ͑57͒ and evaluated in Eq. ͑58͒. a result, tunneling becomes possible in a region of voltages 2 ⌫Ͻ Ͻ⌫ ⌫ Therefore − V , where is the width at half maximum of the plane-wave spectral function in a well,7 i.e., ϱ 1 m3 ␻ m␻ = W͑0͓͒W͑0͒ +2W͑2k ͔͒͵ d␻ ln , 1 ⌫͑␰ ,T͒ ␶͑ex͒ ␲3 2 F ␤␻ 2 A͑E,␰ ͒ = p . ͑73͒ 8 kF −ϱ sh kF p ␲ ͑ ␰ ͒2 ͓⌫͑␰ ͒ ͔2 2 E − p + p,T /2 ͑ ͒ 70 ͑ ␰ ͒ From the well-known relation A E, p ͑ ␲͒ ͑ ␰ ͒ ͑ ␰ ͒ which can be, to the leading order, further simplified to =− 1/ Im Gret E, p , where Gret E, p is the retarded Green’s function, one can show that the spectral width ⌫ is 1 ͑␲k T͒2 k T = B W¯ ͑0͓͒W¯ ͑0͒ +2W¯ ͑2k ͔͒ln B . ͑71͒ just the inverse of the lifetime of a plane wave state, which is ␶͑ex͒ ␲ F 32 EF 2EF the sum of the lifetimes of electron and hole quasiparticles in 17,21 The ratio of 1/␶͑ex͒ to 1/␶͑D͒ is therefore found to be the the following manner: Ӷ␰ same as that in the case of kBT p, which has been plotted 1 1 ⌫͑␰ ͒ ͑ ͒ in Fig. 4. p,T = + . 74 ͑ ͒ ͑ ͒ ␶ ͑␰ ,T͒ ␶ ͑␰ ,T͒ Combination of 1/␶ D in Eq. ͑67͒ and 1/␶ ex in Eq. ͑71͒ e p h p thus yields The principle of detailed balance demands 1 ͑␲k T͒2 k T n͑␰ ,T͒ 1−n͑␰ ,T͒ =− B ͓3W¯ 2͑0͒ +2W¯ 2͑2k ͒ −2W¯ ͑0͒W¯ ͑2k ͔͒ln B . p = p , ͑75͒ ␶ ␲ F F ␶ ␶ e 32 EF 2EF e h ͑72͒ ͑␰ ͒ where n p ,T is the thermal occupation number at tempera- ␰ Ӷ ⌫͑␰ ͒ Notice the difference between the above result and the one ture T. If we assume p kBT and approximate p ,T by ⌫͑␰ ͒ obtained in Ref. 20. p =0,T , we see from the above equations that the half width at half maximum of the tunneling conductance peak is V. COMPARISON WITH EXPERIMENTAL RESULTS expressed in terms of the electron quasiparticle lifetime as IN 2D follows: Consider two identical 2D electron liquids in closely 2 ⌫ = . ͑76͒ spaced quantum wells between which a small potential dif- ␶ ͑0,T͒ ference V is maintained. We expect a small tunneling current e between the layers. However, as Fig. 5 shows, no tunneling We can now attempt a comparison between the experi- is possible in the absence of impurities and electron interac- mental values of ⌫ from Ref. 7 and the theoretical values of ␶ ͑ ͒ tions. This is because under those unrealistic assumptions 2/ e 0,T . This is shown in Fig. 6. It must be kept in mind both the energy and the momentum of the electron must be that, in order to perform a meaningful comparison, one must conserved during tunneling, and there are simply no states first subtract from the experimental data a ͑presumably͒ satisfying these conditions. temperature-independent constant due to residual disorder.

075112-8 LIFETIME OF A QUASIPARTICLE IN AN ELECTRON LIQUID PHYSICAL REVIEW B 71, 075112 ͑2005͒

The value of this constant is determined by the condition that help to produce better agreement with experiments, it re- ⌫ tend to zero for T→0. Even after this subtraction we see mains a great challenge for experimentalists to device the that the theoretical curve lies well below the experimental conditions that will eventually allow them to probe the truly data. Furthermore, the shortcomings in Refs. 17 and 18 as intrinsic behavior of the electron liquid. revealed in this paper imply that the “excellent agreement” with experiment claimed in those papers is overly optimistic, as pointed out earlier by Reizer and Wilkins.20 ACKNOWLEDGMENTS We note that all derivations presented in this paper are to We gratefully acknowledge support by NSF Grant Nos. the accuracy of the leading logarithmic term. Calculations including higher order terms might bring in a better agree- DMR-0074959 and DMR-0313681. We especially thank ment with the experimental data. However, it seems too op- Gabriele Giuliani for many valuable discussions. timistic to believe that the huge difference ͑roughly a factor ͒ 4 between theory and experiment is totally due to such APPENDIX higher order contributions. The size of the discrepancy sug- gests that there might be other factors playing a role, such as In this appendix, we give the details of the derivation of the ﬁnite width of the quasi-two-dimensional system, Eq. ͑23͒. To this end, we denote the left-hand side of Eq. ͑23͒ electron-impurity scattering, electron-phonon scattering, and as A3 and A2 for 3D and 2D cases, respectively. In 3D, A3 surface roughness. While the inclusion of these effects may can be rewritten as

␲ ␲ 2 pq cos ␪Ј q2 1 A = e2͵ d␪Ј sin ␪Ј͵ d␾Ј␦ͩ␻ + + ͪ , ͑A1͒ 3 2 2 2 ͓ ␪ ␪Ј ␪ ␪Ј ͑␾ ␾Ј͔͒ 0 0 m 2m p + k + ks −2pk cos cos + sin sin cos −

where ͑␪,␾͒ and ͑␪Ј,␾Ј͒ are the spherical angles of k and p, ␲ 2͵ ␾Ј␦͑␻ ␾Ј 2 ͒ respectively, relative to q. Carrying through the ␾Ј integra- A2 = e d + pq/m cos + q /2m tion, one obtains 0 1 pqx q2 1 ␲ 2͵ ␦ͩ␻ ͪ ϫ ͫ A3 =2 e dx + + ͱp2 + k2 −2pk cos͑␾ − ␾Ј͒ + k −1 m 2m s 1 1 + ͬ. ͑A5͒ ϫ . ͱ 2 2 ͑␾ ␾Ј͒ ͱ͑ 2 2 2 ␪ ͒2 ͑ ␪͒2͑ 2͒ p + k −2pk cos + + ks p + k + ks +2pk cos x −4 pk sin 1−x ͑A2͒ The integral in the above equation is trivial due to the ␦ Carrying out the integration over ␾Ј yields function, and it leads to ␲ 2␪͓⍀ ͑ ͒ ␻͔␪͓␻ ⍀ ͑ ͔͒ 2 me + q − − − q A3 = , ͱ͓ 2 2 2 ͔2 ͓ 2 ͑ ˆ ͒2͔͓ 2 2͔ pq p + k + ks − k · q − k − k · q 4p − q 2␪͓⍀ ͑ ͒ ␻͔␪͓␻ ⍀ ͑ ͔͒ me + q − − − q ͑A3͒ A2 = ͱp2q2 − ͑m␻ + q2/2͒2 ␻Ӷ 2 where we have used the fact that 2m ks . Putting in this ϳ ϳ ϳ 1 expression the approximate equalities k p kF and k·q ϫ ͩ 2 ͓ ␰ ␰ ␻Ӎ ͱ 2 2 ͱ͓ 2 ͑ ͒2͔͓ 2 2͔ −q /2 which follows from the condition k − q+k = 0 p + k + k · q − k − k · qˆ 4p − q + ks due to the ﬁrst ␦ function in Eq. ͑18͔͒ one easily arrives at ͑ ͒ 1 Eq. 23 in the 3D case. + ͪ. In 2D, A can be explicitly written as ͱ 2 2 ͱ͓ 2 ͑ ͒2͔͓ 2 2͔ 2 p + k + k · q + k − k · qˆ 4p − q + ks ␲ 2 ͑A6͒ 2͵ ␾Ј␦͑␻ ␾Ј 2 ͒ A2 = e d + pq cos /m + q /2m 0 1 ϫ ͑A4͒ ͱ 2 2 ͑␾ ␾Ј͒ Substituting, as in the 3D case, the approximate equalities p + k −2pk cos − + ks ϳ ϳ ϳ 2 ͑ ͒ k p kF and k·q −q /2 one ﬁnally arrives at Eq. 23 in or 2D.

075112-9 Z. QIAN AND G. VIGNALE PHYSICAL REVIEW B 71, 075112 ͑2005͒

1 D. Pines and P. Nozières, The Theory of Quantum Liquids ͑Ben- ͑1965͒. jamin, New York, 1996͒, Vol. 1. 11 L. Kleinman, Phys. Rev. B 3, 2982 ͑1971͒. 2 G. F. Giuliani and J. J. Quinn, Phys. Rev. B 26, 4421 ͑1982͒. 12 D. R. Penn, Phys. Rev. B 22, 2677 ͑1980͒. 3 K. K. Choi, D. C. Tsui, and K. Alavi, Phys. Rev. B 36, 7751 13 H. Fukuyama and E. Abrahams, Phys. Rev. B 27, 5976 ͑1983͒. ͑ ͒ 1987 . 14 A. V. Chaplik, Zh. Eksp. Teor. Fiz. 60, 1845 ͑1971͓͒Sov. Phys. 4 A. Yacoby, U. Sivan, C. P. Umbach, and J. M. Hong, Phys. Rev. JETP 33, 997 ͑1971͔͒. Lett. 66, 1938 ͑1991͒. 15 C. Hodges, H. Smith, and J. W. Wilkins, Phys. Rev. B 4, 302 5 Y. Kawaji, S. Yamada, and H. Nakano, Appl. Phys. Lett. 56, ͑1971͒. 2123 ͑1990͒. 16 ͑ ͒ 6 Y. Berk, A. Kamenev, A. Palevski, L. N. Pfeiffer, and K. W. G. Fasol, Appl. Phys. Lett. 59, 2430 1991 . 17 West, Phys. Rev. B 51, 2604 ͑1995͒. T. Jungwirth and A. H. MacDonald, Phys. Rev. B 53, 7403 ͑ ͒ 7 S. Q. Murphy, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, 1996 . 18 Phys. Rev. B 52, 14825 ͑1995͒. L. Zheng and S. Das Sarma, Phys. Rev. B 53, 9964 ͑1996͒. 19 8 M. Slutzky, O. Entin-Wohlman, Y. Berk, A. Palevski, and H. D. Menashe and B. Laikhtman, Phys. Rev. B 54, 11561 ͑1996͒. Shtrikman, Phys. Rev. B 53, 4065 ͑1996͒. 20 M. Reizer and J. W. Wilkins, Phys. Rev. B 55, R7363 ͑1997͒. 9 J. J. Quinn and R. A. Ferrell, Phys. Rev. 112, 812 ͑1958͒. 21 G. D. Mahan, Many-Particle Physics ͑Plenum, New York, 1981͒. 10 R. H. Ritchie and J. C. Ashley, J. Phys. Chem. Solids 26, 1689 22 I. D’Amico and G. Vignale, Phys. Rev. B 68, 045307 ͑2003͒.

075112-10