Density of states of the interacting two-dimensional electron gas

Density of states of the interacting two-dimensional electron gas

E. Kogan,1,2,*and B. Rosenstein2,1,†

1

Jack and Pearl Resnick Institute, Physics Department, Bar Ilan University, Ramat Gan 52900, Israel

2_{National Center for Theoretical Sciences and Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan}

共Received 23 October 2003; published 26 March 2004兲

We study the influence of electron-electron interactions on the density of states共DOS兲 of a clean two-dimensional electron gas. We find that the linear cusp in the DOS around the Fermi level, which was obtained previously, has an additional logarithmic factor. The cusp crosses over to a pure logarithmic dependence further away from the Fermi surface.

DOI: 10.1103/PhysRevB.69.113105 PACS number共s兲: 71.10.Ca

It was established more than 20 years ago by Altshuler and Aronov1 and Altshuler, Aronov, and Lee2 that in low-dimensional diffusive systems the electron-electron interac-tion leads to the suppression of the single-particle density of states 共DOS兲 at the Fermi level. In two dimensions their theory predicted a logarithmic cusp at the Fermi level due to the diffusion pole divergences in the vertex renormalizaion. The theory was later extended by Rudin, Aleiner and Glazman3 to larger values of the energy measured from the Fermi surface.

Recently it was shown by Khveshchenko and Reizer4and by Mishchenko and Andreev5 using the diagrammatic ap-proach with the random phase approximation共RPA兲 dynami-cal susceptibility, that large electron-electron interaction in-duced correction to the DOS exists even in the absence of disorder. Both groups obtained a linear cusp at the Fermi level␦␯(⑀)/␯0⬃兩⑀兩/EF 共with the slope differing by a factor

of 2兲, independent of the strength of the electron-electron interaction rs. More recently Rollbuhler and Grabert6using

a path integral technique 共applied for relatively small cou-plings rs⬍1) found numerically that the slope does depend

on the coupling and flattens away from the Fermi surface. Due to the apparent discrepancy between the two ap-proaches we reconsider the problem in the framework of the diagrammatic method. Our aim is to spell explicitly and to clarify all the approximations which are made to obtain the transparent analytical formulas for the DOS correction. We also obtain the results which are partially different from those of Refs. 4 and 5.

The Hamiltonian of the two-dimensional electron gas is

H⫽

兺

p␴ ␧p a_p†_␴a_p␴⫹1 2 _{p p}

兺

_{⬘␴␴⬘q} ap⫹q␴ † _a p⬘⫺q␴⬘ † ⫻V共q兲ap⬘␴⬘ap␴, 共1兲 where ␧p⫽ p2 2m, V共q兲⫽ 2␲e2 q . 共2兲

The Green’s function G( p,E) is given by the equation

G⫺1共p,E兲⫽G₀⫺1共p,E兲⫺⌺共p,E兲, 共3兲

where G0( p,E) is the Green’s function in the absence of electron-electron interaction:

G₀⫺1共p,E兲⫽E⫺ p

2

2m; 共4兲

the self-energy⌺(p,E) in this paper will be calculated in the RPA at T⫽0: ⌺共p,E兲⫽

冕

d 2_q (2␲)2

冕

d␻ 2␲ G0共p⫹q, E⫹␻兲 V共q兲 ␧共q,␻兲. 共5兲 The DOS is ␯共E兲⫽⫺_␲2Im

冕

d 2_p (2␲)2G共p, E⫹i0兲. 共6兲 Expanding the Green’s function with respect to the self-energy and reversing the order of integrations, one obtains the electron-electron interaction induced correction to the DOS共Ref. 1兲:

␦␯共E兲⫽⫺_␲2 Im

冕

d 2_p

(2␲)2␦G共p,E⫹i0兲⬅Im X共E⫹i0兲.

共7兲

We will work in the Matrubara formalism, calculate X for the imaginary frequency, X共i⍀兲⫽⫺2 ␲

冕

d2q (2␲)2

冕

_⫺⬁ ⬁ d␻ 2␲ V共q兲 ␧共q,i␻兲 ⫻

冕

d 2_p (2␲)2G0 2 共p,i⍀兲G0关p⫹q,i共⍀⫹␻兲兴, 共8兲 and, at the end, will make an analytical continuation.

Using the fact that G₀2( p,i⍀)⫽i⳵G0( p,i⍀)/⳵⍀, we can write the integral over p of the three Green’s functions as

冕

d2p (2␲)2G0 2_{共p,i⍀兲G} 0关p⫹q,i共⍀⫹␻兲兴 ⫽⫺i

冉

_⳵⳵_{⍀ ⫺⳵␻}⳵

冊

⌸共q,i␻,i⍀兲, 共9兲 PHYSICAL REVIEW B 69, 113105 共2004兲

where the polarization function ⌸(⍀,q,␻) is given by the equation ⌸共q, i␻, i⍀兲⫽

冕

d 2_p (2␲)2G0共p, i⍀兲G0关p⫹q, i共⍀⫹␻兲兴. 共10兲

The last equation in polar coordinates takes the form

⌸共q,i␻,i⍀兲 ⫽ 1 共2␲兲2

冕

₀ 2␲ d␪

冕

0 ⬁ pd p 1 i⍀⫹␮⫺ p 2 2m ⫻ 1 i共⍀⫹␻兲⫹␮⫺ p 2 2m⫺ pq cos共␪兲 m ⫺ q2 2m . 共11兲

Since the main contributions come from fermionic momenta close to Fermi momentum, one can make the first approxi-mation by replacing p in the term pq cos(␪)/m by

冑

2m␮

⫽mvF and ignoring the term q2/2m. After that the

integra-tion over p can be easily performed:

⌸共q, i␻, i⍀兲⫽ m 共2␲兲2

冕

₀ 2␲ _d␪ i␻⫺vFq cos共␪兲 L共␻,⍀,q兲, 共12兲 where

L共␻,⍀,q兲⫽log共⍀⫹␻兲⫺i␮⫹ivFq cos共␪兲

⍀⫺i␮ . 共13兲

Explicitly, presenting the logarithm as

L共␻,⍀,q兲⫽1 2log 共⍀⫹␻兲2_{⫹关␮⫺v} Fq cos共␪兲兴2 ⍀2_⫹␮2 ⫹i

冋

tan⫺1␮ ⍀ ⫺tan⫺1 ␮⫺vFq cos共␪兲 ⍀⫹␻

册

, 共14兲

and taking into account that⍀,␻,vFqⰆ␮, we make the

sec-ond approximation:

L⫽␲i关⌰共⫺⍀兲⌰共⍀⫹␻兲⫺⌰共⍀兲⌰共⫺⍀⫺␻兲兴. 共15兲

After that, the integral over ␪ in Eq. 共12兲 can be easily cal-culated, and the polarization function takes a form

⌸共q, i␻, i⍀兲⫽m 2 sgn共␻兲

冑

␻2_⫹v F 2 q2关⌰共⫺⍀兲⌰共⍀⫹␻兲 ⫺⌰共⍀兲⌰共⫺⍀⫺␻兲兴. 共16兲

Taking appropriate derivatives one obtains

冕

d2p (2␲)2G0 2_{共p,i⍀兲G} 0关p⫹q,i共⍀⫹␻兲兴 ⫽ ⫺m兩␻兩i 2关␻2⫹v_F2q2兴3/2关⌰共⫺⍀兲⌰共⍀⫹␻兲 ⫺⌰共⍀兲⌰共⫺⍀⫺␻兲兴. 共17兲

The dielectric constant ␧(q,␻) is related to the polariza-tion operator

P共q,i␻兲⫽⫺2

冕

d⍀

2␲⌸共q,i␻,i⍀兲 共18兲 by the equation

␧共q, i␻兲⫽1⫺V共q兲P共q, i␻兲. 共19兲

It is easier, however, not to use the approximate equation.

共16兲 for the polarization function, but to insert in Eq. 共18兲

exact Eq.共10兲 and 共as it is traditionally done兲 integrate over

⍀ first, to obtain P共q,i␻兲⫽2

兺

p np⫺np_⫹q ␧p⫹q⫺␧p⫺i␻ , 共20兲

where n_p is the Fermi distribution function. Obvious algebra gives P共q,i␻兲⫽ 1 ␲2Re

冕

_p⬍p F pd p

冕

d␪ 1 pq cos␪ m ⫹ q2 2m⫺i␻ ⫽2 ␲Re

冕

p⬍p_F pd p 1

冑

冉

q2 2m⫺i␻

冊

2 ⫺p 2_q2 m2 共21兲 Integrating over p, P共q, i␻兲⫽2m 2 ␲q2Re

冋

q2 2m⫺i␻⫺

冑

冉

q2 2m⫺i␻

冊

2 ⫺vF 2_q2

册

_, 共22兲

and expanding the radical in small q, one obtains

␧共q,i␻兲⫽1⫹2e 2_m q

冋

1⫺ 兩␻兩

冑

␻2_⫹v F 2 q2

册

. 共23兲

Now we are ready to return to the calculation of the DOS. Substituting Eqs. 共17兲 and 共23兲 into Eq. 共8兲, considering ⍀

⬎0 for definiteness, and subtracting an ‘‘inessential’’

con-stant, one has

BRIEF REPORTS PHYSICAL REVIEW B 69, 113105 共2004兲

X共i⍀兲⫽e 2_mi 2␲2

冕

⫺⍀ 0 ␻d␻

冕

0 ⬁ qdq ⫻ 1 q⫹2e2m

冋

1⫺ ␻

冑

␻2_⫹v F 2_q2

册

⫻ 1 关␻2_⫹v F 2 q2兴3/2. 共24兲

Equation. 共24兲 coincides with those obtained in Refs. 4 and 5共apart from the fact that we are considering imaginary energy兲. But we have noticed that the double integral can be calculated in a more rigorous way than it was done there. After we introduce the dimensionless variable q¯⫽vFq/兩␻兩,

and change the order of integrations, the integral takes a form X共i⍀兲⫽⫺ i 4␲2vF 2

冕

₀ ⬁ ¯ dq¯q 关1⫹q¯2_兴3/2

冕

₀ ⍀ ⫻ d␻ q ¯ 2e2mvF ␻⫹1⫺ 1

冑

1⫹q¯2 . 共25兲

The integral over ␻ can be easily calculated and we obtain the following ‘‘scaling’’ form:

X共i⍀兲⫽⫺i␯0rS

23/2␲f

冉

⍀

2

冑

2rSEF

冊

, 共26兲

where rs⫽

冑

2e2/vF, ␯0⫽m/␲ is the DOS of the noninter-acting two dimensional electron gas and the function f is

f共x兲⫽

冕

0 ⬁ dq¯ 关1⫹q¯2_兴3/2log

冋

1⫹ xq¯ 1⫺ 1

冑

1⫹q¯2

册

. 共27兲 For xⰇ1, f共x兲⫽log x. 共28兲

To obtain the density of states we should substitute ⍀→i⑀, where⑀⫽E⫺␮ is the energy measured from the Fermi sur-face, and take the imaginary part. Thus we obtain, for 兩⑀兩

ⰇrsEF, ␦␯共⑀兲 ␯0 ⫽ rS 23/2␲log

冉

兩⑀兩 rSEF

冊

. 共29兲

To obtain asymptotic of f (x) for xⰆ1 we can chose ar-bitrary␩ satisfying xⰆ␩Ⰶ1 and present the integral in Eq.

共27兲 as some of two integrals: from 0 to␩and from␩ to⬁. In the first integral we can expand the integrand with respect to q and in the second we can expand the logarithm in with respect to x. Thus we obtain

f共x兲⫽

冕

0 ␩ dq¯ log

冋

1⫹ 2x q ¯

册

⫹x

冕

␩ ⬁ dq¯ 关1⫹q¯2_兴3/2 q ¯ 1⫺ q

冑

1⫹q¯2 . 共30兲

After a simple algebra in a leading approximation with re-spect to x we obtain f共x兲⫽⫺2xlog x. 共31兲 Thus, for兩⑀兩ⰆrsEF, ␦␯共⑀兲 ␯0 ⫽⫺ 兩⑀兩 4␲EF log

冉

兩⑀兩 rSEF

冊

. 共32兲

Equations共29兲 and 共32兲 are our main results. The correc-tion is smaller than ␯0, consistent with the perturbative as-sumption. At small兩⑀兩 the DOS has a linear downward cusp modified by a logarithmic factor. This factor also gives a weak dependence of the cusp upon the strength of the cou-pling. The quasilinear segment crosses over to the logarith-mic one at the energy scale兩⑀兩⫽rsEF. This last statement is

in good agreement with numerical results of Ref. 6.

*Electronic address: [email protected] †_{Electronic address: [email protected]}

1_{B.L. Altshuler and A.G. Aronov, Zh. E´ ksp. Teor. Fiz. 77, 2028,}

共1979兲 关Sov. Phys. JETP 50, 968 共1979兲兴.

2_{B.L. Altshuler, A.G. Aronov, and P.A. Lee, Phys. Rev. Lett. 44,} 1288共1980兲.

3_{A.M. Rudin, I.L. Aleiner, and L.I. Glazman, Phys. Rev. B 55,}

9322共1997兲.

4_{D.V. Khveshchenko and M. Reizer, Phys. Rev. B 57, R4245}

共1998兲.

5_{E.G. Mishchenko and A.V. Andreev, Phys. Rev. B 65, 235310}

共2002兲.

6_{J. Rollbuhler and H. Grabert, Phys. Rev. Lett. 91, 166402共2003兲.}

BRIEF REPORTS PHYSICAL REVIEW B 69, 113105 共2004兲