Indian 10umal of Pure & Applied Physics Vo l. 1, 7. OCLOber1999. pp. 764-771,

Self-energy of an electron due to electron-phonon interaction in alkali metals : Its dependence on phonon dispersion relation and

realistic electron-electron screening S P Tewari & Charu Kapoor

Dcpartmcnt of Physics & Astrophys ics. Uni versity of Delhi . Delhi 110007.

Received 23 October 1998: rev ised 12 February 1999: accepted 1,0 June 1999

The Q/ectron mass-enhancement factor. A. due to el ectron-phonon interaction . has heen ev'lluated considerin g

realisti c description . both . for electrons and phonons. Ex rli cit analytica l cxrressions for A havc heen lleri ved 1'01'

disr ersionless. highly disr ersive and intermediately disr ersi ve rhonons when the electron screening is gi ven by Thomas­Fermi model. Differen t types of electron screening whi ch incorrorate electron-electron interacti on much beyond Thom;]s­Fermi have al so been considered. Computation or A for all alka li metal s h;]vc been made and comrared with the corresponding experi mental and other theoreti cal result s.

I. Introduction

The elec tron-phonon interactions in metals have an important in f luence upon electron states near

Fermi energy ; the actual sel f-energy L(p,{J,,) of the electron (p is the momentum of electron , p" is the energy of electron) due to electron-phonon interac t ions' is a small energy . However. its deri vati ve

a'ifuplI is large. so that. it makes a large contributi on to the electron effecti ve mass, which is'

1 _~ _ _ ~ I RC[( p .f' >l 1 _I) 1\- - " 17,, -

111 Oflo ... ( I )

Sel f-energy of the electron is dependent on the description of elec trons , phonons and elec tron­phonon interacti on. There is hardl y any systematic study on the dependence of self-energy on the type of screening present amongst electrons in a metal. Screening is one of the most important concepts in many-body theory; many models have attempted to explain thi s effect by vari ous expressions for

dielec tri c con stan t £ (Iql) == £(q), Iql = Ip' -pi , q is the elec tron momentum transfer and is equal to the momentum of phonons . Thomas-Fermi theory

prov ides a stati c mode l for £( q). The correlation efrec t was accou nted ror by Lindhard or Random Ph ase Approx imation (RPA) dielcc tri c runcti on' which was improved rurther by Hubbard' to account for the ex istence or exchange and correlation hole

around the electron. Singwi el ai .' and Biswas and

Tewari ' improvised the dielectric function £(q) further . The authors considered these models to account for the q dependence of the plasma

frequency for calculatin g the mass-enhancemen t. Ie was eva luated for Debye type phonon di spersi on and al so considering more realisti c phonon picture. All calculations were performed at T = () K.

2. Mathematical Formal ism

Electron se lf-energy due to electron-phonon interaction at zero temperature is '

.. . (2)

where, q is the phonon momentum, p the electron momentum, p" and q" electron and phonon energies respective ly, Dl q) the phonon Green's Function and

{.irql } the screened matrix element.

A lso Ip ' l = Ip + ql . . .0)

Consider

TEWARI & KAPOOR: SELF-ENERGY OF ELECTRON 765

For free electrons. Ep' = ,,' ~ / 2m

Hence

A = 1/1 IT i- sgn(pli + 1I11)

Now

IY! I'u

J sgn(pl)+ lfll)d(11i =2 fdfJ l) II

Substituting Egs (4) .(5) and (6) in Eg . (2)

... (4)

.. . (5)

... (6)

2.2 Realistic dispersion relationship

As a crystal is essentially a periodic discrete structure, the phonon di spersion relation turns out to be as follows:

( qa) <0.:1 = lQ') sin -2h .

... ( 12)

where, 2a is inter-particle distance, q is the phonon

27r '1' momentum i.e. q = h K where IKI = K = - , I\.

A' being the wavelength of phonons .

In this case

since states around Ik rl contribute strongly to the integral. the upper limit of the g integration is taken =

as 2 ti k, , k, being Fermi wave-vector.

!!.L In Po + n.wo s in (qa / 2tz)

W'I I Po - hillo sin(qa / 2tz) . . . ( 13)

Now consider different phonon dispersion relationships :

2.1 Debyc dispersion relation

In this model one ca n ha ve

h W = c I q 1= c, (I 'II I

Q 2

Also __ '11-. , E (I q I) E (q) W·

'I'

... (8)

.. . (9)

where, Q 'II and W 'I I are bare and dressed phonon

freq uencies, respecti ve ly.

Hence , in thi s case

Q I' + c, q = __ '11- In l--'-"":'("":" _--":"'~I . . . ( 10) W", P II - (; 1 q

Substitutin g Eq. ( 10) in Eq. (7) and using Eq. ( I ). O Il C gcr.~

... ( I I )

Substituting Eq. ( 13) in Eg. (7) and using Eq. ( I), one gets

b111/111 =

... ( 14)

Us ing jellium mode l fo r the electron-phonon matrix element which is consistent wit h different forms of screening used by the authors later on, unlike some other studies" , and Thomas-Fermi model for screeni ng, one obtain s Egs . ( 15 ) and ( 16) correspond in g to Egs. ( II ) and ( 14) respectively, as :

bl11/rn = --~--2mZ2e

4

N [ 4k ; 1 k l·cl" M q; {4(tik r )2 + (tzq, )" }

. .. ( 15)

... ( 16)

where

gtjl { 4~(" r-2-:-·~-,-, r

INDIAN J PURE APPL PHYS. VOL 37. OCTOBER 1999

g", +: 'II c(q)

(t/(/ ) ~ c(q) = I +--.~­

q-

~)

'fl. 6rcNc: -q . =q = ---., ., EF

... ( 17)

, 4rcN(' ~ and . ()) ;,(O) = --- . where N is the electron number

/1/

density, (' the elementary charge of electron, m the electron mass, M the ionic mass , (;, the longitudinal "peed of phonons, Z the valency of the ion and E" the Fermi energy .

In order to gain an insi ght in the consideration of presence of dispersion in the phononic elementary exc itation, the (Iispersion relation given by Eq. (12) has been ex panded in the form of a series, i.e.

5 I , \ I S'''' (')" = q- - S (( + - . q + .. .I' 5'

... (18)

wh erc

When the first term in Eq. ( 18) is considered in Eq. ( 16) and solved, one gets the following equation for 811l/1ll :

... ( 19)

wh ich reduces to Eq.( 15) prov ided one identifies LU ,

with '2(', /(/ .

T hi s result is ex pected. as for Ko « I, the cli~creteness of the lattice hardl y plays any rol e and the resull corres ponds to the continuum Illodel of the crys tal used in developing Debye approximation.

If one takes in to account in Eq.( I 8) the seconcl terlll al so, one introcl uces hi gh dispersion and one finds that it is possible to so lve for 8111/ 111 in thi s case ii/ SO The expression for 8111/ /11 is given as follows

14 /. ? ? I . 1 II "'F+q ; 2 kj +- 11 -

6 ~ .\ . .. 2 2 (/., . (4k F + (/.,. )

. .. (20)

Similarly , if the terms up to (( In Eq.( 18) are considered, and if one restricts the expansion up to (/ in the sin ' (qarr./h) term in the denominator of the right hand side of Eq. (16), and solves, one gets the following explicit equation for 8171/ /11 :

, (f - { ? ? --, 4(fIk,..) - - 2(1iq , ) -I l Oti -

11l[ 4k~ 7Q} ]+(t,q ,) 2[ 42k~. ~ 1 } 1 q, (4k,.. +q , _

., .(2 1)

Thus, it is possib le to obtain ex pli ci t analytical equ ati ons for di spersionless, highly d ispersive and intermediatel y di spersi ve phonon dispersion re lations. However, when rea l di spersion re lationship is considered, one gets an equation fo r whi ch all integral has to be evaluated numeri cally as is ev ident from Eq. (16) and it is not possible to obtain an analytical equation .

The va ri ous eq uati ons for 8/11//11 obtai ned so far have been deri ved assuming electron screening to be given by Thomas-Fermi mode l, wherei n. the bare Coulomb potential changes to Yukawa potential due to screened interaction. But, the screelli ng in a metal is much more in vo lved than what is given by Thomas-Fermi mode l and the screening length is no longer independent of q but is dependent upon it , beca use of plasmon exc itations, exchange ilnL! corre lations present amongst the e lect rons.

,.

,

TEWARI & KAPOOR: SELF-ENERGY OF ELECTRON 767

In the Random Phase Approximation, the eq uation fo r plasma dispersion relationship is no longer independent of (I like in the case of Thomas­Fermi approximation and is g iven as follows :

ill I' ... (22)

However, no alkali metal corresponds to r, « situation , which corresponds to RPA ; r, for these

metals are in th e range 3.25 < r , < 5.62. Therefore,

even thoug h the e lectron gas in an alkal i metal is hi ghl y degenerate, ye t. it does not correspond to the approximation of hi gh electron dens ities and there fore one has to go beyond RPA in order to take into account the presence of exc han ge inte ractions.

correction factor is taken to be independent of frequency . Biswas and Tewari ' suggested a me thod for an approximation which yields local fi e ld correction factor dependent on both , wave vector and the frequency of the modes . The ex press ion for plasma dispersion relationship turn s out to be :

where

() - 2

'A2= 1.8818 A

... (25)

S I . t t d by and uc 1 an Improveme n was sugges e Hubbard , where, contribution o f interacti on of e lectrons with anti-parallel spins has been taken into account. The equati on of plasma dispersion re lationship , in contrast to Eq . ( 18), is as follows :

[

l) (/ I c/ 1 (j) = (j) (0) 1+--.----

I' I' I 0 c 4 . c C/ 'I' k r

where , (12

'I'

4111e 2 k I:

lrh

... (23)

Thou gh Hubbard's die lect ric function is more rea li st ic than that g iven by RPA , it too overestimates short ran ge corre lati o ns between e lectro ns, but much less than that g iven by RPA .

Singwi el ((/ .' suggested a better approximation, which resulted in mod ifying the loca l field correction from that g iven by Hubbard mode l and more-or-Iess re moved the negative value that g(r) , static pair corre lati on fun c ti o n of e lectron s, shows in a lkali metals, e ven unde r Hubbard's approximation. Thus, the ir die lectri c fun ct ion correspond to a rea li sti c desc ripti on of the pol ari sat ion present in metal s . in this approximation , the ex press io n for plasma dispersion re lati o nship turns out to be:

[ 9 {/ I c/j

(V = (0 (0) I + - -- - - y -- ... (24) I' I' I () ( " 2 k ".

/.,,1 / .

whe re , y is a number dependent on r, and the structure factor' . In the above theori es, the local field

2 _ 2 Tr ' ~ - 2k r +q,

Hence, one can study the effect of screening on

0/11//11 for a g iven phonon di spe rs ion re lation ship . It may be noted that such a systematic and rea li sti c study has not been reported so far , both, for phonons and electrons in alkali metals .

3. Results and Discussion

The general equation for e lectron mass­

enhancement factor 0111/117. due to e lec tron-phonon interaction in the j e llium model , considering any sc reening and realisti c longitudinal phonon dispersion relationship spec ified by Eq. ( 12), is g iven byEq . (16) .

If one considers a non-di spers ive Debye ph onon dispe rsion re lati o nship , one gets Eq. ( 19). Whe l~ sine term, in the denominator of the integrand of expression in the right hand s ide of Eq. ( 16), is

expanded in a series, in te rms of (qan/h). as g iven by Eq. ( 18), it is poss ible to obtain analyti ca l equati on

for om/m unde r Thomas-Fermi sc reenlllg approximation for different truncated forms of si ne series.

Thus, one gets explicit eq uati ons for mass­enhancement factor fo r diffe re nt descripti ons of phonon di spersion re lati onship : Eq. ( 19) for non­di spe rsive, Eq. (20) for hi gh ly dispe rsive and Eq. (21) for intermediately dispersive re lati on.

7ClX INDIAN.J PURE APPL PHYS. VOL 37. OCTOBER IY9l)

However, the real istic phonon dispersion relation­ship. as mentioned earlier, involves full sine series and so one has to resort to numerical computations of the terms in the equation .

Further, if one considers screening beyond that given 1.8

hy Thomas- Fermi , as di scussed earlier, it is not possible to ohtain analyti cal equations, even when one considers the truncated fo rm of phonon dispersi on relationship. 1.7

i. c .. for each one of the screenings given by RPA and ot hers" . the appropriate integral has to be worked out.

These have been eva luated and the values of b1111111

determined fo r all alkali metals : Li to Cs, the charac-teristic parameters of which are presented in Table I, for ready reference, which are now discussed in the follow­ing paragraphs.

In Fig. I (a), 171 *1/11 (= I + (b/nill/)) , are plotted against r, va lues covering alkali metals: from Li to Cs, for two desc ripti ons of phonon dispersion: full sine function and, when the first term in the expansion of sine function is raken for electron screening, given by Thomas-Fermi and RPA , along with the results obtained by earlier workers.''!'.

In Fi g. I (a), the five va lues of 11/ */11/ for Li , Na, K, Rb and Cs have been joined just to facilitate to observe the va ri ation in the different va lues. The continuous curves do not represent 111*//11 for val ues of 1", different from the va lue for alkali metals. The continuous and dashed curves are for Thomas-Fe rmi screening when full sine ;lIlci the first term in the ex pansion of sine are, respec­t i ve ly. considered. As is evident. the latter consideration yields Illuch lower values for 171 */11/ in comparison with that give n by the ronnel' approximation . The difference helweenlhe non- dispersive and dispersive phonon di s­persion re lationsh ip is signi fican!.

T ;lh lc I - V,llu es of r.,'" and Fermi-energy r:F in different alkali metals

M et;1i r, / 110 r:F (eV)

L i 3.25 4.74

N;I 3t )) 3.24

K 4.X(, 2. 12

Rh 5.20 I .X5

Cs 5.62 1.59

r,'" is Ihe r;l tiius of a sphere in units 0,' 1(1). the Bohr radiu s. as­

signed to ;In elect ron in a lllel ;1i

1.6

1.5

E ~t.4 E I

1.3 --1.2

1.1

1 •. 0 !:----'------~---'---_ _!_--.l........---...J 3 6

Fig . I (a) -- Compari son of computed va lues of tota l mass of the

electron . mol'. in the presence of electron-phonon intera ·Iion. to its

bare mass. m. for the five alkali metals : Li. Na. K. Rh and Cs.

considering the di spersion less and rea li stic phonon dispersion

relati onship. for the two types of elew'on screenings:

Thomas-Fermi and RPA. with the correspond ing ex perimental

and other theoretical results.

- -- reali stic dispersive phonon dispersion relati onship and

Thomas-Fermi screen ing: -- -- -- dispersi on less pho­

nons and Thomas- Fermi screening: - - - •• --- realis­

tic di spersive phonon dispersi on relationship and R;lndom Phase

Approximation (RPA) screening : --- • - -- dispersion­

less phonons and RPA screening : .r ex perimenta l results) and .

theoretical results) l For lithium. the ex periment;;1 va llie is too

large to he shown in the graph. It s va lue is 2.22 ± n.m.

TEW ARI & KAPOOR: SELF-ENERGY OF ELECTRON 7f,l)

1 .7

1. 6

1.5 !

~ 1.4

E'

1.3, I

't . 1. 1 r-

I

I 1.0

3 4

Fii!. I (h) - S~llle ~s Fi g. I (a). hut. the electron screening is now

t;lk cn to hc that gi ven hy Huhhard. Singwi ('/ (//. anc! Biswas &

Tewari (BT) models. referred to in the lext.

1--- . • --- rea li stic di spcrsive phonons and Singwi et al.

electron screening: --- . --- dispersionless phonons and

Singwi ct al. electron screening: --- --- --- realis­

ti c dispersive phonons anc! Huhhard electron screening: --­

dispersion less phonons and Huhhard electron sc reening : --­

••• --- ••• --- rea li sti c di spersive phonons and BT elec­

tron scrcening :mel --- •••• --- •••• --- dispersion­

less phonons and BT electron screeningl . Experimenta l and other

theore ti c li result s arc S:lIlle as givcn ill Fig. 1(:1)

In Fig. I (a), for a g iven alkali metal, the diffe rence

between the two di spersion relation ships in RPA is

much lower than that obtained for T homas-Fermi

model. Further, there is a significant decrease when the

screening is changed from Thomas-Fermi to RPA for

consideration of any phonon di spe rsion re lati onship. In Fig. I (b ) are shown results of sim il ar exe rcises for

Hubbard , Singwi et al. ~ and Biswas and Tewari .l sc reen­

ings. Like in the earlier studies. as shown in Fi g. I (a).

the reali s ti c phon o n di s pe rs ion re lati o nship y ie ld s

hi ghe r values of m*/m for a g iven alkali metal , in com­

parison with that given by non-di spersive re lation under

the consideration of a particular screening.

Thus, the effect of the presence of di spers ion of

phonons, or otherwise, and the type of screening pl ay

significant roles in deciding the value of 111*/111 . The points shown by solid circles (.) in the figure are

the results of earli e r calculations from Grimvall ' . the

e lectron- phonon matri x element is taken to be g i ven by

pseudo-potenti al. The po ints shown by crosses (x) are

so called, experimental points , dete rmined from the

Landau Fermi-liquid parameters . These po ints for Li

and Cs are pretty high and do not correspond to anyone

of the reali stic calcu lations. Tn order to assess the effect of chan ging the fo rm of

phonon di spe rsion relatio nship, from non-di spe rsive to

highly to inte rmediately dispe rsive, similar calcul at ions

have been performed for differe nt screenings . In Fig. 2, are shown , the re lati ve change 0 under

different considerations of phonon di spe rsion re lat ion­

ship, g iven as fo llows :

D = [A., - A,,] x 100 A,

where, A" and A,) are the values of 1'Il*lm when full s ine

funct ion is considered , and , when only the first term,

first two terms and first three terms in the expa nsion o f

s ine function are respec ti ve ly conside red . There is no systemat ic variati on inm*/111 for diffe rent

sc reenings and for diffe rent phonon dispersi on re lati on­

ship, except , that the values change noti ceably unde r di ffe rent conside rati ons.

From the study, one can conc lude that the va lues of

mass- e nhancement of e lectrons due to e lectron-phonon

inte raction is quite sensiti vc to the type o f e lectron-elec­tron screening and the phonon dispe rsion re lat ionshi p.

The rea li st ic conside rati ons of both these fac to rs yie ld

no INDIAN J PURE APPL PHYS. VOL 37. OCTOBER 1991J

8 - ( d)

~ x x

)( x

0 0

~ ~ • • 0

81- ( c)

x

x

0 x x

0 0

x 0 0 •

• • •

•

0 o ~

1 8 (b) x

0 x

• x x ~

0

x 0 • •

0 •

0

8 .- (a 1

0 x x x x ~

~ ~ ~ ~

o I I I I I I 3 4 5 6

ri g. 2 __ Percentage dev iation Dol' 11l */1ll with respect to the reali sti c Jispersive phonons ror di spersionless. highl y cl ispersive anc! in lerl1lediately dispersive phonons. as descrihed in the tex \. under dirferent electron screenings. 1'01' the ri ve alkali met:ll s: I .I" di spersion ­

less pllonons : . highl y dispersive phonons : 0 intermediate di spersive phononsl. a) RPA . b) Huhhard. c) Singwi et al. and (d) BTl

TEW ARI & KA POOR: SELF-ENERGY OF ELECTRON 77 1

in • -40~

- 60~ __________ ~1 ____________ ~1 ____________ ~ 3 4 5 6

Fig . . \ - Percentage dev i:lIion or n" or m"/m with respect 10 the experiment.1i va l lies ror di spersion less. highl y dispers ive, intermediatel y

dispersi ve .lIld reali stic di spersive phol1 ol1s, as descrihed in the teX l.under dirferent electron screeni ngs. ror the ri ve :libli metals : I . dispersio lil ess pllllnons. 0 hi ghl y di spersive phonoll s. X intermediate dispersive phonnns. () rea l istic di spersive phonons. (a) RPA

and (h) Huhhard l

772 INDIAN J PURE APPL PHYS. VOL 37. OCTOBER 1999

quite different va lues of the mass-enhancement factor in

alkali meta ls from those reported earli er.

Unlike alkali metals which are not slIperconducting

clown to the lowest poss ible temperatures, there are other rnaterial s7x

, which are sun rconcluctin r!" clue to electron­o

( b)

-20 f-

-20 -

-40 f-o ~ •

phonon interacti on. Due considerati on has to be paid in

the latter case, for evaluat ion of electron rnass-enhance­

ment factor for the realistic description of elementary

exc itations for both-electrons and 0 honons. o ~ •

2 •

o

9 0

• 0 x

•

o

o x

•

- 60L-___________ L-1 __________ -J' ____________ ~ 3 4 5 6

Fi !!, 4 - Pcrccnt ;l gc dCvr.lti on of D· of m·/m with rcspect to the ex perimcnt;]1 va lucs for dispcrsionlcss , highl y di spcrsi vl: . intcrmcdiatcly dispcrsivc anci rcali stic cii spcrsivc phonons. as desc ribcd inlhc text. under different elcctron screcnings. for thc fi vc alka li mctal s: I. dispcr's ionlcss phonons. 0 highl y di spcrsivc phonons. X intermediate dispersi vc phonons. 0 rcali sti c dispcrsivc honons. (a) Si ngwi ct ;d, ;lIld (h) BT l

TEWARI & KAPOOR: SELF-ENERGY OF ELECTRON 773

Acknowledgement One of li S (Charu Kapoor), gratefull y acknowledges

that thi s work was funded by CSTR, New Delhi , vide

award No . (9/4S(744)/92-LMR- I).

References I Mahan G D. Many-particle physics. (Pl enum Press. New

York ). 1990. p. 5~~.

2

:. 4

Tewari S P & Kapoor Cham . Solid S({f/I.' CO/llIII/III . 1(1) ( 11)91) 20 I .

Biswas B & Tewari S P. Phrs Rev /3. 22 ( 19RO) oR I.

Sham LJ . Proc R S()cA. 2R3 ( 1%5) 33.

5 Grimvall G. Pln'sica Scripta. 12 ( 1975) 337.

7

~

Kittel C. lntroduction to solid state physics. VII Ed ition. (John Wiley. New York). 1996. p. 157.

Tewari S P & Kapoor Chanl . Pln's LI.' II A. 250 ( 199R) 1 () 1 .

Cote Michel e/ (II. Phrs /?ev Lell . R I (11)9H) 697.