Article信州大学工学部紀要 26: 17-25 (1969)departmental bulletin pape

KATO,  Yuji

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KATO  Yuji

Shinshu University Institutional Repository

Orbital Susceptibility of Impurity Systems

           tt t/tt ttt t t t         0rbital Susceptibility of Impurity Systems'           /tt t       ttt tt t /tt                         , Yuji KATo*                          (Received Apri1 30, l969) , ' ,,   The formulation of orbital suscePti'bility in the 'system contalning theimPurities is studied taking account of the interference between two Greenfunctions. The formu!ation is very similar to the drdinary Boltzmann-B!ochtype theoiry df electrical:conductivity on the basis of two-time Green function.The sum fule'for condUctiVity due to the Edwards' approximation is not satisfied.                         '                                                   tt            t . t tt. ../t:. t. t t t..1                            g.1. Introq,uction   The impurity effect on the'orbit-ai diamagnetic susceptibility of free electronshas been discussed by several workers. ,Peierlsi),has shown that the effect ofimpurities on the diamagnetic susceptibility is neglected under the conditionhlr <kT, where T denotes the. mean free time of the free eiectron. Kohn andLuming2) have carried out the calculation for the susceptibility of a simplemodel of a dilute alloy and have obtained the result that the susceptibility isthe same as that of a £ree electron gas assuming the perturbation to be suth-ciently weak. ' .   Recently, Ando3) has calculated the orbital susceptibility o'£ dilute alloysand has proved that the contribution of impurities to the susceptibility can beneglected under the condition h!r < rp, where n is a characteristic energy ofthe order of the Fermi energy. They have calculated the susceptibility bythe use of Nakajima's formuia4) and Matsubara and Toyozawa's technique5) forimpurity band conduction, and have estimated it in the Iowest order approxi-mation. In this estimate, they have retained only the terms with a 6-singular-ity6) in the term 1!(E-H) and have neglected the interference between twoGreep funct.ions in respect of the random average of impurity sites, where Hfepresents' the totai Harniltonian of the sy'stem. Furthermore, it has tieehconcluded that the effect of interference between two 'Green functions do6isnot contribute to the result obthined by the }owest order 'approximation e'xLcepting for the change of the expression of a damping constant of electron'icstates.* Professor of Physics, Faculty ol Engineering,'Shinshu University, Nagano.    IR the succeeding section, after the two-time Green function method forthe evaluation of electrical qonductivity7), a calculation of.the diamagneticsusceptibility is given to take into account of the interference between twoGreen functions. This formulation is similar to that of the ordinary Boltzmann-Bloch type theory for the conductivity. In･ g3 the evaluation of susceptibilityis made. The section 4 wili be offered to the calculation of the electrical con-ductivity by the use of Edwards' approximation for the product of two Greenfunctions8), however, a sum rule for the conductivity9) is not satisfied so far,as the Edwards' approximation is assumed.                               '            . g2. Forrnulation of the problem .              t tt tt tt    We consider an assembly of electrons containing the imp.urities, whoseelectrons can be represented by the one-electron model. According tg Nakajima4),the orbitai diamagnetic susceptibility has been written as                                 1 S2<q, - q)                           Z.T-E7,im q2 -'. , <1)                                                                 '                     '             'Here, S2(q, - q) is obtained by thel relations                                '         '                     '            tt              S"p(q, - ij'> == !",d2 <]',(q, - ih2)1'.(- q', O),>                                                 '  '                             . qr"q'. '                         ww- So(q, -q')6,.+S2<q, -q') 'un･ q,, , . (2>.                                          1                      pt,v= {X, Y, Z} P =: feT'                  i(r, t) - 2 i<q, t) exp (-iqr)                           a                                    '           'and the spa$ial Fourier tranSform of current density operator i(r, t), in therepresentation of the second quantization,                  i(q,'oi = -e.le-:l](k - t;q)a"k-qak' ' '1･ (3)                                   '  '                                       '    ' 'where a'k and ak stand for the creation and annihilation operqtors of a freeelectron, respectively, whose eigenstates are denoted by a wave vector k asthe basis of one electron state. Furtherrnore, <.9> means the expectationvalue of 9 for the canonical ensemble, that is,                              Tr{9 exp (- PH)}                       <9> =. Tr{exp(-PrE()} 'No. 26 Orbital Susceptibil･i'ty of lmpurity Systems 19Let us corisider th'e non-uniform system whose volume .O.g is unity. The totalHamiltonian of the field-free system '' ' ･                             ' '                 H== IZ] eka'kak+ Z]£ p(q) V<q>a"k-aak <4)                      k kqwhere Ek is the free electron energy with vCTave vector k,             t/t            '                          p(q> = £exp(iqri),                                 i                       V(q) = !..., e5P,(- iqr)li(r)drandidenotes the site of an impurity and V<r) the interaction between animpurity and an electron.                                                                       '    By making use of the Buckingham's identityiO) for non-uniform system          '                                         e                  So(q, -q') + S2<q, -q') == iii<n(q - g' >>where n(q) being the Fourier transform of the charge density operator, and ofthe relation ef equation(2), one may e4sily leads to the expressiofi of S2(q,-gt) aS                                    '          '             s,(q, -q') =: -23.e <n(q -q')> - S)il]s,t,t(q, -q'). (s)In order to rewrite (5), it is convenient to introduce the Green function, definedby                                      1･             G<k, k';E) == Im <O akE ww i, --Ha'k O> <E --> -F O> (6>where IO> represents the vacuum state in which no electron exists and Immeans the imaginary part, and to diagonalize the total Hamiltonian H by aunitary transformation5). Let q' =:q, (5) now reduces to                 '       s,(q, -q) = 8.e <n(o)> -g( .e.h )2:i]pu, SdE!dEt(le-gq)･(k･ + g,)               .IC<Er) - .f<E)              × E nv E, G(k', k- q; E)G<k, k' +q; E') , (7>                                                                 'with f(E) the Fermi distribution function.. '' ' .    In order to proceed to the calculation taking account of the interference20 ･,.-.., Y. ･KATo .. No. 26betweeB two Green functions in respect of,,the,random avecage,.-+of impuritysites, we utilize the approximation rnethod obtained･by Edwards8). The twQ.-particle Green function <G(k', k-q; E)G(k, k' + g; E')>. has been written as                                 tt tt               '    <G(k', k - q; E)G(k, k' + q; E') >. = ik,k･+.GO(k'; E)GO(k' + q; E')    +nXGe(k'; E)GO(k' + q; E')i V<q')12 <G(k' + q', k - a; ･E) .       a'                                  , x G(k, k' +q+ q'; E')>. <8)where                    Go(k, E) -LL (E - ,l!l)lkill l7),(k, Ej ･7a''denotes' the total fiumber of impurities, r(k, E) a damping constant ofelectronic state and <･･････>. means an average over the sites'6f random im-purities between two Green functions. Since we need S2(q, -a>' in the limitq-->O as will be shown later, we may consider the value of q quite small forthe time, and so, we can do a replacement 6f some terms in the integrandsof <7). Let o being the wave vector (fe --l.l;-q) and (k' ,+ -Sq), o' between (k -r,tq) and (q''+ gq), ana o'" 6etween(k' + gq) and(q' + Siq), 'We can 'write'the scalar product (k -gg)e(k' -y gq) as                       '1   (k -Sq)･(k' + gq) =- k -iq, k' + gq,coso                                  11  f ;(k --;-q).(g' + -ll-a) cos e" + k 7-l}-ql k' ÷ -li-a sin 0' sin 0" cos (g' - g")                                                                (9)where we have made use-of the consideration that the predominant terms inthe second term on the right-hand side of (7) are the terms which have thewave vector k' nearly equal eo q'. B.y averaging over the azimuthal coordinate(g' ir g"), the second term containing an azimuthal coordinate in <9).vanishesowing to the symmetry of the system. Using the approximation mentionedabove, we may get the equation for two-partile Green function correspondingto the Boltzmann-Bloch equation7) from (8)... , IVIulEi.plyi"g <8),by (k.TS-q).(k' -t/-ll-a) and surnming over /.F 4nd k', weNo. 26 Orbital Susceptibility of lmpurity Systerns 21find the following equation                                          t t tt ttt                          /           '   :ll] ;il,l (k rw -llmq) ' (k' + -l;-q) <G(k'･ k - q; E)G(k, k' + g; E' )>. ' . ,' / . /                                                           '  "= ]Ii](k --ll-q)2Go(k-q; E)Ge(k,;Et) .''･･ ,' ,,.･ .･., . .                                                 2         ', + ]ll,]Go (kr; E)GO (k' + q; E') XK :il,) nl V(g' - k') l cos 0" ', ,'  , .. × (k --ll-q)･(q' + -l;-q) <G(q',k- q; E)G<k, g' + G; Ei')>.･ (io)  ttIt seems to us that a certain constant vectoy kf being of the order of theFermi momentum exists7), so that the summation term over k and q'  in thesecond term on the right-hand side of <10) is approximately given by        '   ' nlv(k'f-k')I2cose-kfk, ]>,](k--l-q).(q'-" lltl-q) ''                       x <G(q', k-q; E)G<k, d+q; E')>. (11) where cosekfk, is a constant independent of q'. Then, the solution of (10) caR be written down at once      ;li) ;il,](k --l;g)'(k' + -ll-q)<G(k', k - q; E)G<k, k' + g; E')>.                   . , . - J(q, E, E') , ･. , a2)                                   rw 'r II ･r(q, E, E')                                               '                               '            I(a, E, E') = ]!il] (k --ll-q)2 G"(k - g; E>Go(k; E') and                                                     tt      r<q, E, E') === llE ]nl Y(umkf- k')l 2 cos 0kfb･ GO(k'; E>GO<k' + g; E').                k'                g3. Evaluation of the orbital susceptibility     A damping constant l' (k, E) in GO <k; E> can be replaced by l" (kf, Ef) under  the conditions lE- Efi S(h2/2m) (2leffep - kp2) ancl lEk - sfl f{:(h2/2m) <2kffep - lep2)  with kf being the wave number of the Fermi levei and kp the inverse of the  range of force due to impurity3). Let G2 (q, E, E') be I(q, E, E')/{1 - r<q, E, E')},                       '22 ''''' Y. KATo '･'' '' No. 26then we expand GL)(q, E, E') as series in q. By substituting (12) into <7), theterm from the constant term in respect of q in G2(q, E, E') vanishes with thefirst term on the. right-hand side of (7>, and the iinear･term of q disappearsbecause of symmetry. The term containing q" (n;}i3) on the right-hand sideof (7) does not contribute to the orbital susceptibility, as it has. been shown by(1), for the reason of limit q->O.   The problern now is tQ find the quadratic terms of q on right-hand side of(12), let G2(2) be the ceeMcient of these terms, ･          ¥(-il-Ge(kE)Ge<kEr) um gEhe{GiSe3(iiiE) Ge(kEt) + gE,2G2GoOiiig,i{!)Ge(kEt)]    G2(2) ,., .                 1 - Z]P(k,)GO(kSE)GO(k,Ef>          ' h,         '  tt¥skGo<kE>Go(kE').;, p(kt) Go (ktE) (+:-,,, 62Ggtilig',E') 6GO(k,Et)  6Et}         + (1 - ;ii, P( kt>Go<k,E)Go(k,E, )l2                                                                (l3)where                     p(k> == nl v(k-f- k) 2cose}rh.Tlte summation over k can be repiaced by integration per unit volume, namely:!il]->ASoOOdEhEki12(A = (2x2)rmi(2m/h2)e/2>. By the sirnilar calculation has been madeby Ando3), G2(2) reduces to                                    t.                                          '             f<El/2 + Et y2)+ IllOl<E, - ExE312 m Et3/2)L )                         -F -34-i-<Esi2 + E's/2){3(Et -E)2 - 4r}2L2 jG2(2) = Arc        f        i + P<Arc)2              (Et - E)2 + 442                                            x,2r(Er - E)(E'cz g, g,IS'r'tEV'2,;-Ei!il,'.,,, . .t3,2,2L2 j{(Et - E)2 + 4ri2}2<14>No.. 26 Orbital Susceptibility of 'Impurity Systems 23with'                         L= r                              (Et - E)2 -i- 4r2and                         21T,2 = r(2r - PAffefl12),where we have made use of the constant value of P in respect of its argument.    Then, we reach the final formula for orbital susceptibility, i.e.,                    z= 2.pt,2BSdEidE' '1(< tlii')m- E4 Erm>G2(2) (is)where ptB denotes the Bohr magneton. In the limit P-->O, namely in the caseof no impurities, x reduces to                                           ,rv?ttB2       A                        Xe == --{i'IUB2 of1/2 =": - 2qlun,um (16)with N the number of electrons in the unit volume of the system. It is not soeasy to find the order estimation for iinpurity effect owing to the slight intricacyof formula <14), however, it seems that the orbital susceptibility of electronswith a small censentration of impurities is quite equal to that of the freeelectrons so far as both r and leT are negiigibly small compared with of, as ithas already been shown by Ando3).                 S4. Sum rule for electrical conductivity   Let us, now, consider the sum rule for electrical conductivity, then we dothe calculation of the conductivity using the Edwards' formula (8). Taking thedirection of q as the z-axis, we expand <G(k', k - q; E)G (k, k' + q; E')>, asseries in q., namely,                                                    '                                         co        <G(k', k- q; E)G(k, k' + q; E'>>.= :Z] q."gh(fe', le; fe, k'; E, E') (17)                                        nt=eand also                                   oo                     Ge(k'+q; E> = Xq."G.O(k'; E) (18)                                   n=ewith putting GO(k;E) as GeO(k;E). Inserting these equations into (8), we obtainthe equations in respect of qe, q, q2, q3 and so on. Since the formula of con-24 --,i,.','･ Y. KATO' .･ No"26ductivity is given from the equation of zero-th order in respect of q, using thesimilar approximation to the second section and defining g(E, E') by                                     '                              t ttttt tt                 g(E, E'>- Z]Z]k･k'gb<k, fe'; k', le; E, E'), G9)                           k h'                                                                     ./ .ttwe have to find the expressioR of g(E, E'). Then, g(E,E') becomes                                                 '         ttt'' g(E, k,> .,.l'･. iil.l,k2GoO(k; E)GoO<k; E') '. ,. ' . ,, <2o>                      .i ma A.qfii2 llili]Goe(h; E)Goo<k; Et)                             tt'                                            'where                  t ttt tt tttl ttt tt                               '               r, .. !n:l:ii2s 1 y<E;(i - cos o)M 2 cos od(cos e).In effect, we may replace the summation over k by integral, as we have madethe same procedure in the previous section, and we then have. /t 1.                 tt  '-..' . gSE.iii..b)lff.i. ZM, (,.!'7,)i/2(ellif2ve+..III,;3/.'){,r,..(S.-mif")iE'l/2. .,,,,            '                             '' Finally, we reach the conductivity propor,tional to . .   ' sdEsdEt l- -(t-i-1)t.[.ltJIF<.-,.:ll--)-,(E, Er), <22>that is, the electr2cal conductivity is proportional to {l]/<1'" - r'>}i/2 which isof .the order of unity. It seems to us that the sum rule9) for the electricalconductivity obtained by using the Edwards' approximation does not satisfied.Theyefore, the approximation represented by the so-called "ladder diagrams"seems to have been a good approximation for conductivity, however, it impliesthe existence of a problem in the Edwards' approxirnatioR.          ''                                                    '   ' Acknowledgement   The author would like to express his sincere thanks to Professor T. Andoof Kinki University for many valuable suggestions and discussions.No. 1) 2) 3)4) 5) 6) 7)8)9)10) 26 Oybital Susceptibility of lmpurity Systems                              References R. Peierls, Z. Phys. 80 (1933), 763. W. Kohn and M. Luming, J. Phys. Chem. Solids 24 (1963), 851. T. Ande, Progr. Theor. Phys. 37 (1967), 231. S. Nakajima, Advances in Phys. 4 (1955), 363;Proc. Phys. Soc. A 69 (1956),Progr. Theor. Phys. 22 G959), 430. T. Matsubara and Y. Toyozawa, Progr. Theor. Phys. 26 (1961), 739. L. Van Hove, Physica 21 (1955), 517. T. Ando and K. Tomita, Progr. Theor. Phys. 36 (1966), 955. S. F. Edwards, Phil. Mag. 3 (1958), 1020. R. Kubo, j. Phys. Soc. Japan 12 (1957), 570. M. J. Buckingham, Nuovo Cimento 5 (1957), 1763.25441;

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Orbital Susceptibility of Impurity Systems

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