Internal Mobility Edge in Doped Graphene: Frustration in a Renormalized Lattice

Transcription

1 PIERS ONLINE, VOL. 4, NO. 3, Internal Moblty Edge n Doped Graphene: Frustraton n a Renormalzed Lattce Gerardo G. Naums Departamento de Físca-Químca, Insttuto de Físca Unversdad Naconal Autónoma de Méxco (UNAM) Apartado Postal , Méxco DF 01000, Mexco Abstract We show that an nternal localzaton moblty edge can appear around the Ferm energy n graphene by ntroducng mpurtes n the splt-band regmen, or by producng vacances n the lattce. The edge appears at the center of the spectrum and not at the band edges, n contrast wth the usual pcture of localzaton. Such result s explaned by showng that the bpartte nature of lattce allows to renormalze the Hamltonan, and the nternal edge appears because of frustraton effects n the renormalzed lattce. The sze n energy of the spectral regon wth localzed states s smlar n value to that observed n narrow gap semconductors. Only very recently a two dmensonal form of carbon was obtaned [1]. Ths materal, known as graphene has attracted a lot of research due to ts amazng electrcal and mechancal propertes [2]. For example, electrons n graphene behave as massless relatvstc fermons that satsfy the Drac equaton [4]. Such property s a consequence of the bpartte crystal structure [5], n whch a lnear dsperson relatonshp appears at the center of the electronc spectrum. Also, one can cte the hgh moblty of ts charge carrers that remans hgher even at hgh electrc-feld nduced concentraton, that translates nto ballstc transport on a submcron scale [3] at 300 K. These and other unusual electronc propertes of graphene makes t a promsng materal for buldng electronc devces. However, from the pont of vew of applcatons, the use of pure graphene pose some problems. The transmsson probablty of electrons across a potental barrer s always unty, rrespectve of the heght and wdth of the barrer. Ths behavor s related to the Klen paradox n relatvstc quantum mechancs [2]. As a result, conductvty can not be changed by an external gate voltage, a feature requred to buld a FET transstor, although a quantum dot has been used to perform the requred task [6]. In spte of all ths research n pure graphene, at the moment there s not so much knowledge n the effects of mpurtes n the electronc propertes and ts potental use to produce gates. In a prevous work [7], the densty of states (DOS) of graphene wth Anderson type of dsorder revealed that the lnear dsperson relatonshp was affected [7], and recently many electrcal propertes of graphene wth dsorder have been obtaned [8]. However, the exstence of a moblty edge has not addressed. Here we show that graphene doped wth mpurtes or wth vacances presents a very unusual property; nstead of havng a localzaton moblty edge at the band lmts as n the usual Anderson localzaton, the localzed states appear at the center of the spectrum, around the Ferm energy. As we wll show, ths s a smple consequence of the bpartte crystal structure, whch produces a frustraton effect n a renormalzed. Hamltonan that removes one of the bpartte lattces. The observed effect can be used n certan applcatons, snce the moblty edge can be tuned wth a gven concentraton of mpurtes. Let us start by consderng the tght-bndng Hamltonan of graphene wth dsorder, whch can be wrtten as H = H 0 + H 1, where H 0 s the pure graphene Hamltonan gven by [9], H 0 = E 0 + γ 0 j + H 1. (1) <, j> E 0 s the self-energy of carbon and γ 0 s the carbon-carbon resonance ntegral, as gven n Ref. [9]. H 1 s the Hamltonan of the perturbaton due to defects, H 1 = δe + δγ 0 j, (2) <, j> where we defne δe E I E 0 and δγ 0 γ I γ 0. Here E I s the self-energy of the defects, and γ I the transfer ntegral between mpurtes (whch are bascally solated n the dlute lmts). When

2 PIERS ONLINE, VOL. 4, NO. 3, δe E 0, the spectrum s dvded n two parts, one centered around E 0 and the other at E 0 + δe. Ths case s known as the splt-band lmt. The states n the sub-band around the carbon self energy E 0, that we call the C -band, are strongly confned on carbon atoms. Furthermore, n the lmt δe E 0, t has been shown that mpurty atoms can be formally removed n a tght-bndng Hamltonan [10], and thus the C -band can be studed by usng a Hamltonan restrcted to C stes only, H CC = E 0 + γ 0 j. (3) C Ths Hamltonan descrbes an electron that can hop from one ste to ts neghbors only f both are carbon atoms (C ). Furthermore, the problem for the C sub-band s smlar to a lattce wth holes, because mpurty atoms act as perfect barrers n the lmt of nfnte self-energy. As a result, the results presented here are also vald for vacances n the lattce. Now let us study the spectrum of H CC. Frst t s convenent to work on a renormalzed Hamltonan H CC, whch takes advantage of the bpartte nature of the C lattce, once the I atoms are removed. The bpartte character of the C lattce means that t can be separated n two nter-penetratng sublattces, A and B. We defne two orthogonal operators that project the wavefunctons nto each sublattce, P A = A, j C, and P B = j B j j (4) Therefore, any egenvector φ of H CC can be wrtten n terms of these projectors, H CC (P A + P B ) φ = E(P A + P B ) φ. (5) Snce H CC produces a hoppng n the wavefuncton between the A and B sublattces, H CC P A φ = EP B φ, and H CC P B φ = EP A φ. (6) From these equatons, one can see that the spectrum s symmetrc around E = E 0, snce f (P A + P B ) φ s an egenvector wth egenvalue E, (P A P B ) φ s also an egenvector wth egenvalue E. We can decuple the sublattces by further applyng H CC to Eqs. (6), H CC (H CC (P φ )) = H 2 CC (P φ ) = E 2 (P φ ), (7) where = A, B. Thus, the projecton of an egenvector n each sublattce s a soluton of the squared Hamltonan. Observe that the egenvalues of HCC 2 are postve defnte, and ther egenstates are, at least, doubly degenerate. Ths spectrum can be regarded as the foldng of the orgnal spectrum of H CC around E = 0, n such a way that the two band edges of H CC, are mapped nto the hghest egenvalue of HCC 2, whle the states at the center of the orgnal band are now at the mnmum egenvalue of the squared Hamltonan (E 2 ). The mportant property of the renormalzed Hamltonan HCC 2 s that the states at the bottom of the spectrum have an antbondng nature (the phase between neghbors s π), and we can expect that the frustraton of the wavefuncton can prevent the spectrum from reachng ts mnmum egenvalue n a contnuous form when frustraton s present [11, 12]. In fact, frustraton acts as an effectve potental whch leads to localzaton snce the wave-functon tends to avod regons of hgher-frustraton. The moblty edge appears when the energy cost n localzaton s less than that of havng ampltude n frustrated bonds. As we wll show next, ths frustraton augments wth dsorder. To do ths, we observe that the Hamltonan HCC 2 s equvalent to a renormalzaton of stes B n the lattce, whch leads to a trangular lattce wth an effectve nteracton, as shown n Fg. 1(a). The new lattce contans odd rngs, and when mpurtes are present, there are holes, as ndcated n Fg. 1(b). The correspondng Schröednger equaton derved from HCC 2 s, ( (E E0 ) 2 Z γ0 2 ) c (E) = γ0 2 c j (E), (8) (j, ) A where c (E) s the ampltude of the wave-functon at ste for an egenenergy E, and the notaton (j, ) A means that the sum s taken only for carbon atoms whch are frst neghbours n the new trangular lattce,.e., those carbon atoms that were second neghbours n the orgnal lattce.

3 PIERS ONLINE, VOL. 4, NO. 3, Notce that such atoms belong to only one of the bpartte sublattces A or B. Due to the symmetry of the problem, we can solve for any sublattce, say for example sublattce A. Fnally, Z s the coordnaton number at ste. Ths number goes from 0 when a carbon atom s surrounded by mpurtes, to 3 n the lattce wthout defects. Then we can perform a varatonal procedure to estmate the ground state of Eq. (8). After multplyng Eq. (8) by c (E) and summng over, (E E 0 ) 2 = Z γ0 c 2 (E) 2 + γ0 2 c (E)c j (E), (9) (j, ) A Fgure 1: Renormalzaton of the graphene lattce. Atoms n the A sublattce are shown wth dfferent color than those n the B sublattce. The new lattce that appears after renormalzng the B, s represented wth double bonds. The frst contrbuton s an effectve self-energy whle the second depends on the number of bonds and the ampltude and phase of the wave-functon. For example, n pure carbon Z = 3. Also, the lattce s perodc from where we can wrte c j (E) = ce φj where c s an ampltude (n fact c = 1/ N where N s the number of atoms), and φ j s a phase. The mnmal egenvalue s thus obtaned from Eq. (8) when the phase dfference between stes n the A sublattce s π. Thus, the ground state has an antbondng nature and c (E)c j(e) = 1/N. Usng that there are three second neghbours for each atom, t follows that E = E 0. As a consequence, ths shows that there s no gap for pure graphene, as expected. However, the prevous case reveals an nterestng fact, the zero gap s obtaned due to the balance between the postve renormalzed self-energy Z and the antbondng contrbuton. In pure graphene, both contrbutons match exactly to produce a gapless spectrum. Now consder the case of a fnte concentraton x of mpurtes or holes. Snce an mpurty belongs to one of the bpartte sublattces, say A, there are two effects. The frst s a reducton n the average coordnaton number and the second s that some bonds are deleted. Ths coordnaton effect s estmated as follows. The frst term of Eq. (9) can be wrtten as an average term plus a correlaton of ampltude-coordnaton, Z γ0 c 2 (E) 2 = Z γ0 2 + V γ0 2 δz δc 2 (E) (10) where t was used that Z can be wrtten as an average Z plus a fluctuaton part δz. A smlar procedure can be made for c (E) 2 c 2 (E) + δc 2 (E). The average coordnaton number can be obtaned by observng that around a gven carbon atom, there are four possble confguratons: t can be surrounded by one, two and three mpurtes, or t can be completely surrounded by carbon atoms. For each confguraton, there s a dfferent coordnaton number Z, snce mpurtes act as holes. As a result, the coordnaton number Z has a bnomal probablty dstrbuton P (Z) = CZ 3 xz (1 x) 3 Z where CZ 3 s a combnatoral factor. It follows that Z s the frst moment of the bnomal dstrbuton: Z = Z=3 Z=0 ZP (Z) = 3(1 x). The contrbuton of the last term n Eq. (10) leads to the producton of mpurty states, snce t s the correlaton between ampltude and self-energy fluctuatons. Thus, the system has a moblty edge when ths term lowers the energy compared wth the energy requred for havng an extended state wth ampltude n frustrated bonds.

4 PIERS ONLINE, VOL. 4, NO. 3, Fgure 2: Renormalzaton of the lattce wth defects. The mpurtes are shown wth dark color. There are two cases: the mpurtes can fall n the A sublattce or n the B, as ndcated n the fgure. In the frst case, sx bonds are deleted n the renormalzed sublattce, whle only three dsappear n for other case. The other effect s the removal of bonds that changes the second term of Eq. (9). We can estmate ths effect as follow. For low concentraton of mpurtes x 1, most of them are solated, snce the probablty of havng two mpurtes as neghbours goes as x 2. Thus, we wll consder that mpurtes are solated. Two stuatons are possble. Ether an mpurty belongs to the renormalzed sublattce, or t can reman as shown n Fg. 2. For each mpurty ste that s renormalzed, 3 bonds are lost. In the other case, 6 bonds are lost for each mpurty. Snce they are randomly dstrbuted n sublattces A and B, the concentraton of mpurtes s x on each sublattce. As a result, the number of mssng bonds s (6+3)xN, from a prevous total of 3N. Usng ths count n Eq. (9), and assumng no self-energy ampltude correlaton Eq. (10) for an antbondng tral state, we obtan the approxmate poston of the moblty edge (E d ), (E d E 0 ) 2 3γ 2 0(1 x) γ 2 0(3 9x) = 6γ 2 0x, whch leads to a symmetrc moblty edge separated an energy from the center of the band, ± 6xγ 0. (11) As a check of these deas, n Fg. 3 we present the normalzed logarthm of the nverse partcpaton E (ev) Fgure 3: Logarthm of the nverse partcpaton rato as a functon of the energy for pure graphene (lne) compared wth the dope case wth x = 0 01 (trangles) and x = 02 (squares) around the center of the spectrum of the carbon sub-band, for a lattce wth 5184 stes. Observe the rse at the center of the spectrum for the doped case. A band of degenerate states s also observed for pure graphene. The zero corresponds to the Ferm energy.

5 rato, PIERS ONLINE, VOL. 4, NO. 3, α(e) = log IP (E) log N where IP (E) s the nverse partcpaton rato, defned as IP (E) = N =1 c (E) 4, whch s a well-known measure of localzaton. For extended states, α(e) 1, whle t tends to be bgger values when localzaton s present. Fg. 3 shows a comparson between pure graphene case and the doped cases, for a tgth-bndng smulaton usng an average of 10 lattces wth N = 5184 stes. It s worthwhle mentonng that a band of degenerated states appears n the center of pure graphene, whch has not been reported prevously by other workers. They are a consequence of the local topology of the lattce, as also happens n the square [10] and Penrose lattces, and are due to a decouplng of the A and B sublattce at the center of the spectrum. Fg. 3 shows that the IP (E) s n general bgger for the doped case, but at the center of the spectrum there s a clear rse n ts value, ndcatng a greater degree of localzaton. In Fg. 4 we compare Eq. (11) wth the numercal value of obtaned from the localzaton plot, whch shows a good agreement wth the predcted value. Fgure 4: Theoretcal value of the moblty edge predcted from Eq. (11), ndcated wth a sold lne, and the value obtaned from a drect dagonalzaton of the Hamltonan (squares). The numercal results were obtaned from an average of 10 lattces wth N = 5184 stes. The value of γ 0 s around [9] 0.9 ev or γ 0 = 20 Kcal/mol. For a 1% dopng, the sze of the whole localzed regon s around ev. Snce lght absorbed when the band-gap energy s n the lmt of the vsble spectrum 1.77 ev (700 nm), the localzed regon n doped graphene can be consdered as smlar n sze as the energy gap n narrow-band-gap semconductors. In concluson, we have shown that doped graphte n the splt band regmen presents a moblty edge at the center of the spectrum, an ths can be useful for many devces snce the poston of the moblty edge can be controlled by dopng. ACKNOWLEDGMENT I would lke to thank M. and H. Terrones for useful suggestons. Ths work was supported by DGAPA UNAM project IN108502, and CONACyT F and REFERENCES 1. Novoselov, K. S., et al., Scence, Vol. 306, 666, Katsnelson, M. I., Materals Today, Vol. 10, 20, Gem, A. K. and K. S. Novoselov, Nature Materals, Vol. 6, 183, Semenoff, G. W., Phys. Rev. Lett., Vol. 53, 2449, Slonczewsk, J. C. and P. R. Wess, Phys. Rev., Vol. 109, 272, Novoselov, K. S., et al., Scence, Vol. 306, 271, Hu, W. M., J. D. Dow, and C. W. Myles, Phys. Rev. B, Vol. 30, 1720, Peres, N. M. R., F. Gunea, and A. H. Castro-Neto, Phys. Rev. B, Vol. 73, , 2006.

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