Triplet excitations in graphene-based systems
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Triplet excitations in graphene-based systems
Vladimir Posvyanskiy, ∗ Logi Arnarson, and Per Hedeg˚ard
Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
In this article we investigate the excitations in a single graphene layer and in a single-walled carbonnanotube, i.e. the spectrum of magnetic excitations is calculated. In the absence of interactions inthese systems there is a unique gap in the electron-hole continuum. We show that in the presenceof Coulomb correlations new states, magnons, appear in this forbidden region. Coulomb interactionis examined in the context of Pariser-Parr-Pople (PPP) model which takes into account long rangenature of interaction. The energy of new bound states depends on the strength of Coulomb forces.The calculations are performed for arbitrary electron-hole ( e − h ) momentum q what allows to findthe magnons dispersion law ε ( q ) , effective mass m ∗ and velocity v gr . Finally, we determine thecritical values of system parameters when this type of excitations can exist. I. INTRODUCTION
Since the first isolation of graphene [1], a single two-dimensional (2D) atomic layer of graphite, it has at-tracted a lot of attention of both experimentalists andtheoreticians. Such characteristics as high conductivitymakes graphene a candidate for variety of modern na-noelectronics applications. Mainly, graphene differs fromusual 2D materials in that electrons have a linear rela-tivistic like dispersion law and zero band gap. Becauseof these unusual properties a number of new effects ap-pear in this material. Thus, one of the examples ofsuch unusual behavior are graphene plasmons. They arecollective oscillations of electronic density which can befound only in a spin singlet state. Currently, plasmons ingraphene are under intensive investigation [2, 3]. Thereare a lot of works showing that dispersion law and proper-ties of plasmons for Dirac electrons differ markedly fromthe plasmons in conventional 2D materials. For instance,in 2D semiconductors at long wavelength the plasma fre-quency ω p ∼ n , whereas in graphene ω p ∼ n . Thisis a direct consequence of the quantum relativistic na-ture of graphene [4–6]. It is important to point out thatrecent experimental results [7] agree well with theoret-ical predictions [4]. Another significant feature of thegraphene plasmons is long lifetime caused by the peculiarway of damping [8, 9]. Unlike conventional 2D materials,Landau damping occurs due to interband transitions ingraphene. The edges of this region can be moved by ma-nipulation of the doping level. As the authors of ref.10claim, for sufficiently large doping values low plasmonlosses are possible in graphene.Another type of excitation is the exciton, an electronand a hole coupled by their Coulomb attraction. Excitonstates in different semiconductors and π -bonded planarorganic molecules have been studied for decades. Theirenergies lie in the band gap close to the bottom ofconduction band. However, recently, it has becomeclear that they can also be found in semiconductorsingle walled carbon nanotubes (SWCNTs) [11–13] or ∗ [email protected] in gapped graphene [14–16]. Low dimensionality ofsuch structures results in strong Coulomb interactions.That opens the way for carbon-based optoelectronicdevices operating at room temperature [17, 18]. A lotof experimental and theoretical investigations [19] havebeen done in this area. Exciton states can be both spin -singlet or triplet. It is known that excitons in SWCNTsform a complex series of 16 exciton states. However,many authors maintain that only singlet exciton withodd parity is optically active and all the others are dark[20, 21]. Therefore, there is a great deal less literatureabout the structure of low lying triplet excitons. Despiteof this, using different experimental techniques a satellitepeak in photoluminescence spectrum attributed to thetriplet dark exciton was measured independently bydifferent groups [22–24]. It was discovered that magneticexcitons have longer lifetimes [24, 25] than singlet ones,which makes them extremely important for photoelec-tronics applications and spin transport experiments [26].During the last decade there was a discussion about theexistence of a neutral spin triplet mode in a graphenesheet [27–29]. Some authors showed the existence of aspin-1 collective mode in undoped graphene using theHubbard model and found its dispersion law [27]. How-ever, in ref.28 such solution was not found. Therefore,the issue remains open. In this article we prove that thetriplet particle-hole excitations, magnons, do exist bothin undoped and doped graphene. This problem is solvedtaking into account all the matrix elements of the Hamil-tonian. In addition, we are interested in these excitationsin the case of doped graphene. We carry out a full analy-sis of magnons spectrum properties what allows us to findconditions for existence of the spin-1 mode in the system.What is more, Coulomb interaction is considered in thecontext of the PPP-model.The origin of the triplet excitations is very similar tothe triplet excitons mentioned above or, e.g., to Stonerexcitations observed in ferromagnets. Mainly, the exis-tence of magnons in doped graphene is made possible bythe gap in the two particle spectrum [27, 30]. Indeed, the e − h spectrum of graphene differs from the conventional2D materials [30]. In metals, e.g., the e − h continuum isformed by intraband transitions only. At the same time,in large band gap semiconductors there is a room onlyfor interband excitations because the chemical potentiallies in the band gap. In graphene the situation is a bitmore complicated. In undoped graphene the Fermi sur-face shrinks to the point where two cones of valence andconduction bands are connected. Therefore, there areonly interband transitions. However, moving the chemi-cal potential away from zero leads to appearance of intra-band transitions lying in the low energy region. Becauseof relativistic electronic properties of graphene these twoprocesses form a window in the e − h spectrum. The sizeof the gap depends on the value of doping. For directtransitions ( | q | = 0) the gap size equals to 2 µ .We show that magnons are formed only in the presenceof Coulomb interaction. Hence, the model describing cor-relation effects plays an important role in our calcula-tions. We compare the results obtained with Hubbardand PPP models. It is shown that they differ, especially,when screening effects are not too strong. In addition,our calculations allow to get information about the effec-tive mass and velocity of magnons. II. MODEL
It is known that graphene has a honeycomb latticeformed by two triangular sublattices i.e. A and B , and aunit cell consists of two carbon atoms (Fig.1).We consider a Pariser-Parr-Pople (PPP) model which A B d d d a a xy FIG. 1. Crystal structure of graphene. Two basis vectors a = n √ a , a o and a = n √ a , − a o form a unit cell. Blueand red circles indicate atoms belonging to the sublattices A and B , respectively. Each atom at site B is connectedto the three nearest neighbor atoms of type A by the vec-tors: d = n a C − C , √ a C − C o , d = n a C − C , − √ a C − C o and d = {− a C − C , } . a = √ a C − C and a C − C ≈ . A (distancebetween nearest neighbors). If R ( A ) i is a position of atom A in the unit cell i , then the position of B atom in the sameunit cell is: R ( B ) i = R ( A ) i − d . has successfully been used in different π - conjugated sys- tems [31, 32]. The PPP Hamiltonian describing our sys-tem is written in the following way:ˆ H = ˆ H tb + ˆ H U + ˆ H V . (1)The tight-binding term is expressed like:ˆ H tb = X ij X α α σ ( t α α ij ˆ c † iα σ ˆ c jα σ + h.c. ) , (2)where operators ˆ c † iασ (ˆ c iασ ) create (annihilate) an elec-tron with spin σ ( σ = ↑ , ↓ ) in the unit cell i on the atombelonging to the α ( α = A/B ) sublattice. t α α ij is thehopping amplitude between nearest neighbor sites.The second term describes on-site Coulomb repulsion:ˆ H U = U X iα ˆ n iα ↑ ˆ n iα ↓ , (3)where U is the strength of on-site interaction andˆ n iασ = ˆ c † iασ ˆ c iασ .Finally, the last term is a long range interaction termdefined by:ˆ H V = X ij X ′ α α V α α ij (ˆ n iα − n jα − , (4)where ˆ n iα = P σ ˆ c † iασ ˆ c iασ is the number of electrons onthe site iα . V ij is the value of off-site Coulomb interac-tion. Prime in the second sum means that α = α when i = j .There are various ways to interpolate the long range partof the interaction. In this article we use the Ohno inter-polation formula [33]: V α α ij = U r | R ( α i − R ( α j | a , (5)where R ( α ) i determines the position of iα atom and a isa numeric parameter connected with screening effects.Indeed, in a long range limit V ij = U a | R i − R j | . Wherefromthe relation between a and dielectric constant of the sub-strate can be found: a = e πǫ U ǫ r , (6)where e is an electron charge, ǫ r is dielectric constant ofa substrate and ǫ is vacuum permittivity.From the expressions (4) and (5), obviously, that if i = j and α = α , then V ii = U and we obtain the Hubbardterm.The main advantage of this approximation is that forspecified value of ǫ r there is only one parameter U inthe Hamiltonian what makes it much easier to analyzethe properties of the system.In this article the band structure of graphene is de-scribed by the tight-binding model, from which it isknown that ε ( k ) = βt | f ( k ) | , where f ( k ) = 1 + e ı ka + e ı ka and β = + / − denotes either conduction or valencebands, respectively. The transformation between the realspace and the momentum representations is: c † iασ = 1 √ N X k β e − ı kR i U − k βα c † k βσ , (7)where U − k βα are the matrix elements of the unitary ma-trix: ˆ U − k = (cid:18) − e − ıφ k e ıφ k (cid:19) , (8)and φ k = arg( f ( k )).Because of a magnon being a bosonic excitation, it mustbe described by bosonic operators. For these purposes,unlike ref.34, we introduce three new operators creatinga triplet e − h pair with S z = 1, S z = 0, S z = − (cid:16) B , (cid:17) † = c † ↑ c ↓ (cid:16) B , (cid:17) † = √ (cid:16) c † ↑ c ↑ − c † ↓ c ↓ (cid:17)(cid:16) B , − (cid:17) † = c † ↓ c ↑ (9)and one operator which creates a singlet e − h pair: (cid:16) A , (cid:17) † = 1 √ (cid:16) c † ↑ c ↑ + c † ↓ c ↓ (cid:17) . (10)In the definitions above 1 and 2 are general quantumnumbers. In general, these operators do not describe qa c-c ω / t ω=-0.017+1.278q-0.55q FIG. 2. Particle-hole spectrum and a spin-1 mode for un-doped graphene. Blue area is a region of interband transi-tions in the absence of Coulomb interaction, while black dotsdenote the magnons spectrum computed using the Hubbardmodel ( U t = 4). Dashed black line is a dispersion law of themagnons. bosons. However, with respect to the ground state, theFermi Sea state, and considering that the operator c † σ creates an electron with spin σ above the chemical po-tential, µ , while c σ creates a hole with spin − σ belowthe chemical potential, usual commutation relations arefulfilled: h F S | (cid:20)(cid:16) B , (cid:17) † , B ′ , ′ + (cid:21) | F S i = − δ ′ δ ′ , h F S | (cid:20)(cid:16) B , − (cid:17) † , B ′ , ′ − (cid:21) | F S i = − δ ′ δ ′ , h F S | (cid:20)(cid:16) B , (cid:17) † , B ′ , ′ (cid:21) | F S i = − δ ′ δ ′ , h F S | (cid:20)(cid:16) A , (cid:17) † , A ′ , ′ (cid:21) | F S i = − δ ′ δ ′ . (11)Therefore, it can be shown that the on-site interactionterm of the Hamiltonian, H U , can be presented in theinvariant form: H U = U N X ′ ′ T ′ ′ (cid:18) − K † ′ · K ′ + 12 (cid:16) A , ′ (cid:17) † A , ′ (cid:19) , (12)where just for simplification: = { k β } , = { k − q β } , ′ = { p − q β } , ′ = { p β } , T ′ ′ = P α U α ( U α ) ∗ U ′ α ( U ′ α ) ∗ .and K † ij = (cid:16) B i,j + (cid:17) † + (cid:16) B i,j − (cid:17) † (cid:16) B i,j + (cid:17) † − (cid:16) B i,j − (cid:17) † ı √ (cid:16) B i,j (cid:17) † . (13)The same procedure could be done and with H V : H V = 1 N X ′ ′ F ′ ′ (cid:18) − K † ′ · K ′ − (cid:16) A , ′ (cid:17) † A , ′ (cid:19) , (14)where the following notations were introduced: F ′ ′ = X ′ α α V α α q U α ( U α ) ∗ U ′ α ( U ′ α ) ∗ ,V α α q = X j V α α j e − ı qR j . Now, let us consider the ground state of unexcitedgraphene is the Fermi Sea state | F S i . In this articlewe are going to find the spectrum of S z = 1 excitations.It can be done without loss of generality because of therotation invariance of the Hamiltonian. We look at a sub-space of the total many-body Hilbert space which con-sists only of these excitations. Therefore, we specify the ω / t qa c-c c) ω / t qa c-c d) ω=0.20+0.16q-5.44q ω=0.191+0.009q-0.359q ω / t a) qa c-c ω / t qa c-c b) ω / t Interband transitionsIntraband transitions
FIG. 3. Triplet excitations spectra calculated with the Hub-bard model. a) and b) are low and high energy magnon modesin the doped graphene ( µt = 0 .
18) for U t = 4, respectively.c) Spectrum of magnons in the armchair (5,5) SWCNT for U t = 3 (red circles). Blue circles denote the spectrum for U t = 0. d) Magnons dispersion relations for graphene (blackcurve) and for the (5,5) SWCNT (red curve). states living in that subspace as superposition of e − h pairs: | ψ q i ( T ) = X k β ′ β ′′ a k β ′ β ′′ (cid:16) B k β ′ , k − q β ′′ + (cid:17) † | F S i , (15)where a k β ′ β ′′ are the coefficients which in general arecomplex. Similarly, it is possible to construct a wavefunction describing singlet excitations of graphene corre-sponding to the plasmon mode: | ψ q i ( S ) = X k β ′ β ′′ a k β ′ β ′′ (cid:16) A k β ′ , k − q β ′′ (cid:17) † | F S i . (16)Therefore, finding the spectrum of magnetic exci-tations reduces to solving the Schrodinger equation H | ψ q i ( T ) = ε q | ψ q i ( T ) for given values of q . By substi-tution of (1) and (15) one gets a system for coefficients a k ββ ′ . In this article the spectrum is calculated for apiece of graphene composed of 1225 unit cells (or 2450carbon atoms). We determined such size of the sample toeliminate the impact of size quantization effects. So, toobtain the spectrum of magnons the eigenvalues problemfor the square (1225 × III. RESULTS AND DISCUSSION
First of all, let us consider the case of undopedgraphene sheet. Without Coulomb correlations the e − h U /t ω / t Formation of magnons in graphene and in SWCNTsFormation of magnons in SWCNTs only
Interband transitions (U ) gr (U ) tub FIG. 4. Magnons energy dependence as a function of strengthof Coulomb interaction for | q | = 0 for graphene (black curve)and for (5,5) carbon nanotube (red curve). continuum consists only of the interband transitions.However, Coulomb interaction couples an electron anda hole what can lead to appearance of bound states.This situation is presented in the Fig.2. There is a curvebeneath the region of e − h excitations, which corre-sponds to magnons. Thus, our calculation confirmedthe Baskaran and Jafari proposal in the ref.27 on theexistence of a magnetic collective mode.Now, if one dopes graphene, there is a window in thetwo particle spectrum and, as stated above, at small mo-menta and in the presence of Coulomb interaction wecan expect to find new states in it. In Fig.3a,b thespin-1 spectrum is presented. It was found numericallyby solving the Schrodinger equation for Hubbard model( V ij = 0, U = 10 . eV ) and for different values of q . Itis seen there are two new branches. The first one is inthe long wavelength region (panel a)), while the secondone, high energy branch, is located at small momenta(panel b)). For better insight into the system behavior,it is useful to plot energy dependence of the bound statefrom the strength of Coulomb interaction for specific mo-mentum value. The black curve in the Fig.4 shows thatenergy of this bound state diminishes with rise of U .It is immediately seen from (12). The triplet compo-nents of interaction give a negative contribution to theenergy of the excited states, while a singlet one has apositive sign. Therefore, under influence of Coulomb in-teraction low and high energy magnon modes are formedby the shift of the states from the interband and intra-band transitions regions, respectively. The size of thisshift is proportional to the value of U . Therefore, wecan speak of a critical value (cid:0) U crit (cid:1) gr above which thenew states could be formed in the initially forbidden area.It is necessary to mention, that in ref.35 authors alsodemonstrated the existence a magnon mode. However,it appears at much smaller values of interaction than inour calculations. Probably, such discrepancy is the con-sequence of considering all the matrix elements in thesolution of the Schrodinger equation. ω / t qa c-c ε r =4.4ε r =5.9 ε r =8.9 ε r =17.8 ε r =∞ a) ω / t qa c-c ω=0.206+0.009q-0.528q ω=0.236+0.135q-2.709q ω=0.219+0.093q-1.719q ω=0.254+0.184q-4.101q b) FIG. 5. a) PPP-spectrum of magnons in the doped graphenefor different values of q and ǫ r . b) Low energy magnons dis-persion laws for different values of ǫ r . Next, we consider carbon nanotubes. We would like toshow that there are the same excitations in the metallicsingle-walled carbon nanotubes (mSWCNTs). The onlydifference between this case and the one described aboveis different graphene and mSWCNT dispersion laws.It is known that SWCNT consists of a graphene sheetthat is rolled over a chiral vector C nm = n a + m a =( n, m ), where n and m are some integers. There arethree classes of the SWCNTs: armchair (n,n), zig-zag(n,0) and chiral (n,m). The condition for been metallicis n − m = 3 q , with integer q [36, 37]. In this article weconsider only armchair nanotubes, but the calculationsfor zigzag one could be performed in the same manner.In the SWCNTs one of the component of the momentumis quantized, consequently, it is possible to show that theenergy dispersion relation for armchair carbon nanotube U =3.6tU =4tU =3.3t ε r n No Bound StatesMagnons Formation
FIG. 6. The diapason of parameters n , U and ǫ r wheremagnons can exist. Solid line – U t = 4; dashed – U t = 3 . U t = 3 .
3. Above these edges a magnon modecan appear with given values of parameters. is [36]: ε ν ( k ) = s (cid:18) ak (cid:19) cos (cid:16) πνn (cid:17) + 4 cos (cid:18) ak (cid:19) + 1 , (17)where k - is the continuous component of the wavevector, while ν corresponds to the discrete part of thewave vector (band index). Considering ν = 0, we get theparticle-hole spectrum similar to graphene. Therefore,it is instinctively clear that there should be a magnonmode, as well (Fig.3c,d). However, there are a numberof differences from the graphene magnons. The mainone is lower value of critical Coulomb interaction. InFig.4 the red curve shows that the bound state appearsin the armchair nanotube at (cid:0) U crit (cid:1) tub ≈ .
4, whilein graphene (cid:0) U crit (cid:1) gr is twice that. Another sharpdifference is that in the SWCNTs magnons have muchstronger energy dependence from momentum than ingraphene (Fig.3d). That is why, a magnon mode in thetubes is dumped at smaller values of the wave vector,comparing with the case of graphene.Finally, we shall study the effect of long rangeCoulomb interaction. Concentrating only on the lowenergy magnons, from Fig.5 it is seen that long range in-teraction shifts up the bound states from the their initialenergy, calculated for V ij = 0. This shift is the smallestfor large values of dielectric constant ǫ r . It is explainedby (5) from which we have V ij ∼ U p ǫ r | R ij | . Forlarge ǫ r the screening effects become stronger reducingthe role of long range interactions. The values of thedielectric constants of most semiconductors lie in therange ǫ r ∼ −
16. For such substrates, as can beseen, difference between the Hubbard model and thePPP-model is quite appreciable. Nevertheless, in aqualitative sense the structure of the spectrum is notchanged. Therefore, substituting U by some effectiveCoulomb interaction V eff the spin-1 spectrum can becalculated using the Hubbard model without loss ofaccuracy.As shown above, there is a critical value of on-siteCoulomb interaction below which the bound states arenot formed. However, importantly that condition forappearance of these states depends not only on U , butalso on ǫ r and µ (Fig.6). One of the most essential fea-tures is growth of the value of doping needed for magnonformation when the strength of on-site interactiondecreases. Taking ǫ r = 15 and U = 4 t = 10 . eV , thedoping level corresponding to the bound state formationshould be, roughly, n = 0 . µ = 0 . eV ) whatis quite sensible and experimentally achievable value.However, for suspended graphene ( ǫ r = 1) the dopingmust be much higher.In Fig.5b we present the magnons dispersion laws. Asit is shown, the frequency dependence from momentumbecomes stronger when the role of H V -term increases.Approximating obtained data we can find the values of xy a) Distance, a c-c P r o b a b i l i t y , % b) P r a b a b i l i t y , % ε c)FIG. 7. a) Spatial probability distribution of electron localization in graphene crystal for a fixed position of a hole calculatedusing Hubbard Hamiltonian ( U t = 4). The radius of the circle is proportional to the probability of finding electron on the site.Blue/red circles are again atoms in sublattices A/B . A hole is localized on the atom of type A and is denoted by (cid:13) . b)Fullprobability to find an e − h pair in the graphene crystal as a function of distance between an electron and a hole positions.Blue circles denote the case when both electron and hole are in the same sublattices. Pink squares describe the case when anelectron and a hole belong to different sublattices. The probabilities were calculated for Hubbard model ( U t = 4). c)Probabilitydependence of localization both an electron and a hole on the same site as a function of ǫ r . magnons velocity and effective mass. Thus, for the curvecorresponded to ǫ r = 17 . v mag = ∂ω∂q ≈ ∗ ms , what is three orders less thenthe Fermi velocity in graphene. Magnon effective massis m ∗ = (cid:16) ∂ ω∂q (cid:17) − = 1 . m e .Finally, the last thing which is analyzed is how themagnons appear in the real space. To do it one has tocalculate the probability to find an e − h pair somewherein the graphene lattice. For instance, if the hole positionis fixed and it is localized on the atom of type α = A/B ,then the probability to find an electron in the position R i on the same type of atom is: P αα = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k a k ββ ′ e − ı kR i e − ıφ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (18)and the probability to find an electron on the differenttype of atom is: P α α = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k a k ββ ′ e − ı kR i e − ıφ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)Evidently, we have P AA = P BB and P AB = P BA .Fig.7a shows the spatial probability distribution of find-ing an electron somewhere in the crystal when the holeposition is fixed (its position is marked on the graph as (cid:13) ). One sees that the distribution function has the samesymmetry as the graphene lattice. As we can see, there is the largest probability to find an electron on the samesite as a hole what corresponds to a spin-flip. This sit-uation is quite similar to that observed in a 1D chain ofaligned spins when one spin-flip event causes a magnon.As it is shown on Fig.7b the probability distribution isexponentially decreasing as we move away from the po-sition of a hole. This fact proves that magnons are reallylocalized in the sample we used for our calculations andthat the choice of its size was well founded. Finally, itis quite interesting to know how long range interactionaffects the probability of magnon observation (Fig.7c). Itshows that the probability that both an electron and ahole are on the same site increases with rise of ǫ r andin the high ǫ r limit it achieves the result obtained us-ing the Hubbard model. Thus, it is shown that for largedielectric constants the magnons wave function is morelocalized in space. IV. CONCLUSION
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