APS/123-QED States near Dirac points of rectangular graphene dot in a magnetic field S. C. Kim1, P. S. Park1, and S.-R. Eric Yang1,2 ∗ 1Physics Department, Korea University, Seoul Korea and 2Korea Institute for Advanced Study, Seoul Korea (Dated: January 8, 2010) In neutral graphene dots the Fermi level coincides with the Dirac points. We have investigated in the presence of a magnetic field several unusual properties of single electron states near the Fermi level of such a rectangular-shaped graphene dot with two zigzag and two armchair edges. 0 Wefind that a quasi-degenerate level forms near zero energy and thenumberof states in this level 1 can be tuned by the magnetic field. The wavefunctions of states in this level are all peaked on 0 the zigzag edges with or without some weight inside the dot. Some of these states are magnetic 2 field-independentsurfacestateswhiletheothersarefield-dependent. Wehavefoundascalingresult from which the numberof magnetic field-dependentstates of large dots can be inferred from those n of smaller dots. a J PACSnumbers: 8 ] I. INTRODUCTION graphene[9,10]withzigzagedgesalongthex-axisdevelop l l a flat band of zero energy chiral states. These states are a h surfacestatesandarelocalizedstatesatthezigzagedges - withvariouslocalizationlengths[8]. The zigzagedgeand s the LLL states have zeroenergy because their wavefunc- e m A tions are chiral. Effects of a magnetic field on graphene Hall bars have been investigatedrecently, and some zero . t energychiralstates arefoundto be stronglylocalizedon a m B the zigzag edges in addition to the usual LLL states[11– y 14]. - d One may expect that the degeneracy of chiral states n x with zero energy will be split when quantum confine- o menteffectisintroducedinagraphenedot[15–17]. How- c FIG.1: Finitegraphenelayerwithzigzagandarmchairedges. ever, the splitting of these energies may be unusual [ There are equal number of A and B carbon atoms. The in some graphene dots[18–21]. Recently the magnetic graphenelayerhasreflectionsymmetriesabouthorizontaland 1 field dependence of these levels in a gated graphene verticallinesthat gothroughthecenterofthelayer. Amag- v dot was investigated experimentally[21]. Effects of var- netic field is present perpendicularto the layer. 1 ious types of edges have been also investigated: zigzag- 0 edged dots, armchair-edged dots[19, 22–24], and rectan- 2 gular graphene dots with two zigzag and two armchair 1 . Graphene dots have a great potential for many ap- edges[25, 26] have been studied. Armchair edges cou- 1 plications since they are the elemental blocks to con- ple states near K and K points of the first Brillouin 0 ′ struct graphene-basednano devices. It is possible to cut zoneandgenerateseveralmixedchiralzigzagedgestates 0 1 graphenesheet[1]inthedesiredshapeandsize[2],anduse with nearly zero energies. In the rectangular dots the v: it to make quantum dot devices. In such devices it may number of these states, Nl, may be determined from the be possible to realize experimentally zigzag or armchair condition that the x-component of wavevectors satisfies i X boundaries. 1/L < k < π/3a, where k = πn 2π, a = √3a y x,n x,n Lx − 3a 0 r Graphene systems possess several unusual physical is the length of the unit cell, and n = 0, 1, 2,... (the a properties associated with the presence of the Dirac nearest neighbor carbon-carbon distance±is a±= 1.42˚A, 0 points. For example, compared to ordinary Landau lev- the horizontal length of the dot is L = √3Ma with x 0 els of quasi-two-dimensional semiconductors the lowest M numberofhexagonsalongthe x-axis,andthevertical Landau level (LLL) of graphene is peculiar since it has lengthisL =a (3N+2)withN thenumberofhexagons y 0 zero energy that is independent of magnetic field[3–7]. and (N +1) carbon bonds along the y-axis. See Fig.1). Moreover, wavefunctions of the LLL are chiral, i.e., the This condition implies that the integer n is given by probability amplitude of find the electron on one type carbon atoms is zero. There are other graphene systems √3M 2M + n M. (1) with zero energy states. Semi-infinite[8] or nanoribbon (3N +2)π 3 ≤ ≤ Theeffective massapproximationwavefunctionsofthese surface states are derived in Ref.[26]. correspondingauthor,[email protected] We investigate how properties of rectangular dots ∗ 2 dots since the number of carbon atoms increases rapidly withthe size. We havefoundascalingresultfromwhich onecaninferresults forlargerdotsfromthoseofsmaller dots. Fordifferentrectangular-shapedgraphenedotsand values of φ it can be described well by the following di- mensionless form L L ℓ2 x y N (φ)= f( ), (3) D A a (L +L ) h 0 x y FIG.2: Numberofnearlyzeroenergystatesthatareinduced where ℓ = 33/4a0 is the magnetic length and f(x) is a by a magnetic field follow a scaling curve when plotted as a (4πφ)1/2 function of 1 for different rectangular shaped scalingfunction,seeFig.2. Thetotalnumberofhexagons (M+√3N+2/√3)φ inthedotisN =L L /A . Ournumericalresultshows graphene dots. Here δ=0.01eV. h x y h thatthedependenceofN (φ)onφisinitiallynon-linear D intheregimewherethediameterofthecyclotronmotion is comparable to the system length, 2ℓ L . x,y ∼ change in the presence of a magnetic field in the regime where Hofstadter-butterfly effects[18, 19, 21] are negligi- II. NUMBER OF STATES IN THE ble. In neutral graphene dots the Fermi level has zero QUASI-DEGENERATE LEVEL energy, and, consequently, magnetic, optical, and STM propertiesareexpectedtodependonthenumberofavail- Our Hamiltonian is able states near zero energy. Our investigation shows that the wavefunctions of nearly zero energy states are all peaked on the zigzag edges with or without appre- H = − X tijc†icj, (4) <i,j> ciable weight inside the dot. This may be understood aosf tmhiexisnqguaorfeLdLoLt athnrdousughrfaicnetesrtvaatlelesybsycaatrtmercihnagi,rwedhgicehs wheretij =tei~eRR~R~jiA~·d~r are the hopping parametersand c+ creates an electron at site i. Here we use a Lan- is unique to the square dot (This will be explained in i Sec.IV). dau gauge A~ = B( y,0,0). The summation < i,j > − Our study shows that the number of states within the is over nearest neighbor sites and t = 2.5eV. The eigen- energy interval δ around zero energy is given by statewith eigenenergyǫn is denotedbyφǫn(R~), whereR~ labels each lattice point. Because of electron-hole sym- N (φ)=N +N (φ) (2) metry eigenvalues appear in pairs of positive and nega- T l D tivevalues(ǫ, ǫ),andtheprobabilitywavefunctionsofa − (the energy δ is typically less than the quantization en- pair of states, (φ (R~)2, φ (R~)2), are identical. Our ergy of a rectangular dot, which can be estimated using numerical result|sǫanre co|ns|ist−eǫnnt wi|th this. the Dirac equation: γkmin ∼ tLxa,y, where γ = √3ta/2 Figs.3(a) and (b) display the energy spectra near zero with the hopping energy t). Here magnetic edge states energy for φ = 0 and 0.01. At φ = 0 there are approx- are not included since their energies are larger than δ. imately 20 states within ǫ < 0.01eV, consistent with n | | N (φ) is the number states at zero magnetic field that theanalyticalresultofEq.(1). Atφ=0.01thenumerical D merge into the energy interval δ around zero energy as valueisincreasedto24. Fig.4(a)showshowsomeenergy the dimensionless magnetic flux φ = Φ increases. This levels ǫ at φ = 0 change as a function of the magnetic Φ0 n effectprovidesameanstocontrolthenumberofstatesat flux φ. These energy levels do not anticross. We ob- the Fermi level. There are other states with nearly zero serve that nearly zero energies at φ = 0 do not change energies at φ = 0, which remain so even at φ = 0. The noticeably in magnitude as φ varies. There are N such l 6 localization lengths of these states are shorter than the localizedsurface states. Onthe other hand, we find that magnetic length. We denote the number of these states asφincreasesnon-zeroenergiesbecomesmallerandmove by N . closertozero. Thisimpliesthat,foragivenenergyinter- l Ournumericalresultsindicatethatforrelativelysmall valδ,thenumberofstatesinit,NT(φ),increaseswithφ. rectangular-shaped graphene dots with size less than of FromFig.4(b)weseethatitdisplaysanon-lineardepen- order102˚AthenumberofstatesattheFermileveldisplay dence on φ. For a large dot of size 50 50nm2 a similar × anegligiblemagneticfielddependenceforvaluesthatare dependence of NT on B is seen, as shown in Fig.5. Non- usuallyaccessibleexperimentally(B <10T corresponds, linear dependence occurs in the regime 2ℓ/Lx 0.5. As ∼ indimensionlessmagneticflux,toφ= Φ <10 4,where a test of our numerical procedures we have verified that Φ0 − Φ0 = hc/e and Φ = BAh with the area of a hexagon thesumofND(φ)=NT(φ)−Nl andthenumberofmag- A = 3√3a2). On the other hand, in larger dots this netic edge states is equal to the total bulk Landau level h 2 0 degeneracy 2D (D = 2LxLyφ is the degeneracy per dependenceissignificant. However,itiscomputationally B B 3√3a20 difficulttoinvestigatelargerectangular-shapedgraphene valley). 3 0.50.4 0.3 2 /Lx 0.2 90 80 2 /NT70 60 50 0 10 20 30 40 B[T] FIG. 5: Results for a dot with size 50×50nm2. Dependence of N /2 on B for δ=0.01eV. T FIG. 3: (a) Eigenenergies ǫ at φ = 0. Quasi-degenerate n states are present near zero energy. Size of the dot is 74× 71˚A2. Aquantizationenergyoforderγk ∼0.03eVcanbe min seen asanexcitation gap nearzeroenergy. (b)Eigenenergies ǫ at φ=0.01. n FIG. 6: Size of the dot is 74×71˚A2. (a) The probability wavefunction of thestate with ǫ =−0.07eV at φ=0. The 1027 length unit is a. (b) Profile of z-component of pseudospin: sizesofred(blue)dotsrepresentprobabilitiesofoccupyingA (B) carbon atoms. Note that blue (red) dots are dominant in the upper (lower) of dot. When the probabilities are less than 0.00001, the radius of the dots is set to the smallest value. Theupperandlowerhorizontaledgesrepresentzigzag edges. (c) The probability wavefunction of the state with ǫ1027 =−2.0×10−3eVatℓ/Lx =0.12(φ=0.01). (d)Profile of z-component of pseudospin for n=1027 at φ=0.01. FIG.4: (a)Someenergylevelsǫ changeasφincreaseswhile n somedonot. (b)Thetotalnumberofstateswithintheenergy interval δ around zero energy at a finite value of φ. Size of thedot is 74×71˚A2. tem length 2ℓ L . Note that in this regime many x,y ∼ cyclotronorbits getaffected bythe presenceofthe edges and corners of the rectangular dot. Since we must also assume that Hofstadter effect is negligible the validity The ratio between the number of nearly zero energy regime of Eq.(3) is a 2ℓ < L . Note also that the 0 x,y ≪ states induced by the magnetic field and the number scaling function f(x) should be different for each δ. of hexagons, ND(φ), should depend on a dimensionless Nh quantity consisting of a combination of ℓ, a , L and 0 x Ly, which are the important parameters of rectangu- III. WAVEFUNCTIONS OF lar graphene dots. The lengths Lx and Ly should ap- QUASI-DEGENERATE STATES IN A pear asLx+Ly sothat for rectangular-shapedgraphene MAGNETIC FIELD sheetsN /N remainsthesamewhenL andL areex- D h x y changed. These considerationslead us to the dimension- We first show how the wavefunction of a non-zero en- less variable a0(Lℓx2+Ly) = 43πφ(M+√31N+2/√3), see Eq.(3). ergy state at φ=0 changes into a state with nearly zero We are especially interested in the regime where the di- energy as φ increases. Consider the probability wave- ameter of the cyclotron orbit is comparable to the sys- function for n = 1027 at a finite φ = 0.01, as shown in 4 FIG. 7: Size of the dot is 74×71˚A2. (a) The probability wavefunction of the state with ǫ1045 =5.38×10−6eV at φ= 0. (b) Profile of z-component of pseudospin for n = 1045 FIG. 9: (a) The probability wavefunction of the state with at φ = 0. (c) The probability wavefunction for n = 1045 ǫ1053 = 0.144eV at ℓ/Lx = 0.12 (φ = 0.01). Size of the dot at ℓ/L = 0.12 (φ = 0.01). (d) Profile of z-component of is74×71˚A2. (b)Profileofz-componentofpseudospinofthe x pseudospin for n=1045 at φ=0.01. same state. changed from 5.38 10 6eV to 4.5 10 6eV when φ is − − × × changed to 0.01 from zero. The pseudospin profiles are shown in Figs.7(b) and (d). Fig.8 shows a probability wavefunction at a smaller value of ℓ/L = 0.09 (corre- x,y sponding to φ=0.02),and we see that the wavefunction is lesslocalizedonthe zigzagedges andLLLcharacteris more pronounced in comparisonto result of ℓ/L =0.12 x (Fig.6(c)). AllN (φ)stateshavesimilarpropertiesmen- D tioned above with finite probabilities of finding an elec- troninsidethedot. TherearealsoN zigzagedgesstates ℓ that are more strongly localized on the edges with local- ization lengths comparable to a. When an electron is in one of these states the probability of find the electron away from the edges is practically zero. The energies of FIG.8: Sizeofthedotis74×71˚A2andℓ/L =0.09(φ=0.02) x these states are less then 10 10eV. We can summarize (a) The probability wavefunction of the state with ǫ = − 1027 our results as follows: all nearly zero energy states are −2.04×10−5eV. (b) Profile of z-component of pseudospin of localizedonthezigzagedgeswithorwithoutsomeweight thesame state. These results should compared with those in Fig.6. inside the dot. When the magnetic length is much smaller than the systemsizemagneticedgestatescanbeformed,seeFig.9. The probability wavefunction of a magnetic edge state withǫ =0.144eVisshowninFig.9(a). Itisamixture Fig.6(c). It is localized on the zigzag edges with a fi- 1053 ofordinarymagnetic andzigzagedge states. The proba- nite probability inside the dot. On the armchair edges bilitywavefunctiondecaysfromthearmchairedgeswhile the wavefunction is vanishingly small. The wavefunc- it is strongly peaked on the zigzag edges. Pseudospin tion has changed significantly from the φ = 0 result, expectation values on each zigzag edge display opposite see Fig.6(a), and also its energy has changed from - 0.07eVto 2.0 10 3eV.Thevaluesofthez-component chiral behavior. On armchair edges the chiralities are − − × more or less evenly mixed, see Fig.9(b). of the pseudospin, Fig.6(b) and Fig.6(d), are larger on the zigzag edges at φ = 0.01 compared to the result at φ = 0. The probability wavefunction of another state with nearly zero energyis shownin Fig.7(c) at φ=0.01. IV. DISCUSSIONS AND CONCLUSIONS Weseethattheresultissomewhatdifferentfromthezero field result of Fig.7(a), which displays a localized state We now explain qualitatively how mixed states of with the localization length comparable to the unit cell Fig.6(c) and Fig.7(c) can arise. An infinitely long lengtha. Nowthereisafiniteprobabilitytofindanelec- zigzag nanoribbon in a magnetic field has nearly zero tron inside the dot while the probabilities on the zigzag energy surface states that are localized on the edges edgesarereduced. Notethatthe energyofthis statehas in addition to ordinary lowest Landau level states, see 5 carbonatoms inside the dotsince LLL states ofdifferent chiralities are coupled by the armchair edges. Our nu- merical result is indeed consistent with this expectation, see Fig.6 (d). As the ratio ℓ/L takes smaller values x,y the nature of these states become more like that of LLL states (see Fig.8). We have investigated, in the presence of a mag- netic field,quasi-degeneratestatesofrectangular-shaped 0 graphenedotsneartheDiracpoints. Someofthesestates are magnetic field independent surface states while the other states are field dependent. We find numerically FIG. 10: Cross section of probability wavefunctions of a nanoribbon with infinitely long zigzag edges along the x-axis thatthewavefunctionsofthesestatesareallpeakednear in the presence of a perpendicular magnetic field (a Landau thezigzagedgeswithorwithoutsignificantweightinside gaugeisused). |ψ |2 representsalocalizedsurfacestate. Two the dot. The physical origin of the presence of a signifi- 1 examplesofLLLstates|ψ2|2 and|ψ3|2 arealsoshown. These cantweightisthecouplingbetweenKandK’valleysdue states all havenearly zero energies. to the armchair edges. This effect is expected to survive small deviations from perfect armchair edges as long as they provide coupling between different valleys. Experi- mentally the dependence of N on φ may be studied by Fig.10. The properties of these states are given in D measuring STM properties[27] or the optical absorption Refs.[6, 11, 12, 31, 32]: LLL states of valley K (K) are ′ spectrum as a function of magnetic field[28–30]. In fab- of B (A) type and localized surface states have a mixed ricating rectangular dots a special attention should be character between A and B. The surface states can have given to the direction of armchair edges since the prop- various localization lengths but the minimum value is erties of dot may depend on it[33]. of order the carbon-carbon distance a [12]. The arm- 0 Acknowledgments chair edges couple K and K valleys[6, 11, 26], and, con- ′ sequently, surface and LLL states of a nanoribbon can be coupled and give rise to mixed states with significant ThisworkwassupportedbytheKoreaResearchFoun- weight on the zigzag edges and inside the dot, as shown dation Grant funded by the Korean Government (KRF- in Fig.6(c) and Fig.7(c). In addition, these mixed states 2009-0074470). In addition this work was supported by should display a significant occupation of both A and B the Second Brain Korea 21 Project. [1] For recent reviews see: T. Ando, J. Phys. Soc. Jpn. 74, Rev. 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