States near Dirac points of rectangular graphene dot in a magnetic field
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States near Dirac points of rectangular graphene dot in a magnetic field
S. C. Kim , P. S. Park , and S.-R. Eric Yang , ∗ Physics Department, Korea University, Seoul Korea and Korea Institute for Advanced Study, Seoul Korea (Dated: November 1, 2018)In neutral graphene dots the Fermi level coincides with the Dirac points. We have investigatedin the presence of a magnetic field several unusual properties of single electron states near theFermi level of such a rectangular-shaped graphene dot with two zigzag and two armchair edges.We find that a quasi-degenerate level forms near zero energy and the number of states in this levelcan be tuned by the magnetic field. The wavefunctions of states in this level are all peaked onthe zigzag edges with or without some weight inside the dot. Some of these states are magneticfield-independent surface states while the others are field-dependent. We have found a scaling resultfrom which the number of magnetic field-dependent states of large dots can be inferred from thoseof smaller dots.
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I. INTRODUCTION BA xy FIG. 1: Finite graphene layer with zigzag and armchair edges.There are equal number of A and B carbon atoms. Thegraphene layer has reflection symmetries about horizontal andvertical lines that go through the center of the layer. A mag-netic field is present perpendicular to the layer.
Graphene dots have a great potential for many ap-plications since they are the elemental blocks to con-struct graphene-based nano devices. It is possible to cutgraphene sheet[1] in the desired shape and size[2], and useit to make quantum dot devices. In such devices it maybe possible to realize experimentally zigzag or armchairboundaries.Graphene systems possess several unusual physicalproperties associated with the presence of the Diracpoints. For example, compared to ordinary Landau lev-els of quasi-two-dimensional semiconductors the lowestLandau level (LLL) of graphene is peculiar since it has zero energy that is independent of magnetic field[3–7].Moreover, wavefunctions of the LLL are chiral , i.e., theprobability amplitude of find the electron on one typecarbon atoms is zero. There are other graphene systemswith zero energy states. Semi-infinite[8] or nanoribbon ∗ corresponding author, [email protected] graphene[9, 10] with zigzag edges along the x-axis developa flat band of zero energy chiral states. These states aresurface states and are localized states at the zigzag edgeswith various localization lengths[8]. The zigzag edge andthe LLL states have zero energy because their wavefunc-tions are chiral. Effects of a magnetic field on grapheneHall bars have been investigated recently, and some zeroenergy chiral states are found to be strongly localized onthe zigzag edges in addition to the usual LLL states[11–14].One may expect that the degeneracy of chiral stateswith zero energy will be split when quantum confine-ment effect is introduced in a graphene dot[15–17]. How-ever, the splitting of these energies may be unusualin some graphene dots[18–21]. Recently the magneticfield dependence of these levels in a gated graphenedot was investigated experimentally[21]. Effects of var-ious types of edges have been also investigated: zigzag-edged dots, armchair-edged dots[19, 22–24], and rectan-gular graphene dots with two zigzag and two armchairedges[25, 26] have been studied. Armchair edges cou-ple states near K and K ′ points of the first Brillouinzone and generate several mixed chiral zigzag edge stateswith nearly zero energies. In the rectangular dots thenumber of these states, N l , may be determined from thecondition that the x-component of wavevectors satisfies1 /L y < k x,n < π/ a , where k x,n = πnL x − π a , a = √ a is the length of the unit cell, and n = 0 , ± , ± , ... (thenearest neighbor carbon-carbon distance is a = 1 . A ,the horizontal length of the dot is L x = √ M a with M number of hexagons along the x-axis, and the verticallength is L y = a (3 N +2) with N the number of hexagonsand ( N + 1) carbon bonds along the y-axis. See Fig.1).This condition implies that the integer n is given by √ M (3 N + 2) π + 2 M ≤ n ≤ M. (1)The effective mass approximation wavefunctions of thesesurface states are derived in Ref.[26].We investigate how properties of rectangular dots FIG. 2: Number of nearly zero energy states that are inducedby a magnetic field follow a scaling curve when plotted as afunction of M + √ N +2 / √ φ for different rectangular shapedgraphene dots. Here δ = 0 . eV . change in the presence of a magnetic field in the regimewhere Hofstadter-butterfly effects[18, 19, 21] are negligi-ble. In neutral graphene dots the Fermi level has zeroenergy, and, consequently, magnetic, optical, and STMproperties are expected to depend on the number of avail-able states near zero energy. Our investigation showsthat the wavefunctions of nearly zero energy states areall peaked on the zigzag edges with or without appre-ciable weight inside the dot. This may be understoodas mixing of LLL and surface states by armchair edgesof the square dot through intervalley scattering, whichis unique to the square dot (This will be explained inSec.IV).Our study shows that the number of states within theenergy interval δ around zero energy is given by N T ( φ ) = N l + N D ( φ ) (2)(the energy δ is typically less than the quantization en-ergy of a rectangular dot, which can be estimated usingthe Dirac equation: γk min ∼ t aL x,y , where γ = √ ta/ t ). Here magnetic edge statesare not included since their energies are larger than δ . N D ( φ ) is the number states at zero magnetic field thatmerge into the energy interval δ around zero energy asthe dimensionless magnetic flux φ = ΦΦ increases. Thiseffect provides a means to control the number of states atthe Fermi level. There are other states with nearly zeroenergies at φ = 0, which remain so even at φ = 0. Thelocalization lengths of these states are shorter than themagnetic length. We denote the number of these statesby N l .Our numerical results indicate that for relatively smallrectangular-shaped graphene dots with size less than oforder 10 ˚ A the number of states at the Fermi level displaya negligible magnetic field dependence for values that areusually accessible experimentally ( B < T corresponds,in dimensionless magnetic flux, to φ = ΦΦ < − , whereΦ = hc/e and Φ = BA h with the area of a hexagon A h = √ a ). On the other hand, in larger dots thisdependence is significant. However, it is computationallydifficult to investigate large rectangular-shaped graphene dots since the number of carbon atoms increases rapidlywith the size. We have found a scaling result from whichone can infer results for larger dots from those of smallerdots. For different rectangular-shaped graphene dots andvalues of φ it can be described well by the following di-mensionless form N D ( φ ) = L x L y A h f ( ℓ a ( L x + L y ) ) , (3)where ℓ = / a (4 πφ ) / is the magnetic length and f ( x ) is ascaling function, see Fig.2. The total number of hexagonsin the dot is N h = L x L y /A h . Our numerical result showsthat the dependence of N D ( φ ) on φ is initially non-linearin the regime where the diameter of the cyclotron motionis comparable to the system length, 2 ℓ ∼ L x,y . II. NUMBER OF STATES IN THEQUASI-DEGENERATE LEVEL
Our Hamiltonian is H = − X t ij c † i c j , (4)where t ij = te i e ~ R ~Ri~Rj ~A · d~r are the hopping parameters and c + i creates an electron at site i . Here we use a Lan-dau gauge ~A = B ( − y, , < i, j > is over nearest neighbor sites and t = 2 . ǫ n is denoted by φ ǫ n ( ~R ), where ~R labels each lattice point. Because of electron-hole sym-metry eigenvalues appear in pairs of positive and nega-tive values ( ǫ, − ǫ ), and the probability wavefunctions of apair of states, ( | φ ǫ n ( ~R ) | , | φ − ǫ n ( ~R ) | ), are identical. Ournumerical results are consistent with this.Figs.3(a) and (b) display the energy spectra near zeroenergy for φ = 0 and 0 .
01. At φ = 0 there are approx-imately 20 states within | ǫ n | < . eV , consistent withthe analytical result of Eq.(1). At φ = 0 .
01 the numericalvalue is increased to 24. Fig.4(a) shows how some energylevels ǫ n at φ = 0 change as a function of the magneticflux φ . These energy levels do not anticross. We ob-serve that nearly zero energies at φ = 0 do not changenoticeably in magnitude as φ varies. There are N l suchlocalized surface states. On the other hand, we find thatas φ increases non-zero energies become smaller and movecloser to zero. This implies that, for a given energy inter-val δ , the number of states in it, N T ( φ ), increases with φ .From Fig.4(b) we see that it displays a non-linear depen-dence on φ . For a large dot of size 50 × nm a similardependence of N T on B is seen, as shown in Fig.5. Non-linear dependence occurs in the regime 2 ℓ/L x ∼ .
5. Asa test of our numerical procedures we have verified thatthe sum of N D ( φ ) = N T ( φ ) − N l and the number of mag-netic edge states is equal to the total bulk Landau leveldegeneracy 2 D B ( D B = L x L y √ a φ is the degeneracy pervalley). FIG. 3: (a) Eigenenergies ǫ n at φ = 0. Quasi-degeneratestates are present near zero energy. Size of the dot is 74 × A . A quantization energy of order γk min ∼ . ǫ n at φ = 0 . ǫ n change as φ increases whilesome do not. (b) The total number of states within the energyinterval δ around zero energy at a finite value of φ . Size ofthe dot is 74 × A . The ratio between the number of nearly zero energystates induced by the magnetic field and the numberof hexagons, N D ( φ ) N h , should depend on a dimensionlessquantity consisting of a combination of ℓ , a , L x and L y , which are the important parameters of rectangu-lar graphene dots. The lengths L x and L y should ap-pear as L x + L y so that for rectangular-shaped graphenesheets N D /N h remains the same when L x and L y are ex-changed. These considerations lead us to the dimension-less variable ℓ a ( L x + L y ) = π φ ( M + √ N +2 / √ , see Eq.(3).We are especially interested in the regime where the di-ameter of the cyclotron orbit is comparable to the sys- N T / B[T] x FIG. 5: Results for a dot with size 50 × nm . Dependenceof N T / B for δ = 0 . × A . (a) The probabilitywavefunction of the state with ǫ = − . φ =0. Thelength unit is a . (b) Profile of z-component of pseudospin:sizes of red (blue) dots represent probabilities of occupying A(B) carbon atoms. Note that blue (red) dots are dominantin the upper (lower) of dot. When the probabilities are lessthan 0.00001, the radius of the dots is set to the smallestvalue. The upper and lower horizontal edges represent zigzagedges. (c) The probability wavefunction of the state with ǫ = − . × − eV at ℓ/L x = 0 .
12 ( φ = 0 . n = 1027 at φ = 0 . tem length 2 ℓ ∼ L x,y . Note that in this regime manycyclotron orbits get affected by the presence of the edgesand corners of the rectangular dot. Since we must alsoassume that Hofstadter effect is negligible the validityregime of Eq.(3) is a ≪ ℓ < L x,y . Note also that thescaling function f ( x ) should be different for each δ . III. WAVEFUNCTIONS OFQUASI-DEGENERATE STATES IN AMAGNETIC FIELD
We first show how the wavefunction of a non-zero en-ergy state at φ = 0 changes into a state with nearly zeroenergy as φ increases. Consider the probability wave-function for n = 1027 at a finite φ = 0 .
01, as shown in
FIG. 7: Size of the dot is 74 × A . (a) The probabilitywavefunction of the state with ǫ = 5 . × − eV at φ =0. (b) Profile of z-component of pseudospin for n = 1045at φ = 0. (c) The probability wavefunction for n = 1045at ℓ/L x = 0 .
12 ( φ = 0 . n = 1045 at φ = 0 . × A and ℓ/L x = 0 .
09 ( φ = 0 . ǫ = − . × − eV. (b) Profile of z-component of pseudospin ofthe same state. These results should compared with those inFig.6. Fig.6(c). It is localized on the zigzag edges with a fi-nite probability inside the dot. On the armchair edgesthe wavefunction is vanishingly small. The wavefunc-tion has changed significantly from the φ = 0 result,see Fig.6(a), and also its energy has changed from -0.07eV to − . × − eV. The values of the z-componentof the pseudospin, Fig.6(b) and Fig.6(d), are larger onthe zigzag edges at φ = 0 .
01 compared to the result at φ = 0. The probability wavefunction of another statewith nearly zero energy is shown in Fig.7(c) at φ = 0 . a . Now there is a finite probability to find an elec-tron inside the dot while the probabilities on the zigzagedges are reduced. Note that the energy of this state has FIG. 9: (a) The probability wavefunction of the state with ǫ = 0 . ℓ/L x = 0 .
12 ( φ = 0 . × A . (b) Profile of z-component of pseudospin of thesame state. changed from 5 . × − eV to 4 . × − eV when φ ischanged to 0.01 from zero. The pseudospin profiles areshown in Figs.7(b) and (d). Fig.8 shows a probabilitywavefunction at a smaller value of ℓ/L x,y = 0 .
09 (corre-sponding to φ = 0 . ℓ/L x = 0 . N D ( φ ) states have similar properties men-tioned above with finite probabilities of finding an elec-tron inside the dot. There are also N ℓ zigzag edges statesthat are more strongly localized on the edges with local-ization lengths comparable to a . When an electron is inone of these states the probability of find the electronaway from the edges is practically zero. The energies ofthese states are less then 10 − eV. We can summarizeour results as follows: all nearly zero energy states arelocalized on the zigzag edges with or without some weightinside the dot.When the magnetic length is much smaller than thesystem size magnetic edge states can be formed, see Fig.9.The probability wavefunction of a magnetic edge statewith ǫ = 0 . IV. DISCUSSIONS AND CONCLUSIONS
We now explain qualitatively how mixed states ofFig.6(c) and Fig.7(c) can arise. An infinitely longzigzag nanoribbon in a magnetic field has nearly zeroenergy surface states that are localized on the edgesin addition to ordinary lowest Landau level states, see FIG. 10: Cross section of probability wavefunctions of ananoribbon with infinitely long zigzag edges along the x-axisin the presence of a perpendicular magnetic field (a Landaugauge is used). | ψ | represents a localized surface state. Twoexamples of LLL states | ψ | and | ψ | are also shown. Thesestates all have nearly zero energies. Fig.10. The properties of these states are given inRefs.[6, 11, 12, 31, 32]: LLL states of valley K (K ′ ) areof B (A) type and localized surface states have a mixedcharacter between A and B. The surface states can havevarious localization lengths but the minimum value isof order the carbon-carbon distance a [12]. The arm-chair edges couple K and K ′ valleys[6, 11, 26], and, con-sequently, surface and LLL states of a nanoribbon canbe coupled and give rise to mixed states with significantweight on the zigzag edges and inside the dot, as shownin Fig.6(c) and Fig.7(c). In addition, these mixed statesshould display a significant occupation of both A and B carbon atoms inside the dot since LLL states of differentchiralities 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 x,y takes smaller valuesthe nature of these states become more like that of LLLstates (see Fig.8).We have investigated, in the presence of a mag-netic field, quasi-degenerate states of rectangular-shapedgraphene dots near the Dirac points. Some of these statesare magnetic field independent surface states while theother states are field dependent. We find numericallythat the wavefunctions of these states are all peaked nearthe zigzag edges with or without significant weight insidethe dot. The physical origin of the presence of a signifi-cant weight is the coupling between K and K’ valleys dueto the armchair edges. This effect is expected to survivesmall deviations from perfect armchair edges as long asthey provide coupling between different valleys. Experi-mentally the dependence of N D on φ may be studied bymeasuring STM properties[27] or the optical absorptionspectrum as a function of magnetic field[28–30]. In fab-ricating rectangular dots a special attention should begiven to the direction of armchair edges since the prop-erties of dot may depend on it[33]. Acknowledgments
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