Diamagnetic Levitation and Thermal Gradient Driven Motion of Graphite

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arXiv:1904.09741v2 [cond-mat.mes-hall] 10 May 2019 rvnmto fgaht.Uigteobtldiamag- orbital susceptibility the Using netic thermal- graphite. the of and motion driven levitation diamagnetic the of study ntemgei susceptibility. change magnetic photothermal the the in by irra- motivated photo the spot, towards diated moved the is where graphite levitation, mo- diamagnetic levitating the optical in an control demonstrated tion also experiment same egto rpiepeei hsgeometry. this in piece graphite levitation a the of of height measurement detailed a performed eti respace. free ob- a diamagnetic where in a ject 1, support to a Fig. gradient generates field in poles magnetic magnetic shown of pattern as alternating magnet the NdFeB of arrange- checkerboard ment a been is levitation have diamagnetic the applications exten- various also proposed. and was studied graphite sively of levitation diamagnetic ia oino h ia-ieelectrons. Dirac-like the or- of anoma- the motion from its originates bital and susceptibility substances, magnetic lous natural among terials diamagnetism. tiny their to due magnet, powerful iigfrog living a was bismuth 1930’s. and in graphite demonstrated setup. of first experimental levitation stable appropriate also The an can under superconductors, materials of levitate diamagnetic effect Meissner normal-state exam- the while known is best this of The called ple is levitation. it and diamagnetic magnet, the a over freely diamagnetism levitate strong even can with Materials field. magnetic hw htee ic fwo n plastic and wood of piece a even that shown nti ae,w rsn ealdtheoretical detailed a present we paper, this In rpiei n ftesrnetdaantcma- diamagnetic strongest the of one is Graphite imgeimi rpryo aeilt ee a repel to material of property a is Diamagnetism imgei eiainadtemlgain rvnmotion driven gradient thermal and levitation Diamagnetic eprtr ie n siaetegnrtdfrea func a as force generated t the moves estimate gradient and inversely temperature side, diamagnetism the temperature that its demonstrate of exhibit we latter because nally, t the temperatures, compare that low We show in and object. graphite, levitating maximum stacked the the randomly and of magnets, geometr susceptibility the the the of to and period magne certain susceptibility a of the at arrangement maximized on periodic height a levitation over the graphite levitating susceptibi of magnetic derived quantum-mechanically the ing 11–25 etertclysuytedaantclvtto n h t the and levitation diamagnetic the study theoretically We 3,4 .INTRODUCTION I. yia xeietlstpue for used setup experimental typical A n cell and 13,18–20,24,25 χ 1 acltdfo h standard the from calculated eateto hsc,OaaUiest,Oaa5004,J 560-0043, Osaka University, Osaka Physics, of Department 5 r bet eiaewt a with levitate to able are 1 20,25 twsmr recently more was It eetexperiment recent A aaoFujimoto Manato 2 6–10 Dtd a 3 2019) 13, May (Dated: n also and 20 The The 1 n iioKoshino Mikito and or ant n bantelvtto egta a as checker- height levitation the of the over function obtain object and magnet, diamagnetic board levitating a equilibrium of the position compute we model, band checker- a magnets. on of graphite array of board levitation Diamagnetic 1. FIG. h aiu egti hnlnal proportional linearly and then magnets, is is the to height height of levitation maximum period the certain the that a find at We maximized object. the of eaue.I h iudntoe eprtr (77K), tem- temperature low nitrogen in liquid achieved the is In length peratures. levitation large a fore in- in- temperature diamagnetism the to reduced strong proportional a versely the to There leads with in coupling stacked terlayer 2(b)], rotations. are in-plane [Fig. layers random graphite graphene stacked successive con- which randomly also we a 2(a)], sider [Fig. structure stacking (Bernal) a as force susceptibility. generated of the function estimate tem- and high the side, to perature object levitating the moves gradient nadto oteodnr rpiewt AB with graphite ordinary the to addition In χ ial edmntaeta h temperature the that demonstrate we Finally . iy ecmueteeulbimposition equilibrium the compute we lity, .W n httelvtto egtis height levitation the that find We y. eodnr Bsakdgaht n a and graphite AB-stacked ordinary he ag eiainlnt particularly length levitation large a s ino susceptibility. of tion s n netgt h eedneof dependence the investigate and ts, emldie oino rpie Us- graphite. of motion hermal-driven rprinlt h eprtr.Fi- temperature. the to proportional elvttn bettwrstehigh the towards object levitating he z χ egti hnlnal proportional linearly then is height h hcebadpro,adtesize the and period, checkerboard the , y x 1 apan b R fgraphite of 26 n there- and , 2

tion is valid when the twist angle θ between adjacent layers is not too small (θ 1◦). If a small twist angle happens to occur somewhere≫ in the random stack, these two layers are strongly coupled to form flat bands32,33, and do not participate in the large diamagnetism given by the nearly-independent graphene part. By neglecting this, the susceptibility at the charge neutral point is explicitly written as26 1 χ = − , (1) 1+ kBT/∆ where ∆ is a characteristic energy scale defined by g g v 2 e2 ∆= v s 0.03 meV, (2) 6 c 4πǫ0d ≈ and c is the light velocity. The susceptibility is nearly FIG. 2. Atomic structure of (a) AB-stacked graphite and proportional to 1/T in k T ∆ (T 0.35 K). As (b) randomly stacked graphite. B plotted in Fig. 3(b), the susceptibility≪ ≫ of random- stacked graphite is much greater than that of AB- stacked graphite particularly in the low-temperature for example, the maximum levitation length is found regime. The real system should have some disorder to be about 5 mm, which is 10 times as large as the potential, and then we expect that χ saturates at k T Γ, where Γ is the broadening broadening near typical levitation height of the AB-stacked graphite. B ∼ The paper is organized as follows. In Sec. II, we the Dirac point of graphene. briefly introduce the magnetic susceptibility of AB- stacked graphite and randomly-stacked graphite. We III. MAGNETIC LEVITATION then calculate the magnetic levitation of general diamagnetic objects in the checkerboard magnet array in Sec. III. We consider the thermal-gradient force in We consider magnetic levitation of graphite in the magnetic levitation in Sec. IV. A brief conclusion is geometry illustrated in Fig. 4(a). Here the N-pole given in Sec. V. The susceptibility calculation for the and S-pole of square-shaped magnets of size b are AB-stacked graphite is presented in Appendix A. alternately arranged in a checkerboard pattern. We place a round-shaped graphite disk of radius R and the thickness w right above a grid point where four II. MAGNETIC SUSCEPTIBILITY magnet blocks meet. We assume the surface of the magnet top and the graphite disk is perpendicular to the gravitational direction, z. The graphite is We calculate the magnetic susceptibility of attracted to the grid point where the magnetic field graphite using the quantum mechanical liner- is the weakest. response formula27, and the standard band When the magnetic field distribution B = model.28–31 The detail description of the calculation B(x,y,z) is given, the total energy U of the graphite is presented in Appendix A. Figure 3(a) plots disk is given by the susceptibility χ of AB-stacked graphite as a function of temperature. Throughout the paper, we 1 2 U = Mgz + w dxdy − χBz(x,y,z) , (3) define χ as the dimensionless susceptibility in the ZS 2µ0 SI unit (the perfect diamagnetism is χ = 1). In − Here g is the gravitational acceleration, M is the decreasing temperature, χ slowly increases nearly in mass of the disk, S is the area of the disk, χ is the a logarithmic manner, and finally saturate around magnetic susceptibility of graphite, and z is the ver- T 50 K. The logarithmic increase is related to the ∼ tical position of the disk. We assumed the thickness quadratic band touching in the in-plane dispersion, of graphite is thin enough. The equilibrium position and the saturation caused by the semimetallic band z = zlev (i.e., the levitation length) is obtained by structure of graphite, as argued in Appendix A. solving ∂U/∂z =0, or The susceptibility of random-stacked graphite is 2 approximately given by that of an infinite stack of dBz ρg =2µ0 . (4) independent monolayer graphenes. This simplifica- dz S χ 3

(a) 15 (a) Bz [T] 3 AB-stacked 0.4 2 A 0.2 10 ) 1 B -4 C 0 D E (10 0 -0.2 y [mm] y −χ 5 -1 -0.4

-2 0 0 100200 300 400 -3 T(K) -3 -2-1 0 1 2 3 x [mm] (b) 200 0.6 0.8 Randomly stacked (b) (c)

150 AB-stacked 0.6 /mm] 2 ) 0.4 A -4 [T] z

(10 0.4

B [T dz /

100 B 2 z −χ −χ

0.2 C dB B 0.2 50 D C A E E D 0 0 0 0 0.20.4 0.6 0.8 1.0 0 0.20.4 0.6 0.8 1.0 z [mm] z [mm] 0 100200 300 400 T(K) FIG. 4. (a) Distribution of Bz on xy-plane, at 0.2 mm height over an inﬁnite checkerboard magnet array with FIG. 3. (a) Temperature dependence of the total suscep- b = 3 mm. (b) Plot of Bz as functions of z, at the ﬁve tibility χ of AB-stacked graphite with µ = 0. (b) Similar points A = (1.5, 1.5), B = (0.5, 0.5),C = (0.2, 0.2), D = plot for the randomly stacked graphite (blue dashed), (0.05, 0.05) and E = (0.001, 0.001) (in units of mm) compared to that of AB-stacked graphite (black solid). which are indicated in the panel (a). (c) Similar plot 2 for dBz /dz.

Here ρ 2.2 g/cm3 is the mass density of graphite, and ≈ is the average over the graphite area. For S where B is the amplitude of the surface magnetic the magnetich· · · i levitation, therefore, the squared mag- 0 field, and b , b and b , are the side lengths, and the netic field gradient dB2/dz matters more than the x y z z N and S poles of the magnet correspond to the faces absolute field amplitude itself. of z = b /2 and b /2, respectively. The total mag- Now we consider an infinite checkerboard arrange- z − z netic field Bz(x,y,z) is obtained as an infinite sum of ment of square blocks of NdFeB magnet. The z- (1) component of the magnetic field generated by a sin- Bz over all the blocks composing the checkerboard gle block can be calculated by the formula,34 array. We take B0 = 500 mT as a typical value for NdFeB magnet, and assume the square and long (1) B0 shape, i.e., bx = by b, and bz . Bz (x,y,z)= [F1( x,y,z)+ F1( x, y, z) ≡ → ∞ − 2π − − − Figure 4(a) shows the actual distribution of Bz + F ( x, y,z)+ F ( x, y, z)+ F (x,y,z) at 0.2 mm height from the surface for b = 3 mm 1 − − 1 − − − 1 + F (x, y, z)+ F (x, y,z)+ F (x, y, z)], magnet array. Figures 4(b) and (c) are the plots of 1 − 1 − 1 − − 2 (5) Bz and dBz /dz as functions of z, respectively, at the five points A = (1.5, 1.5),B = (0.5, 0.5), C = with (0.2, 0.2),D = (0.05, 0.05) and E = (0.001, 0.001) in units of mm. Here the coordinate origin is taken to a F1(x,y,z)= grid point on the upper surface of the arranged mag- bx by nets. At any xy-points, the magnetic field Bz expo- x + 2 y + 2 arctan , nentially decays in z, and its decay length is shorter 2 2 2 2 bz bx by bz when closer to the origin, and so does dBz /dz. z + 2 x + 2 + y + 2 + z + 2 2 r The averaged squared magnetic field Bz S can (6) also be well approximated by an exponentialh functioni 4 in z as, (a) 4 χ = −50 x 10-4 − B2 αB2e z/λ, (7) h z iS ≈ 0 where α is the dimensionless constant of the order 3 (mm) −20 x 10-4 of 1, and λ is the length scale determined by the lev geometry. Then Eq. (4) is explicitly solved as 2 αλ z λ log 0 (8) lev ≈ λ −10 x 10-4 with the characteristic length, 1 Levitationheight, z −5 x 10-4 2 χ B0 λ0 = | | . (9) −2 x 10-4 2µ0ρg 0 0 1020 30 The negative solution of Eq. (8) indicates that the Magnetic grid size, b (mm) graphite does not levitate. (b) 0.8 If the graphite radius R is much greater than the 2 b = 3 mm magnet grid size b, in particular, Bz S is replaced χ = −5 x 10-4 by the average value over whole xy-plane,h i and then λ 0.6 depends solely on b (not on R). In this limit, we have (mm) α 0.7 and λ ηb with η 0.11. Figure 5(a) plots lev ∼ ∼ ∼ the levitation height zlev as a function of the grid size b, calculated for different χ’s in this limit, The solid 0.4 curves are the numerical solution of Eq. (4), and the dashed curves are the approximate expression Eq. (8) with α =0.7 and η =0.11. The curves with dif- 0.2 ferent χ’s are just scaled through the length param- Levitationheight, z eter λ . Here we take χ = 2, 5, 10, 20, 50( 10−4) 0 − × (ρ is fixed), which give λ0 is 0.9, 2.2, 4.5, 9.0, 22.4mm, 0 respectively. As is obvious from its analytic form, the 0 2 4 6 approximate curve peaks at b = αλ0/(ηe) 2.3λ0 Disk radius, R (mm) (e is the base of the natural logarithm), where≈ the levitation height takes the maximum value, FIG. 5. (a) Levitation height zlev as a function of the magnetic grid size b, calculated for different χ’s with the (max) αλ0 limit of R ≫ b. Solid curves are the numerical solution z = 0.26λ0. (10) lev e ≈ of Eq. (4), and the dashed curves are the approximate expression Eq. (8). (b) R-dependence of the levitation We see that the approximation fails for b greater height in χ = −5 × 10−4 and the magnetic grid with than the peak position. This is because Eq. (7) is b = 3 mm. The horizontal dashed line indicates the 2 not accurate z < λ, where the actual Bz S be- asymptotic value in R →∞. comes higher than the approximation. Theh i gradi- 2 2 ′ ent d Bz S/dz at z = 0 is given by B0 /(η b) with ′ h i − η 0.05, and it gives the vanishing point of zlev at ∼ ′ b = λ0/η 20λ0. This is much further than the end the characteristic length becomes λ0 = 2.23 mm, ∼ (max) of the approximate curve, b = αλ0/η 6.2λ0. The and we have z = 0.65 mm at b = 7.5 mm. For approximate formula is still useful for qualitative≈ es- the randomly-stacked graphite at 77K, on the other timation of the maximum levitation length z(max). hand, the susceptibility is about χ = 45 10−4, The important fact is that all the length scales of giving z(max) =5.9 mm at b = 68 mm. − × the system, such as the levitation height and the grid The levitation also depends on the size of the disk. period, are scaled by a single parameter λ0 [Eq. (9)], Figure 5(b) shows the levitation height as a function which is proportional to χ. If χ is doubled, therefore, of the disk radius R, at χ = 5 10−4 and b = 3 mm. we have the same physics with all the length scale In increasing R, the levitation− × length first monoton- doubled. For the typical susceptibility of AB-stacked ically increases, and then eventually approaches to graphite at the room temperature, χ = 5 10−4, the asymptotic value (dashed line) argued above, af- − × 5

2 Photo-irradiated area Randomly stacked AB-stacked SN AB-stacked (exp.) N S

(mm) 1.5

lev b = 3 mm T+ ΔT T L R = 1.5 mm x 1 L

0.5 FIG. 7. Setup of the thermal-driven motion of levitating Levitationheight, z graphite. A square-shaped graphite piece levitates on the magnet checkerboard and it is partially heated by photo irradiation. 0 0 100200 300 400 T(K) IV. THERMAL GRADIENT DRIVEN MOTION FIG. 6. Temperature dependence of the levitation height of the AB-stacked graphite (blue dashed) and randomly- stacked graphite (black solid) of radius R = 1.5 mm, in A graphite piece much larger than the magnet size the magnetic grid with b = 3 mm. The orange dots repre- b freely moves along the horizontal direction while sent the levitation length of AB-stack graphite measured 20 floating over the magnets, because it covers a num- in the experiment. ber of magnetic periods and the total energy hardly depends on the xy position. Now we consider a situ- ation illustrated in Fig. 7, where a part of the levitating graphite piece is heated by photo irradiation. We ter some oscillation. The monotonic increasing re- assume that the irradiated area is fixed to the rest gion corresponds to the disk size smaller than the frame of the magnets, and consider the movement of magnet grid size b. There a smaller disk has a smaller graphite against it. In the experiment, it was shown levitation, because as seen in Fig. 4(c), dB2/dz near that the graphite is attracted to the photo irradi- z 20,25 the origin quickly decays in z, so a tiny disk can ated region. This can be understood that the levitate only in a small distance to catch the finite graphite minimizes the total energy by moving to 2 the high temperature area, where the diamagnetism dBz /dz. is smaller so that the energy cost is lower under the In Fig. 6, we show the temperature dependence of same magnetic field. the levitation length zlev of the AB-stacked graphite We can estimate the magnitude of the thermal and randomly stacked graphite, with a disk radius gradient force as following. We consider a L L R =1.5 mm. In a fixed geometry, zlev is proportional square-shaped graphite piece with thickness w,× and to log χ according to Eq. (8). We see that the z of | | lev assume that the graphite in the irradiated area is AB-stacked graphite shows a similar temperature de- instantly heated up to temperature T + ∆T , while pendence to the susceptibility itself [Fig. 3(a)]. The otherwise the temperature remains T , as in Fig. 7. randomly stacked graphite exhibits a log T behavior The length of the high temperature region is denoted because z log χ and χ 1/T . The orange dots lev ∝ | | ∝ by a variable x. i.e., when the graphite is moved to in Fig. 6 indicate the levitation length of AB-stack the left, then x increases. We neglect the heat trans- graphite measured in the experiment.20 We can see a port on the graphite for simplicity. The total energy good quantitative agreement between the simulation of graphite contributed by magnetic field is written and experiment without any parameter fitting. The as simulation underestimates the slope of the temperature dependence, suggesting that the susceptibility 1 2 U = χ(T + ∆T ) Bz wLx in the real system decreases more rapidly in tem- −2µ0 h i perature than in our model calculation. A possible 1 2 reason for this would be the effect of phonon scat- χ(T ) Bz wL(L x), (11) − 2µ0 h i − tering, which increases the energy broadening Γ in higher temperature and reduces the susceptibility. where B2 is the square magnetic field at the lev- h z i 6 itation height averaged over xy-plane. The force f γ0 γ1 γ2 γ3 γ4 γ5 ∆ can be calculated as the derivative of the free energy 3.16 0.39 -0.019 0.315 0.044 0.038 0.049 We can show that the free energy is dominated by the magnetic part, and then the force is obtained as TABLE I. Examples of the band parameters (in unit of ∂U 1 ∂χ 2 eV) for the graphite.35 f ∆T Bz wL. (12) ≈− ∂x ≈ 2µ0 ∂T h i

For example, if we take a AB-stack graphite piece B on the layer 1 and A , B on the layer 2, where of L = 10 mm and w = 0.025 mm, and apply a 1 2 2 B1 and A2 are vertically located, while A1 and B2 temperature difference of ∆T = 10 K under T = 300 are directly above or below the hexagon center of K, then we have f =2 10−3 mg-force, which gives × 2 the other layer. The lattice constant within a sin- the acceleration of 4 mm/s . gle layer is given by a = 0.246 nm and the distance On the other hand, we have much greater force between adjacent graphene layers is d = 0.334 nm. in the random stack graphite in low temperature, 2 The lattice constant in the perpendicular direction because ∂χ/∂T 1/T . For a piece of random- is 2d. The low-energy electronic states can be de- stack graphite of∝ the same shape with T =77 K and 2 scribed by a k p Hamiltonian around the valley ∆T = 10 K, the acceleration becomes 63 mm/s . 28–31 · center K±. , Here, we include band parameters ′ γi (i = 0, 1, , 5) and ∆ , where γi represents the tight-binding··· hopping energy between carbon atoms V. CONCLUSION as depicted in Fig.2, and ∆′ is related on-site energy difference between dimer sites (B1, A2) and non- We have studied the diamagnetic levitation and dimer sites (A1,B2) . The parameters adopted in the thermal-driven motion of graphite on a checker- this work are summarized in Table I.35 board magnet array. We showed that the physics is Let A and B (j =1, 2) be the Bloch functions | j i | j i governed by the length scale λ0 [Eq. (9)], which de- at the corresponding sublattices. If the basis is taken pends on the susceptibility χ and the mass density as A1 , B1 , A2 , B2 , the effective Hamiltonian is of the levitating object as well as the field amplitude written| i as| 10,36–39i | i | i of the magnet. The maximum levitation length and αγ2 vp− λv4p− λv3p+ the required grid size are both proportional to λ0, − vp αγ + ∆ λγ λv p− and therefore proportional to χ. We showed a ran- (k)=  + 5 1 − 4 (A1) H λv4p+ λγ1 αγ5 + ∆ vp− domly stacked graphite exhibits much greater levi- −  λv p− λv p vp αγ  tation length than the AB-stacked graphite, and it  3 − 4 + + 2  is even enhanced in low temperatures because of χ where k = (kx, ky, kz), p± = ~(ξkx iky), kx and inversely proportional to the temperature. We in- k are the in-plane wavenumber measured± from the vestigated the motion of the levitating object driven y valley center Kξ, and ξ = is the valley index. by the temperature gradient, and estimate the gen- We defined λ(k ) = 2cos k d±and α(k ) = cos2k d erated force as a function of susceptibility. z z z z with the out-of-plane wavenumber kz. The parameter v = (√3/2)γ0a/~ is the band velocity of monolayer graphene, and v and v are given by ACKNOWLEDGMENTS 3 4 vi = (√3/2)γia/~ (i = 3, 4). Here γ3 is responsible for the trigonal warping of the energy bands, and γ4 MK acknowledges the financial support of JSPS is for the electron-hole asymmetry. KAKENHI Grant Number JP17K05496. Figure 8 shows the energy bands as a function of kx with ky fixed to 0. Here the subbands labeled by different kz’s are separately plotted with hori- Appendix A: Band Model and magnetic zontal shifts. The lower panel is the magnified plot susceptibility of AB-stacked graphite near zero energy. The band structure of each fixed 40 kz is similar to that of bilayer graphene. where a In this Appendix, we present the detailed descrip- pair of electron and hole bands are touching near the tion of the band model and the susceptibility cal- zero energy with quadratic dispersion. At the zone culation for the AB-stacked graphite. We consider boundary, kz = π/(2d), the energy band becomes a AB(Bernal)-stacked graphite as shown in Fig. 2(a). linear Dirac cone like monolayer graphene’s. We see A unit cell is composed of four atoms, labelled A1, that the electron-hole band touching point slightly 7

kz/( π/2d) = 0 0.20.4 0.6 0.8 1 0.03 15 kz/( π/2d) = 1 AB-stacked 0.9 1 0.8

0.02 10 −χ 0.6 (10

0.5 d / 0.4 z k

0.2 -4 ) 0 −χ 0.01 0 5

Energy(eV) -0.5 0 0 -1 -0.1 -0.05 0 0.05 0.1 0.1 μ (eV)

0.05 FIG. 9. Magnetic susceptibility of AB-stacked graphite as a function of the chemical potential µ, separately plotted for diﬀerent kz’s. The dashed curve is the total sus- 0 ceptibility.

Energy(eV) -0.05 2 given by χB /(2µ0). By integration by parts in -0.1 Eq.(A2), we− have 0 0.050.1 0.15 k / (γ / hv) ∞ x 1 ∂f(ǫ) χ(µ,T )= dǫ χ(µ,T = 0), (A4) Z−∞ − ∂ǫ FIG. 8. The band structure of AB-stacked bilayer graphene as a function of kx (with ky = 0). The sub- which relates the susceptibility at finite temperature bands labeled by different kz’s are separately plotted with that at zero temperature. We include the en- with horizontal shifts. The lower panel is the magnified ergy broadening effect induced by the disorder po- plot of the same bands near zero energy. Blue dashed tential by replacing i0 in Eq. (A2) with a small self- curves indicate the dispersion of the band touching point energy iΓ in the Green’s function. We assume the as a function of kz. constant scattering rate Γ = 5 meV in the following calculations. Figure 9 plots the magnetic susceptibility of AB- stack graphite at fixed kz’s (denoted as χk ) as a disperses in kz as α(kz)γ2, and this is the origin of z the semimetallic nature of graphite. function of the chemical potential with the temperature T = 50K, where the dashed curve is the total For the magnetic susceptibility, we use the general π/(2d) 27 expression based on the linear response theory, susceptibility χ = −π/(2d) χkz dkz. Approximately, ∞ χkz is equivalent toR that of bilayer graphene, which is a logarithmic peak centered at the band touching χ(µ,T )= dǫf(ǫ)ImF (ǫ + i0), (A2) 10,41 Z−∞ point and truncated at energies of λ(k )γ . In ± z 1 increasing kz, the peak becomes higher and it finally with becomes a broadened delta function at the zone edge k = π/(2d), which is an analog of the susceptibil- g g e2 z F (z)= µ v s tr(G G G G ). ity of monolayer graphene.6 The center of the peak − 0 2πL3 ~2 Hx Hy Hx Hy Xk moves as a function of kz, in accordance with the (A3) shift of the band toughing point caused by γ2 [Fig. 8]. We notice that the curves near kz = 0 has an ad- Here µ is the chemical potential of electrons, T is ditional sharp peak on top of the logarithmic back- the temperature, gv = 2 and gs = 2 is the valley ground, which originates from the trigonal warping 10 and spin degeneracy, respectively, L is the system caused by γ3. The total susceptibility exhibits a size, and µ0 is the vacuum permeability. We also broadened peak structure bound by kz = 0 peak defined x = ∂ /∂kx, y = ∂ /∂ky, G(z) = (z and kz = π/(2d) peak, and its total width is of the −1 H H H(ǫ−µ)/kBHT −1 − ) , and f(ǫ) = [1 + e ] . The χ of this order of 2γ2 0.04eV. definitionH is the dimensionless susceptibility in the Figure 3(a)∼ shows the temperature dependence of SI unit. The energy density of the magnetic field is the total susceptibility χ of AB-stacked graphite at 8

µ = 0 (charge neutral). The temperature effect on mically decreases as temperature increases, and this the susceptibility can be understood by using Eq. is understood as thermal broadening of the logarith- (A4), where χ(µ) in a finite temperature is obtained mic peak of χ(µ). When kB T is much smaller than by averaging χ(µ) of zero temperature over an en- the peak width of χ(µ), the susceptibility does not ergy range of a few kB T . In Fig. 3(a), χ logarith- depend much on the temperature, and this explains the nearly flat region in T < 50 K in Fig. 3(a).

1 W. Braunbek, Zeitschrift für Physik 112, 764 (1939). 2786 (2013). 2 E. Beaugnon and R. Tournier, Nature 349, 470 22 Y. Su, Z. Xiao, Z. Ye, and K. Takahata, IEEE Elec- (1991). tron Device Letters 36, 393 (2015). 3 M. V. Berry and A. K. Geim, European Journal of 23 S. Kang, J. Kim, J.-B. Pyo, J. H. Cho, and T.-S. Physics 18, 307 (1997). Kim, International Journal of Precision Engineering 4 M. Simon and A. Geim, Journal of applied physics 87, and Manufacturing-Green Technology 5, 341 (2018). 6200 (2000). 24 C. Niu, F. Lin, Z. M. Wang, J. Bao, and J. Hu, 5 A. Winkleman, K. L. Gudiksen, D. Ryan, G. M. Journal of Applied Physics 123, 044302 (2018). Whitesides, D. Greenfield, and M. Prentiss, Applied 25 M. Ewall-Wice, S. Yee, K. DeLawder, S. R. Mont- physics letters 85, 2411 (2004). gomery, P. J. Joyce, C. Brownell, and H. ElBidweihy, 6 J. W. McClure, Phys. Rev. 104, 666 (1956). IEEE Transactions on Magnetics (2019). 7 H. Fukuyama and R. Kubo, J. Phys. Soc. Jpn. 28, 26 Y. Ominato and M. Koshino, Phys. Rev. B 87, 115433 570 (1970). (2013). 8 M. Sharma, L. Johnson, and J. McClure, Physical 27 H. Fukuyama, Prog. Theor. Phys. 45, 704 (1971). Review B 9, 2467 (1974). 28 P. Wallace, Phys. Rev 71, 622 (1947). 9 H. Fukuyama, J. Phys. Soc. Jpn. 76, 043711 (2007). 29 J. W. McClure, Phys. Rev. 108, 612 (1957). 10 M. Koshino and T. Ando, Phys. Rev. B 76, 085425 30 J. C. Slonczewski and P. R. Weiss, (2007). Phys. Rev. 109, 272 (1958). 11 R. D. Waldron, Review of Scientific Instruments 37, 31 M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 51, 29 (1966). 1 (2002). 12 R. Moser and H. Bleuler, IEEE Transactions on ap- 32 R. Bistritzer and A. MacDonald, Proc. Natl. Acad. plied superconductivity 12, 937 (2002). Sci. 108, 12233 (2011). 13 R. Moser, F. Barrot, J. Sandtner, and H. Bleuler, in 33 G. T. de Laissardiere, D. Mayou, and L. Magaud, 17th International Conference on Magnetically Levi- Phys. Rev. B 86, 125413 (2012). tated Systems and Linear Drives (Maglev) (2002). 34 J. Camacho and V. Sosa, Revista mexicana de f´ısica 14 M. Boukallel, J. Abadie, and E. Piat, in 2003 IEEE E 59, 8 (2013). International Conference on Robotics and Automation 35 J.-C. Charlier, X. Gonze, and J.-P. Michenaud, (Cat. No. 03CH37422), Vol. 3 (IEEE, 2003) pp. 3219– Phys. Rev. B 43, 4579 (1991). 3224. 36 F. Guinea, A. Castro Neto, and N. Peres, Phys. Rev. 15 A. Cansiz and J. R. Hull, IEEE transactions on mag- B 73, 245426 (2006). netics 40, 1636 (2004). 37 B. Partoens and F. Peeters, Phys. Rev. B 74, 075404 16 Q. Li, K.-S. Kim, and A. Rydberg, Review of scientific (2006). instruments 77, 065105 (2006). 38 F. Guinea, A. Castro Neto, and N. Peres, Solid state 17 W. Liu, W.-Y. Chen, W.-P. Zhang, X.-G. Huang, and communications 143, 116 (2007). Z.-R. Zhang, Electronics Letters 44, 681 (2008). 39 M. Koshino and T. Ando, Phys. Rev. B 77, 115313 18 Y. Mizutani, A. Tsutsumi, T. Iwata, and Y. Otani, (2008). Journal of Applied Physics 111, 023909 (2012). 40 E. McCann and M. Koshino, Reports on Progress in 19 G. Kustler, IEEE Transactions on Magnetics 48, 2044 Physics 76, 056503 (2013). (2012). 41 M. Koshino and T. Ando, Phys. Rev. B 75, 235333 20 M. Kobayashi and J. Abe, Journal of the American (2007). Chemical Society 134, 20593 (2012). 21 W. Hilber and B. Jakoby, IEEE Sensors Journal 13,