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2012

Quantum tunneling of the magnetic moment in a free

Michael F. O'Keeffe CUNY Lehman College

Eugene M. Chudnovsky CUNY Lehman College

Dmitry A. Garanin CUNY Lehman College

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Contact: [email protected] arXiv:1108.4189v2 [cond-mat.mes-hall] 13 Oct 2011 ini loe nyaotafie xsadtespin magnetic the for clusters typical ferromagnetic and small is axis and latter molecules fixed The strong to a due anisotropy. -down solved and magnetic about spin-up be rota- to reduced only can mechanical are spin states allowed when a frame is with laboratory tion rotor the rigid in a exactly of problem the odcigleads conducting ftepolm famgei oeueebde na solution in simple embedded molecule obtain magnetic to about a rotating one of microcantilever system allows problems a the also in of axis states fixed spin a two to duction etbtencokieadcutrlcws directions SQUID counterclockwise a cur- and in superconducting Similar clockwise a between of orbital). rent tunneling + for (spin occurs momentum situation conserve angular be to must total order particle in the free rotations noticed a mechanical was in with it macroscpin entangled where a 12, of Ref. tunneling mag- in that the made of was tunneling moment affects netic particle magnetic small a of hti iil ope otertr antcmmn of moment Magnetic rotor. frame the coordinate to rigorous coupled the has rigidly in is rotor rotations that rigid arbitrary for symmetric solution a inside macrospin unsolved remained now. rel- , is until magnetic which system, free spin to two-state evant a of rotations bitrary resonator torsional a pnti rbe statbeb nltclmethods Without analytical rotor by spin. symmetric tractable a a with is for rotor problem only quantum this rigid spin a problem a generic of The that scarce. is been has particles magnetic rmsi ere ffedmmk vnsmerccases symmetric difficult even make more freedom significantly of degrees spin from we odcigleads conducting tween r ret oeisd oi nanocavities solid inside move microresonators to free are a reo.Eprmna oki hsae focused area this clusters in magnetic mechani- work free possess Experimental on that magnets freedom. cal nanoscopic of chanics is tep oudrtn o ehnclfreedom mechanical how understand to attempt First nti ae eso httepolmo two-state a of problem the that show we paper this In hr a enmc eetitrs nqatmme- quantum in interest recent much been has There 13–15 nrylvl r bandadtegon-tt antcmo magnetic ground-state the and obtained are levels energy ASnmes 54.j 64.g 55.t 85.25.Dq 75.50.Tt, 36.40.Cg, 75.45.+j, obtain numbers: is PACS diagram phase th quantum Energ of and values rotor. investigated different the are with of shapes states inertia between of transitions moments manife phase tunneling the spin on on moment freedom magnetic mechanical the of effect The esuytneigo h antcmmn napril hth that particle a in moment magnetic the of tunneling study We unu unln fteMgei oeti reParticle Free a in Moment Magnetic the of Tunneling Quantum 20 .INTRODUCTION I. 5,6 21 eety twsdemonstrated was it Recently, . antcmlcl irtn between vibrating molecule magnetic , n antcmlclsbigdbe- bridged molecules magnetic and , n famcopntneiginside tunneling macrospin a of and , 22,23 7–9 5 efr akBueadWs,Box e ok 10468-158 York, New Bronx, West, Boulevard Park Bedford 250 oee,tepolmfrar- for problem the However, . hsc eatet emnClee iyUiest fNew of University City College, Lehman Department, 11 hoeia eerho free on research Theoretical . 1–3 10 . .F ’eff,E .Cunvk,adD .Garanin A. D. and Chudnovsky, M. E. O’Keeffe, F. M. antcprilsthat particles magnetic , opiain resulting Complications . 17–19 4 magnetic , Dtd coe 4 2011) 14, October (Dated: h re- The . 16 that ymti oo ihasi saaye nSc V. VI. VII. Sec. Section Sec. in in in analyzed presented are studied is conclusions is of spin Our moment state a Ground magnetic with Ground-state IV. rotor symmetric Sec. tunneling II. in a a III. constructed containing Sec. are Sec. rotator rigid in in macrospin a reviewed reviewed of is briefly states macrospin Quantum is tunneling a rotator of rigid Theory a of theory spin). the with associated factor gyromagnetic the eoand zero aigbd n ietteae fta frame that of axes ro- the the direct with coupled and rigidly body is tating that frame coordinate the nri (with inertia L claglrmmnu,dfie ntefie laboratory fixed the in defined momentum, angular ical Here atce h rudsaemgei oeto free spin a total of the a moment of shape magnetic with the ground-state particle on variable. The depends continuous transition a particle. the as of treated momen- order is angular The latter total the the when of tum values different between with transitions states phase exhibits quantum particle second-order the or of energy first- the mo- that angular show We total mechanical mentum. the and conserves spin particle that between momentum free interplay angular a complex for a mag- changes to ground-state situation due zero This a solid, in moment. a results netic in spin embedded the It is down of nanoparticle and tunneling a up spin. When the to of states. due contribution spin relative entirely the is on rotor depends neutral electrically the oino ehnclrttosi ie by Hamil- given the is frame rotations coordinate mechanical such of In tonian body. the of inertia z x ln h rnil xso h esro oet of moments of tensor the of axes principle the along h tutr fteatcei sflos Quantum follows. as is article the of structure The osdrfis h rbe ihu pn echoose We spin. a without problem the first Consider L , y I I UNIAINO II BODY RIGID OF QUANTIZATION II. L , antcmmn.Prilso various of Particles moment. magnetic e x eti optdfrasmercrotor. symmetric a for computed is ment I , t tefi togdpnec fthe of dependence strong a in itself sts z ed. gµ y r rjcin fteoeao ftemechan- the of operator the of projections are I , ftepril xiisquantum exhibits particle the of y B z H sfl oainlfedm Exact freedom. rotational full as µ S ˆ r h rnia oet fietaand inertia of moments principal the are B R eedn ntepicplmmnsof moments principal the on depending , = en h ormgeo and magneton Bohr the being ~ 2 ROTATIONS 2

L ,USA 9, I York, x x 2 S + a eayhn between anything be can L I y y 2 + L I z z 2 ! 10 . ,y x, g being and , (1) 2 coordinate frame, onto the body axes x,y,z. Such a which satisfy choice of coordinates and operators results in the anoma- 24 lous commutation relations ,[Li,Lj]= iǫijkLk (notice HˆSΨ± = E∓Ψ± , (7) the minus sign in the right-hand side), but− does not affect 2 2 the relations L = L(L + 1), [L ,Lz] = 0. where For a symmetric rotor two of the moments of inertia are the same, I = I , and the Hamiltonian can be written E+ E− ∆. (8) x y − ≡ as The tunnel splitting ∆ is generally many orders of mag- ~2L2 ~2L2 1 1 Hˆ = + z . (2) nitude smaller than the distance to other spin energy lev- R 2I 2 I − I x  z x  els, which makes the two-state approximation very accu- The corresponding eigenstates are characterized by three rate at low energies. For example, quantum numbers L,K, and M, ˆ 2 2 HS = DSz + dSy (9) L2 LKM = L(L + 1) LKM ,L =0, 1, 2,... − | i | i Lz LKM = K LKM ,K = L, L +1,...,L 1,L with d D describes the biaxial anisotropy of spin-10 | i | i − − − molecular≪ nanomagnet Fe-8, where the tunnel splitting L JKM = M LKM ,M = L, L +1 ...,L 1,L, Z in the limit of large S is given by25 | i | i − − − (3) where L is the angular momentum operator de- 8S3/2 d S Z ∆= D. (10) fined with respect to the laboratory coordinate frame π1/2 4D (X,Y,Z). The eigenvalues of (2) are degenerate on M:   The distance to the next excited spin level is (2S 1)D, ~2L(L + 1) ~2K2 1 1 − E = + . (4) which is large compared to ∆. LK 2I 2 I − I x  z x  It is convenient to describe these lowest energy spin The general form for the energy levels of a rotating states Ψ± with a pseudospin-1/2. Components of the σ asymmetric rigid body, Ix = Iy = Iz, does not exist, corresponding Pauli operator are although it is possible to calculate6 6 matrix elements of σ = ψ ψ + ψ ψ the Hamiltonian for a given L. x | −Sih S | | Sih −S | σ = i ψ ψ i ψ ψ y | −Sih S |− | Sih −S | σ = ψ ψ ψ ψ . (11) III. TUNNELING OF A LARGE SPIN z | Sih S | − | −Sih −S |

The projection of HˆS onto ψ±S states is Let S be a fixed-length spin embedded in a stationary | i body. Naturally, the magnetic anisotropy is defined with Hˆ = m Hˆ n m n . (12) respect to the body axes. The general form of the crystal σ h | S| i | ih | field Hamiltonian is m,nX=ψ±S

Hˆ = Hˆ + Hˆ , (5) Expressing ψ±S in terms of Ψ± one obtains S k ⊥ | i ˆ ˆ where Hk commutes with Sz and H⊥ is a perturbation ∆ ψ±S HˆS ψ±S =0, ψ±S HˆS ψ∓S = , (13) that does not commute with Sz. The states S are h | | i h | | i − 2 |± i degenerate ground states of Hˆ , where S is the total spin k which gives the two-state Hamiltonian of the nanomagnet. Hˆ⊥ slightly perturbs these states, adding to them small contributions from other mS | i ˆ ∆ states. We will call these degenerate perturbed states Hσ = σx (14) − 2 ψ . Physically they describe the magnetic moment | ±Si aligned in one of the two directions along the anisotropy having eigenvalues ∆/2. ± axis. Full perturbation theory with account of the de- In the absence of tunneling a classical magnetic mo- ˆ generacy of HS provides quantum tunneling between the ment is localized in the up or down state. It is clear ψ states for integer S. The ground state and first | ±Si that delocalization of the magnetic moment due to spin excited state are symmetric and antisymmetric combina- tunneling reduces the energy by ∆/2. In a free particle, tions of ψ , respectively19, | ±Si however, tunneling of the spin must be accompanied by 1 mechanical rotations in order to conserve the total angu- Ψ+ = ( ψS + ψ−S ) lar momentum. Such rotations cost energy, so it is not √ | i | i 2 a priori clear whether the tunneling will survive in a free 1 Ψ− = ( ψS ψ−S ) , (6) particle and what the ground state is going to be. This √2 | i − | i problem is addressed in the following Section. 3

IV. RIGID ROTOR CONTAINING TUNNELING the Z-axis only (that is, in the limit of Ix ) these re- MACROSPIN sults coincide with the results obtained by→ the ∞ instanton method in Ref. 26, where it was shown that, in practice, Consider now a tunneling macrospin embedded in a the renormalization of the magnetic anisotropy and spin free particle having the body z-axis as the magnetic tunnel splitting by mechanical rotations is small. Eq. anisotropy direction. Such a particle is characterized by (19) provides generalization of this effect for arbitrary the total angular momentum, J = L + S. In the body rotations of a symmetric rotator with a spin. According frame this operator may appear unconventional due to to this equation and Eq. (9), when rotations are allowed the different sign of commutation relations for L and S. the effective easy-axis magnetic anisotropy and the tun- However, this problem can be easily fixed11 by the trans- nel splitting can decrease or increase, depending on the formation S S that changes the sign of the com- ratio Ix/Iz. mutation relation→ − for S. Such a transformation does not Projection of Eq. (17) on the two spin states along the change the results of the previous Section because the lines of the previous Section gives crystal field Hamiltonian contains only even powers of S ~2J2 ~2J 2 1 1 ∆ ~2S . It is interesting to notice that while in the labora- Hˆ = + z σ J σ . (20) 2I 2 I − I − 2 x − I z z tory frame [Ji,Sj ] = iǫijkSk, components of the opera- x  z x  z tors J and S defined in the body frame commute with each other11. In addition, operator J2 is the same in the where we have used body and laboratory frames24. This permits description ψ S ψ = S, ψ S ψ =0 . (21) of quantum states of the particle in terms of quantum h ±S | z| ±Si ± h ±S| x,y| ±Si numbers associated independently with the total angu- We construct eigenstates of this Hamiltonin according to lar momentum and spin. The full Hamiltonian is given by the sum of the rota- 1 ΨJK = (C±S ψS C∓S ψ−S ) JK (22) tional energy and magnetic anisotropy energy | i √2 | i± | i | i ~2 2 ~2 2 ~2 2 Lx Ly Lz where Hˆ = + + + HˆS . (15) 2Ix 2Iy 2Iz J2 JK = J(J + 1) JK , J =0, 1, 2,... Expressing the mechanical angular momentum L in | i | i J JK = K JK ,K = J, . . . , J. (23) terms of the total angular momentum J and the spin z| i | i − S, we get Solution of Hˆ Ψ = E Ψ gives energy levels as | JKi | JK i ~2 2 J 2 2 ~2 2 S2 2 ˆ Jx y Jz Sx y Sz H = + + + + + ∆ 2 ~2KS 2 2 Ix Iy Iz ! 2 Ix Iy Iz ! E(±) = E + , (24) JK JK ± 2 I J S J S J S s   z  ~2 x x + y y + z z + Hˆ . (16) − I I I S  x y z  where EJK is provided by Eq. (4) with L replaced by J. The upper (lower) sign in Eq. (24) corresponds to For a symmetric rigid rotor with Ix = Iy this Hamilto- nian reduces to the lower (upper) sign in Eq. (22). For K = 0 each state is degenerate with respect to the sign of6 K. For ~2J2 ~2J 2 1 1 Hˆ = + z K =0, 1, 2,... the coefficients in Eq. (22) are given by 2I 2 I − I x  z x  2 2 ~2 JxSx + JySy JzSz ˆ ′ C± = 1 αK/ S + (αK) , (25) + + HS , (17) ± − Ix Iz q   where α is a dimensionless magneto-mechanicalp ratio, where ~ 2 ~2 ~2S2 2( S) ˆ ′ ˆ 1 1 2 α = . (26) H = HS + S + . (18) I ∆ S 2 I − I z 2I z  z x  x Energy levels in Eq. (24) can be given a simple semi- The last term in Hˆ ′ is an unessential constant, ~2S(S + S classical interpretation. Indeed, the last term in this 1)/(2I )2. The second term provides renormalization of x equation is the tunnel splitting of the levels in the ef- the crystal field in a freely rotating particle. For, e.g., fective magnetic field that appears in the body reference the biaxial spin Hamiltonian given by Eq. (9) it leads to frame due to rotation about the spin quantization axis ~ ~2 1 1 at the angular velocity K/Iz. When S = 0 (which also D D . (19) → − 2 I − I means ∆ = 0) Eq. (24) with J = L gives the energy of the  z x  quantum symmetric rigid rotor without a spin, Eq. (4). This, in turn, renormalizes the tunnel splitting given by In the case of a heavy body (large moments of inertia) Eq. (10). For a particle that is allowed to rotate about the ground state and the first excited state correspond to 4

J = K = 0, and we recover the tunnel-split spin states − )( − − )( in a non-rotating macroscopic body, E = ∆/2. In E JJ E00 , (arb. units) 00± ± è è è the general case, spin states of the rotator are entangled 6 è with mechanical rotations. 0.1 è è Equations (22)-(25) are our main analytical results for 4 1 è the low-energy states of a free magnetic particle. In gen- è 2 è eral, numerical analysis is needed to find the ground state è è of the particle. Special cases of the aspect ratio that will è è 1.5 è è be analyzed below include a needle of vanishing diameter 0 è è è è (which is equivalent to the problem of the rotation about è - è a fixed axis treated previously in the laboratory frame by 2 è two of the authors16), a finite-diameter needle, a sphere, è -4 è and a disk. è è S = 10, Iz/Ix = 0 è è -6 α = 0.1, 1, 1.5

V. GROUND STATE 0 2 4 6 8 10 12 14 J

Minimization of the energy in Eq. (24) on J with the FIG. 1: Dependence of energy on J at K = J and Iz/Ix = 0 account of the fact that J cannot be smaller than K for different values of α. The plot shows second-order quan- immediately yields J = K, that is, the ground state al- tum phase transition on α. ways corresponds to the maximal projection of the total angular momentum onto the spin quantization axis. In semiclassical terms this means that the minimal energy E − )( − E − )( , (arb. units) states in the presence of spin tunneling always correspond JJ 00 è è è è to mechanical rotations about the magnetic anisotropy 6 è axis. This is easy to understand by noticing that the sole 0.1 1 è è reason for mechanical rotation is the necessity to conserve 4 è è the total angular momentum while allowing spin tunnel- è è ing to lower the energy. To accomplish this the particle 2 è è 1.59 needs to oscillate between clockwise and counterclock- è è è è è è wise rotations about the spin quantization axis in unison è è è 0 è è è è with the tunneling spin. If such mechanical oscillation è costs more energy than the energy gain from spin tun- è 2 neling, then both spin tunneling and mechanical motion -2 è S = 10, I /I = 2 è must be frozen in the ground state as, indeed, happens z x è -4 α = 0.1, 1, 1.59, 2 è in very light particles (see below). Rotations about axes è è other than the spin quantization axis can only increase 0 2 4 6 8 10 12 14 J the energy and, thus, should be absent in the ground state. For further analysis it is convenient to write Eq. (24) FIG. 2: Dependence of energy on J at K = J and Iz/Ix = 2 in the dimensionless form, for different values of α. The plot shows first-order quantum phase transition on α. (±) 2 2 2 EJK α J(J + 1) K K 1 K 2 = 2 − λ + 2 1+ 2 α , ∆ 4 " S S # ± 2r S case of an ellipsoid, the transition is second order, see (27) −1 Fig. 1. It occurs at α = 1 1/(2S)2 . This case is in terms of dimensionless parameters α and the aspect − ratio for the moments of inertia equivalent to the rotation about a fixed axis studied in   Ref. 16. For any finite ratio Iz/Ix the transition is first λ = Iz/Ix . (28) order, see Fig. 2. It occurs at the value of α that depends on Iz/Ix. The origin of the transfer from a second-order The range of λ for a symmetric rotator is 0 λ 2. For, transition at λ = 0 to the first-order transition at λ =0 e.g., a symmetric ellipsoid with semiaxes a≤= b≤= c, one can be traced to the term [J(J + 1) K2]S−2λ in6 Eq. has λ =2a2/(a2 + c2). 6 (27). We should notice that for a finite-size− nanomagnet The dependence of the energy levels (24) on J at the analogy with first- and second-order phase transition K = J is shown in figures 1 and 2. It exhibits quan- is, of course, just an analogy. To talk about real phase tum phase transition on the parameter α between states transitions one has to take the limit of S , Ix,z with different values of J. Only for a needle of vanishing when the distances between quantum levels→ ∞ go to→ zero ∞ diameter, I /I 0, which corresponds to a 0 in the and the energy becomes quasi-continuous function of J. z x → → 5

of Ref. 16. The quantum number K determines the E  D ground state, as the energy, Eq. (27), no longer formally depends on J. However, the values of αJ at λ = 0, for 0 J = 0 J ³ Jc = 5 Λ = 1 which ground state transitions occur, are the same as S = 10 those for which E(−) = E(−), and we will use J to -0.1 J K−1 JK describe the ground state of the axial rotor as well. The HdEdΑLD 0.25 first ground state transition occurs from J =0 to J =1 -0.2 0 < 0.2 at α(λ) = α1(0) = α1(0), because Jc = 1 for λ 0.01. L ∼ -0.3 0.15 At α = α2(0) the ground state switches from J = 1 to 0.1 J = 2, and so on. The final transition is to a completely -0.4 0.05 localized spin state J = S in which spin tunneling is 0 frozen for all α > αS(0). For example, when S = 10, 0 1 2 3 4 Α 5 0 -0.5 α1(0) = α1(0) = 1.0025 and α10 =3.2066. 0 1 Αc 2 3 Α 4 Α 5 Needle of finite diameter: The ground state of a needle of finite diameter (a c for an ellipsoid) with λ = 0.1, that is free to rotate≪ about any axis, shows qualita- tively different behavior. As α increases, the ground FIG. 3: Dependence of the ground-state energy on α for a 0 spherical particle. Inset shows the discontinuity of the deriva- state changes from J = 0 to Jc = 3 at α = α3(0.1), as the smallest value of α0 (0.1) for 1 J S occurs for tive of the ground state energy on α. J ≤ ≤ Jc = 3. The J = 1, 2 states never become the ground state. After this, transitions occur to successively higher For a given λ, as α increases the ground state switches J, beginning with J =4 at α = α4(0.1), and eventually from J = 0 to higher J when localizing the spin with J = S for α > αS (0.1). For 0 S = 10, α3(0.1)=1.0588 and α10(0.1)=3.3935. (−) 0 (−) 0 E00 (αJ (λ)) = EJJ (αJ (λ)) . (29) Sphere: As λ increases towards unity, the particle be-

0 comes more symmetric with the moment of inertia having Solution of this equation for αJ (λ) gives (prolate) ellipsoidal symmetry, until it reaches spherical symmetry at λ = 1. The first ground state transition (2S)2(J + λ) α0 = . (30) occurs from J = 0 to J = J = 5 at α = α0(1), and J J[(2S)2 (J + λ)2] c 5 − subsequent transitions occur at α = αJ (1). However, the This first transition occurs for the smallest value of α0 (λ) spin never localizes in the J = S state even for very large J alpha, as α (1) has a pole at J = S, so the last transi- and the transition is from J = 0 to the corresponding J 0 tion occurs to the J = S 1 state at α = αS−1(1). For critical value, Jc. For α < αJc the ground state corre- 0 − S = 10, α5(1) = 1.3187 and α9(1) = 2.4325. sponds to J = 0 and C±S = 1. After the first transition from J = 0 to J = Jc, the ground state switches to Disk: With λ increasing from unity, the symmetry of sequentially higher J at values of α which satisfy the body becomes that of an oblate ellipsoid, and begins to flatten in the plane perpendicular to the anisotropy (−) (−) axis. It is easy to check from Eq. (27) that for 1 < λ 2 EJ−1 J−1(αJ (λ)) = EJJ (αJ (λ)) . (31) the state with J = S always has higher energy than≤ the Solution of this equation for αJ (λ) gives state with J = S 1, even in the limit of α . This means that for an− oblate particle some spin→ tunneling ∞ 2 (2S) T (J, λ) (accompanied by mechanical rotations) survives in the αJ = , (2S)2(2J 1)2 T (J, λ)2 (2S)2 T (J, λ)2 ground state no matter how light the particle is. This − − − (32) purely quantum-mechanical result has no semi-classical with p p analogy. In the case of a disk of vanishing thickness, λ = 2, the first ground state transition occurs from J =0 T (J, λ)=2J 1+ λ . (33) to J = J =6 at α = α0(2), and subsequent transitions − c 6 occur at α = αJ (2) up through J = S 1. For S = 10, The critical αJ has poles at λ = 2(S J)+1. For λ 1 0 − α (2) = 1.5873 and α9(2) = 3.5849. there is no longer a ground state transition− to J =≥S, 6 even for very large values of α. Needle of vanishing diameter: The case of a particle that can only rotate about its anisotropy axis16 is equiva- VI. GROUND-STATE MAGNETIC MOMENT lent in our model to a needle of vanishing diameter (a 0 for an ellipsoid), having λ = 0. It is also equivalent→ to the problem of tunneling of the angular momentum of As has been already mentioned, the magnetic moment a superconducting current in a flux qubit coupled to a is due entirely to the spin of the particle, as Lz represents torsional resonator. In this limit we reproduce results mechanical motion of the particle as a whole, and not 6

ÈΜȐHg ΜBL Λ 10 Λ = 0 S = 10 J = 10 2 8 J = 8 J = 9 S = 10 J = 7 6 J = 6 J = 5 1.5 Μ = 0 Μ ¹ 0 4 J = 4 J = 3 2 J = 2 J = 0 J = 1 1 J = 7 J = 8 0 J = 6 0 1 2 3 4 Α 5 J = 9 0.5 J = 5 ÈΜȐHg ΜBL J = 10 10 Λ = 0.1 J = 10 0 8 S = 10 J = 9 J = 8 0 1 2 3 4 Α 5 J = 7 Α 6 J = 6 J = 5 4 Λ J = 4 0.15 J = 3 2 S = 10 J = 0 0

0 1 2 3 4 Α 5 0.1 Μ = 0 Μ ¹ 0 ÈΜȐHg ΜBL

10 Λ = 1 S = 10 J = 9 J = 0 8 J = 8 0.05 J = 7 6 J = 6 J = 1 J = 5 4 J = 2 J = 3 J = 4 J = 5 0 2 1 1.05 1.1 Α 1.15 J = 0 0 Α 0 1 2 3 4 Α 5 ÈΜȐHg ΜBL FIG. 5: Quantum phase diagram for the ground-state mag- netic moment and the total angular momentum. 10 Λ = 2 J = 9 8 S = 10 J = 8 J = 7 6 J = 6 The dependence of the magnetic moment on α for dif- 4 ferent aspect ratios of the particle is shown in Fig. 4. For 2 α < αJc (λ) the ground state corresponds to J = K = 0, J = 0 0 so the spin-up and spin-down states are in an equal su- perposition which produces zero magnetic moment. At 0 1 2 3 4 5 Α greater values of α the spin states contribute in unequal amounts which leads to a non-zero magnetic moment. As FIG. 4: Ground state magnetic moment for a needle of van- α becomes large, the magnetic moment approaches its ishing diameter (λ = 0), finite-diameter needle (λ = 0.1), maximal value µ = gµ S. Note that the magnetic sphere (λ = 1), and a disk of vanishing thickness (λ = 2). max B moment approaches| its| maximum value even for values of λ that do not admit transitions to J = S states. electronic orbital angular momentum. Thus, Because the ground state is completely determined by the parameters α and λ, we can depict the ground state αK behavior in a quantum phase diagram shown in Fig. 5. µ = gµB ΨJK Sz ΨJK = gµBS . − h | | i − S2 + (αK)2 The curves separate areas in the (α, λ) plane that cor- (34) respond to different values of J and different values of Here g is the spin gyromagnetic factor,p and the mi- the magnetic moment. Notice the fine structure of the nus sign reflects the negative gyromagnetic ratio γ = diagram (lower picture in Fig. 5) near the first critical α. gµB/~. The ground state always corresponds to J = This very rich behavior of the ground state on parame- K− , so these are used interchangeably in descriptions of ters must have significant implications for magnetism of the ground state. rigid atomic clusters. 7

VII. CONCLUSIONS ported in beams of free atomic clusters of ferromagnetic materials1–3. Our results may shed some additional light We have studied the problem of a quantum rotator on these experiments. They may also apply to free mag- containing a tunneling spin. This problem is relevant netic molecules if one can justify the condition of rigidity. to of free magnetic nanoparticles. Direct comparison between theory and experiment may It also provides an interesting insight into quantum me- be possible for atomic clusters (molecules) in magnetic chanics of molecules studied from the macroscopic end. traps. The answer obtained for the energy levels of a symmet- To see that the quantum problem studied in this ric rotator, Eq. (24), is non-perturbative and highly non- paper may, indeed, be relevant to quantum states of trivial. It is difficult to imagine how it could be obtained free nanomagnets, consider, e.g., a spherical atomic from first principles without the reduction to two spin cluster of radius R and average mass density ρ having states. Indeed, for spin S the tunnel splitting itself gen- spin S = 10 that, when embedded in a large body, can erally appears in the S-th order of perturbation theory, tunnel between up and down at a frequency of a few GHz, thus providing ∆ 0.1K. Significant changes in see Eq. (10), so the path from the full crystal-field Hamil- ∼ tonian like, e.g., Eq. (9) to Eq. (24) must be very long. the magnetic moment of such a cluster would occur Equations (22) and (24) represent, therefore, a unique at α 1, which, according to Eq. (26), corresponds to I =∼ 8πρR5/15 10−42kg m2 and R 1nm. For exact solution of the quantum-mechanical problem of a ∼ ∼ mechanical rotator with a spin. Striking feature of this a magnetic molecule like, e.g., Mn12, the moments of −42 2 solution is presence of first- and second-order quantum inertia would also be in the ballpark of 10 kg m . phase transitions between states with different values of However, the natural spin tunnel splitting in Mn12 is the magnetic moment. very small, thus, providing a very large α. Same is Our results provide the framework for comparison be- true for Fe8 magnetic molecules. In this case the spin tween theory and experiment on very small free magnetic tunneling in a free molecule must be completely frozen. clusters. Our main conclusion for experiment is that ro- Even if the molecule cannot be considered as entirely tational states and magnetic moments of such clusters de- rigid, such effect, if observed, would receive natural pend crucially and in a predictable way on size and aspect interpretation within the framework of our theory. ratio. This dependence results in a complex phase dia- gram that separates regions in the parameter space, cor- VIII. ACKNOWLEDGEMENTS responding to different values of the magnetic moment. Broad distribution of the magnetic moments that does This work has been supported by the U.S. Department not simply scale with the volume, has, in fact, been re- of Energy through grant No. DE-FG02-93ER45487.

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