Quantum Tunneling of the Magnetic Moment in the S/F/S Josephson ${\bm \varphi}_{\bf 0}$-Junction

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by  Eugene M. Chudnovsky

Quantum Tunneling of the Magnetic Moment in the S/F/S Josephson ϕ -Junction
0
Eugene M. Chudnovsky
Physics Department, Herbert H. Lehman College and Graduate School, The City University of New York
250 Bedford Park Boulevard West, Bronx, New York 10468-1589, USA
(Dated: January 7, 2016)
We show that the S/F/S Josephson ϕ -junction permits detection of macroscopic quantum tun-
0
neling and quantum oscillation of the magnetic moment by measuring the ac voltage across the
junction. Exact expression for the tunnel splitting renormalized by the interaction with the super-
conducting order parameter is obtained. It is demonstrated that magnetic tunneling may become
frozenatasufficientlylargeϕ . Thequalityfactorofquantumoscillationsofthemagneticmoment
0
6 duetofiniteOhmicresistanceofthejunctioniscomputed. Itisshownthatmagnetictunnelingrate
1 in the ϕ -junction can be controled by the supercurrent, with no need for the magnetic field.
0
0
2 PACSnumbers: 75.45.+j,74.50.+r,74.45.+c

n
a
Quantum tunneling of the magnetic moment has been perconductivityontheferromagnetismistypicallyweak.
J
subject of intensive research due to the fundamental This can be understood from the fact that the exchange
5
interest in the phenomenon and because of its poten- interaction responsible for magnetic ordering is typically

] tial applications for quantum information technology. of the order of hundreds or thousands of kelvin while in-
ll Early work focused on non-thermal magnetic relaxation teractionsresponsibleforconventionalsuperconductivity
a innanoparticles1. Lateron,thefocusswitchedtomolec- are in the ball park of a few kelvin. One should notice,
h
ularmagnets2. Duetoidenticalstructureofthebuilding however, that relativistic interactions responsible for the
-
s blocks of crystals made of magnetic molecules, they per- orientation of the magnetic moment in ferromagnets –
e
mitmacroscopicstudiesofquantumtunnelingandquan- the magnetic anisotropy energy is also in the kelvin, or
m
tumoscillationsofmolecularmagneticmoments. Onthe can be even in the subkelvin, range. Thus, the coupling
t. contrary,thereliableobservationandquantitativeanaly- of the superconducting order parameter to the orienta-
a
sis of the quantum tunneling of the magnetic moment in tion of the magnetic moment can, in principle, produce
m
nanoparticles is hampered by the practical impossibility a formidable torque on the moment.
- to make identical nanoparticles. Best samples available Such a possibility was recently proposed by Buzdin
d
still have distribution of sizes and other parameters of who noticed that spin-orbit interaction in a ferromag-
n
o theparticleswithinatleast20%. Duetotheexponential net without inversion symmetry provides the coupling
c dependence of the tunneling rate on the size, this trans- between the direction of the magnetic moment and
[ latesintothedistributionoftunnelingrateswithinmany the superconducting order parameter10,11. In a non-

1 orders of magnitude. centrosymmetric ferromagnetic junction, that Buzdin
v Early on, the difficulty mentioned above prompted re- called the ϕ0-junction, the time reversal symmetry is
6 searchers to look into the possibility to observe mag- broken, and the current-phase relation becomes I =
1 netic tunneling in individual nanopartices. Such mea- Icsin(ϕ−ϕ˜0), where the phase shift ϕ˜0 is proportional
0
surements of 10-nm ferrimagnetic partices of total spin to the component of the magnetic moment perpendicu-
1
0 S ∼105,depositedonananobridgeofadc-SQUID,were lartothegradientoftheasymmetricspin-orbitpotential.
. pioneeredbyWernsdorferetal.3. Theenergybarrierwas In this Letter we argue that such Josephson junction is
1 controlled by the external magnetic field. At very small ideally suited for the study of quantum tunneling of the
0
barrierstheevidenceofnon-thermalmagneticrelaxation magnetic moment. The magnetic tunneling would show
6
1 below 1K was obtained. After a preliminary success, in the ac voltage across the junction and it can be con-
: however, these efforts were largely abandoned in favor of trolled by the dc supercurrent through the junction.
v detecting spin tunneling in better characterized individ- FollowingBuzdin11weconsideraS/F/SJosephsonϕ0-
i
X ual magnetic molecules. Measurements of transport cur- junction depicted in Fig. 1, with the potential energy
r rent through magnetic molecules bridged between con- (cid:26) I (cid:27)
a ducting leads4 and through molecules grafted on carbon U =EJ 1−cos[ϕ−ϕ˜0(M)]− I ϕ +UM(M) (1)
nanotubes5 produced convincing evidence of the effect. c
HereU isthemagneticanisotropyenergywithMbeing
Observation and control of quantum tunneling of the M
themagnetizationoftheferromagneticlayer,ϕ˜ depends
magnetic moment in nanoscale magnetic clusters, how- 0
onthedirectionofM,andE =I Φ /(2π)istheJoseph-
ever, remains a challenging experimental task. J c 0
son energy, with Φ being the flux quantum. With the
InS/F/Snanostructuresferromagnetismcanaffectsu- 0
gradient of the spin-orbit potential normal to the layer
perconductivity via the magnetic field it generates6–8.
(that is, along the Z-direction), ϕ˜ is given by11
Somewhat stronger influence of ferromagnetism on su- 0
(cid:18) (cid:19) (cid:18) (cid:19)
perconductivity at the F/S interface may occur due to M v
ϕ˜ =ϕ y , ϕ ≡l so , (2)
the proximity effect9. The opposite influence of the su- 0 0 M 0 v
0 F
2
where C and R are the capacitance and the resistance of
the junction,
V

1 ∂U
H =− (6)
z
  eff V ∂M
y

istheeffectivefieldactingonthemagneticmoment, and
α is the damping parameter. To allow for quantum tun-
SC
  SC

neling the junction must be of the smallest possible size,
FM

I
  M
  x
  I
  in which case its capacitance C can be safely neglected.
Quantum tunneling of M is carried out by the instanton
solution of Eqs. (4) and (5) in imaginary time, τ = it
at R=0 and α=0. In the resulting semiclassical equa-
tionsϕisaslavevariablethatfollowstheimaginary-time
Figure 1: Schematic picture of the Josephson ϕ -junction. dynamics of M according to
0
(cid:18) (cid:19)
M I
sin ϕ−ϕ y = , (7)
0M I
whereM isthelengthofthemagnetization,v /v (cid:28)1 0 c
0 so F
characterizes the strength of the spin-orbit interaction,
making the effect of the junction on M mathematically
and l = 4LE /((cid:126)v ), with L being the length of the
ex F equivalent to the effect of the external magnetic field
ferromagneticlayerintheX-directionandE beingthe
ex
energy of the exchange interaction between conducting E (cid:18)I (cid:19) Φ I
B =ϕ J yˆ =ϕ 0 yˆ (8)
electrons and localized ferromagnetic spins. Typically I 0M I 02πM
v /v ∼ 0.1, L (cid:46) 10nm, and E ∼ 100−500K, so ϕ 0 c 0
so F ex 0
should be in the range 0.01<ϕ0 <0.111. At I = 0 the instanton solution of Eq. (5) for M =
In this Letter we are making three main points. The M (sinθcosφ,sinθsinφ,cosθ) is given by1,12,13
0
first is that the interaction between the magnetic mo- √
ment and the superconducting order parameter in the sinh(ω τ) λcosφ
sinφ= 0 , cosθ = (9)
ϕ0-junction renormalizes the tunnel splitting in a man- (cid:113)λ+cosh2(ω τ) (cid:112)1+λsin2φ
ner that can be exactly computed and measured. The 0
second point is that the Josephson ϕ -junction permits
0 where ω = [ω (ω + ω )]1/2, λ = ω /ω , ω =
detection of the quantum tunneling and quantum oscil- 0 (cid:107) (cid:107) ⊥ (cid:107) ⊥ (cid:107),⊥
2γK /(M V). The instanton switches the magneti-
lations of the magnetic moment by measuring the volt- (cid:107),⊥ 0
age across the junction. The third point is that the ϕ - zation from M = −M0yˆ at τ = −´∞ to M = M0yˆ at
0 τ = +∞. Path integration of exp(i dtL/(cid:126)) around the
junctionallowsonetocontrolthemagnetictunnelingrate
instanton with the Lagrangian
by the supercurrent through the junction. To illustrate
the first two points we chose a typical form of U for
M M V
a ferromagnetic layer that corresponds to the XY easy L= 0 φ˙(cosθ−1)−U(ϕ,θ,φ) (10)
γ
magnetization plane with the Y easy axis in that plane,
gives for the tunnel splitting
UM = 21K⊥V (cid:18)MMz0(cid:19)2− 12K(cid:107)V (cid:18)MMy0(cid:19)2, (3) ∆0 =Ae−B, A∼(cid:126)ω0, B =2Sln(√λ+√1+λ), (11)
where S = M V/((cid:126)γ) is the total spin of the ferromag-
V being the volume of the ferromagnetic layer. At I =0 0
netic layer. Strong easy plane anisotropy (in which case
classical degenerate equilibrium corresponds to two op- √
λ (cid:28) 1 and B = 2 λS) is required to allow observation
posite orientations M along the Y-axis, with the energy
barrier betwen them given by U = 1K V. The current of tunneling of a macroscopically large S.
0 2 (cid:107) In the low energy domain the problem can be recast
may reduce the barrier to zero and, thus, it can switch
the direction of M10,11. Here we are interested in the as a two-state problem. Projecting Eq. (1) onto the two
magnetic states with M along the Y-axis one obtains a
quantum switching of M at a finite energy barrier.
two-state Hamiltonian
The equations of motion for ϕ and M are
∆
C(cid:18)Φ0(cid:19)2ϕ¨+ 1 (cid:18)Φ0(cid:19)2ϕ˙ =−∂U (4) H = −EJcos(ϕ−ϕ0σy)− 20σx
σ
2π R 2π ∂ϕ
= −E cosϕcosϕ −b · (12)
J 0 eff 2
with
(cid:18) (cid:19)
dM α dM
=γM×B + M× , (5)
dt eff M0 dt beff =∆0xˆ+2EJsinϕsinϕ0yˆ (13)
3
Here σ is a spin-1 operator satisfying At 2E ϕ2 <∆ the system prepared in a state with a
2 J 0 0
definite orientation of M along the Y-axis begins to os-
(cid:126)dσ =i[H,σ]=σ×b (14) cillatewiththeexpectationvaluesofMy andϕsatisfying
dt eff
(cid:18) (cid:19) (cid:18) (cid:19)
∆ ∆
For the components of σ one has My =M0e−Γtcos (cid:126)efft , ϕ=ϕ0e−Γtcos (cid:126)efft
dσ (26)
(cid:126) dtx = −2EJsinϕsinϕ0σz (15) OscillationsdecayatthedampingrateΓ,withthequality
factor given by
dσ
(cid:126) y = ∆ σ (16)
dt 0 z (cid:18)2π(cid:19)2(cid:18)(cid:126)R(cid:19)
dσ Q= (27)
(cid:126) dtz = 2EJsinϕsinϕ0σx−∆0σy (17) Φ0 ϕ20
At ϕ ∼ 0.1 the estimate is Q ∼ 0.1R(Ω), which can
These equations also hold for the expectation values of 0
be quite high for an insulating ferromagnetic layer. Os-
thecomponentsofσ. Itiseasytoseethattheyconserve
cillations of ϕ should generate the oscillating ac voltage
thelengthofσ (|σ|=1). Theymustbeaccompaniedby
across the junction,
the dynamical equation (30) for ϕ that at C =0 reads
(cid:126) dϕ ∆ (cid:18)∆ (cid:19)
1 (cid:18)Φ0(cid:19)2 dϕ =−E (sinϕcosϕ −cosϕsinϕ σ ) (18) V = 2e dt ≈−ϕ0 2eeffe−Γtsin (cid:126)efft (28)
R 2π dt J 0 0 y
Atϕ ∼0.1and∆ ∼0.1Ktheinitial(t=0)amplitude
0 eff
In the practical limit of |ϕ0|(cid:28)1, equations (15)-(18), of the ac voltage would be in the microvolt range, while
linearized near the ground state σx =1, are the frequency, ∆eff/(2π(cid:126)), would be in the GHz range,
which should not be difficult to detect.
(cid:18) (cid:19)
1 Φ dϕ
0 =−I (ϕ−ϕ σ ) (19) To illustrate the possibility to control the magnetic
R 2π dt c 0 y tunneling by the superconducting current consider the
dσ dσ I Φ case of a uniaxial magnetic anisotropy with an easy axis
(cid:126) y =∆ σ , (cid:126) z = c 0ϕ ϕ−∆ σ (20)
dt 0 z dt π 0 0 y along the Z-direction,
Writing ϕ,σy,z ∝ exp(−iωt) and equating the determi- U =−1K V (cid:18)Mz(cid:19)2 (29)
nant of the resulting equations to zero we get M 2 (cid:107) M
0
(cid:20)−iω (cid:18)Φ0(cid:19)+I (cid:21)[((cid:126)ω)2−∆2]+ Ic2Φ0φ2∆ =0 (21) studied in Ref. 11. With account of Eq. (7) one obtains
R 2π c 0 π 0 0 thefollowingdynamicalequationsforthesphericalangles
describing the orientation of M in space
We look for a solution in the form
(cid:18) (cid:19)
dφ I cosθ
ω = ∆eff −iΓ (22) dt(cid:48) = − sinθ− I sinφ sinθ (30)
(cid:126) 0
dθ I
with (cid:126)Γ(cid:28)∆ . Substituting Eq. (22) into Eq. (21) one = − cosφ, (31)
eff dt(cid:48) I
0
obtains
where t(cid:48) = ω t with ω = γK /M being the fre-
(cid:113) FMR FMR (cid:107) 0
∆ = ∆ (∆ −2E ϕ2) (23) quency of the ferriomagnetic resonance at I =0, and
eff 0 0 J 0
K V
for the tunnel splitting and I = (cid:107) I (32)
0 ϕ E c
0 J
(cid:18)ϕ2(cid:19)(cid:18)Φ (cid:19)2 ∆
Γ= 0 0 eff (24) Note that depending on the values of parameters enter-
(cid:126)R 2π (cid:126)
ing Eq. (32) I can be smaller or greater than I . At
0 c
I = 0 the degenerate ground state corresponds to the
forthedecoherencerateprovidedbythefiniteresistance
magneticmomentparallel(θ =0)orantiparallel(θ =π)
of the junction.
to the Z-axis. At I <I (assuming also that I <I ) the
Eq. (23) shows that interaction of the magnetic mo- 0 c
degenerate ground states are
ment with the superconducting order parameter renor-
malizesthetunnelsplitting, ∆ →∆ . Atasufficiently (cid:115)
0 eff π I (cid:18) I (cid:19)2
large interaction, φ= , sinθ = , cosθ =± 1− (33)
2 I I
0 0
2E ϕ2 >∆ , (25)
J 0 0
At I < I < I the non-degenerate ground state is φ =
0 c
the tunneling freezes, that is, ∆ = 0. Smallness of ϕ π/2,θ = π/2, corresponding to the magnetic moment
eff 0
insures practicality of such a regime. looking in the Y-direction.
4
InthecaseofI <I ,whenI isclosebutsmallerthan tunnel spliting ∆=Ae−B with
0 c
I , the energy barrier between the degenerate states in
0 √
Eq. (33) is small, (cid:18) I (cid:19)1/2 4 2 (cid:18) I (cid:19)3/2
A∼(cid:126)ω 1− , B = S 1−
FMR I 3 I
1 I 0 0
U(I)= K V(cid:15)2, (cid:15)=1− (cid:28)1 (34) (38)
2 (cid:107) I
0 Noticethatthetunnelingrateinthiscasedependsexpo-
We are interested in the quantum tunneling between de- nentially on the superconducting current. However, con-
√
generateclassicalgroundstates: φ=π/2,θ =π/2± 2(cid:15). trary to the previously studied case of biaxial magnetic
Substituting θ = π/2 + β and φ = π/2 + α, with anisotropy, quantum oscillations of the magnetic mo-
|α|,|β|(cid:28)1, into Eqs. (30) and (31) one obtains ment b√etween classically degenerate states φ = π/2,θ =
π/2± 2(cid:15) formed by the uniaxial anisotropy along the
dα (cid:18) β2(cid:19) dβ Z-axis and a superconducting current in the X direc-
= (cid:15)− , =α (35)
dt(cid:48) 2 dt(cid:48) tion (see Fig. 1) do not produce oscillations of ϕ. Con-
sequently, they do not generate any voltage across the
Here we have taken into account (see below) that α ∼ junction. One way to detect such a tunneling would be
√
(cid:15) while β ∼ (cid:15), making α (cid:28) β. Combining the two with the help of a SQUID loop sensitive to a small Z-
√
equations, introducing β¯ = β/ 2(cid:15) and the imaginary componentofthemagneticmomentthatchangessignin
time τ¯=it(cid:48)(cid:112)(cid:15)/2 we get the tunneling event. Nano-SQUIDs permit detection of
the change in the magnetic moment of only a few Bohr
dβ¯ magnetons14. In our case the change would be much
=1−β¯2 (36)
greater. In experiments with single-domain magnetic
dτ¯
nanoparticles and molecules, the energy barrier and the
which has the instanton solution β¯=tanhτ¯, tunnelingratewerecontrolledbyastrongmagneticfield,
whichnegativelyaffectedtheperformanceoftheSQUID.
√ (cid:18)(cid:114)(cid:15) (cid:19) The advantage of the ϕ -junction is that the barrier and
β(τ)= 2(cid:15)tanh ω τ (37) 0
2 FMR the tunneling rate can be accurately controlled by the
superconducting current, with no need for the external
√ √
that switches β between − 2(cid:15) at τ = −∞ to 2(cid:15) at magnetic field.
τ =∞. ´ This work has been supported by the grant No. DE-
Path integration of exp(i dtL/(cid:126)) around the instan- FG02-93ER45487fundedbytheU.S.DepartmentofEn-
ton with the Lagrangian given by Eq. (10) yields for the ergy, Office of Science.
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