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T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Quasi-ferromagnet spintronics in the graphenenanodisc–lead system

Motohiko EzawaDepartment of Applied Physics, University of Tokyo, Hongo 7-3-1,113-8656, Japan

New Journal of Physics 11 (2009) 095005 (22pp)Received 5 May 2009Published 30 September 2009Online at http://www.njp.org/doi:10.1088/1367-2630/11/9/095005

Abstract. A zigzag graphene nanodisc can be interpreted as a quantumdot with an internal degree of freedom. It is well described by the infinite-range Heisenberg model. We have investigated its thermodynamical properties.There exists a quasi-phase transition between the quasi-ferromagnet and quasi-paramagnet states, as signaled by a sharp peak in the specific heat and inthe susceptibility. We have also analyzed how thermodynamical properties areaffected when two leads are attached to the nanodisc. It is shown that lead effectsare described by the many-spin Kondo Hamiltonian. There appears a new peak inthe specific heat, and the multiplicity of the ground state becomes just one half ofthe system without leads. Another lead effect is to enhance ferromagnetic order.Being a ferromagnet, a nanodisc can be used as a spin filter. Furthermore, sincethe relaxation time is finite, it is possible to control the spin of the nanodisc byapplying an external spin current. We then propose a rich variety of spintronicdevices made of nanodiscs and leads: spin memory, spin amplifier, spin valve,spin-field-effect transistor, spin diode and spin logic gates such as spin-XNORgate and spin-XOR gate. Graphene nanodiscs could well be basic components offuture nanoelectronic and spintronic devices.

New Journal of Physics 11 (2009) 0950051367-2630/09/095005+22$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2

Contents

1. Introduction 22. Nanodisc quasi-ferromagnets 4

2.1. Zero-energy sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Coulomb interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. SU(N ) approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4. Thermodynamical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3. Many-spin Kondo effects 83.1. The nanodisc–lead system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Many-spin Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3. The Kondo–Heisenberg system . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4. Functional integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5. Zigzag graphene nanoribbon leads . . . . . . . . . . . . . . . . . . . . . . . . 123.6. Metallic leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. Spintronic devices and applications 154.1. Basic components of spintronic devices . . . . . . . . . . . . . . . . . . . . . 164.2. Some spintronic devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3. Spin logic gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5. Conclusions 21Acknowledgments 22References 22

1. Introduction

Graphene nanostructures [1]–[3] have attracted much attention owing to their potential for futureapplications in nanoelectronics and spintronics. In particular, in graphene nanoribbons [4]–[10]the low-energy bands are almost flat at the Fermi level due to the edge states. Such a peculiarband structure has motivated many researchers to investigate their electronic and magneticproperties.

Another basic element of graphene derivatives is a graphene nanodisc [11]–[17]. This is ananometer-scale disc-like material that has a closed edge. There are many types of nanodiscs andtypical examples are shown in figure 1. Among them, the trigonal zigzag nanodisc is prominentin its electronic and magnetic properties because there exist N -fold degenerate half-filled zero-energy states when its size is N . Furthermore, spins have a strong ferromagnetic order due to theexchange interaction being as large as the Coulomb interaction [12], and hence the relaxationtime is finite but quite large even if the size N is very small. We have called such a system aquasi-ferromagnet [12].

In this paper, we make a further study of the quasi-ferromagnet by exploring thethermodynamical properties of the nanodisc–spin system. The system is well described by theinfinite-range Heisenberg model. A nanodisc is interpreted as a quantum dot with an internaldegree of freedom. It is exactly solvable. Constructing the partition function, we calculate thespecific heat, entropy, magnetization and susceptibility. We find a sharp peak in the specific

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

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Figure 1. Basic configurations of typical graphene nanodiscs: (a) benzene;(b) trigonal zigzag nanodisc (phenalene); (c) trigonal armchair nanodisc(triphenylene); (d) hexagonal zigzag nanodisc (coronene); (e) hexagonalarmchair nanodisc (hexabenzocoronene) [18].

heat and in the susceptibility, which is interpreted as a quasi-phase transition between the quasi-ferromagnet and quasi-paramagnet states.

We then investigate a nanodisc–lead system, where the lead is made of a zigzag graphenenanoribbon or an ordinary metallic wire. (We refer to it as a graphene lead or a metalliclead for brevity.) We perform the Schrieffer–Wolff transformation to derive the many-spinKondo Hamiltonian describing the lead effects. Constructing the partition function, we analyzethermodynamical properties. A new peak appears in the specific heat but not in the susceptibilityfor small-size nanodiscs, N . 10. Near the peak, the internal energy is found to decrease.We show in the instance of the metallic lead that the bandwidth of free electrons in the leadbecomes narrower due to Kondo coupling. We interpret this phenomenon as meaning that somefree electrons in the lead are consumed to make spin coupling with electrons in the nanodisc.Furthermore, the multiplicity of the ground state becomes just one half of that in the systemwithout leads. These are indications of Kondo effects due to the Kondo interaction betweenelectrons in the lead and the nanodisc.

With respect to ferromagnetic order, we find that the lead effect is to enhance the order.This is an important property to fabricate spintronic circuits by connecting leads to nanodiscsin nanodevices. Being a ferromagnet, a nanodisc can be used as a spin filter. Furthermore, sincethe relaxation time is finite, it is possible to control the spin of the nanodisc by applying anexternal spin current. We propose a rich variety of spintronic devices made of nanodiscs andleads: spin memory, spin amplifier, spin valve, spin-field-effect transistor, spin diode and spinlogic gates such as spin-XNOR gate and spin-XOR gate. Graphene nanodiscs could well be thebasic components of future nanoelectronic and spintronic devices.

This paper is organized as follows. In section 2, we summarize the basic notion of trigonalzigzag nanodiscs. The low-energy physics is described by electrons in the N -fold degeneratedzero-energy sector, which form a quasi-ferromagnet due to large exchange interactions. We alsoanalyze the thermodynamical properties of the nanodisc–spin system. In section 3, we derivethe many-spin Kondo Hamiltonian by the Schrieffer–Wolff transformation in the nanodisc–spinsystem coupled with graphene leads and also metallic leads. The partition function is calculatedin two steps. Firstly, we perform a functional integration over the lead-electron degree offreedom. Secondly, we sum up over the nanodisc–spin degree of freedom. We then analyzethe thermodynamical properties of the nanodisc–lead system. In section 4, we propose somespintronics devices made of nanodiscs and leads.

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

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2. Nanodisc quasi-ferromagnets

2.1. Zero-energy sector

We calculate the energy spectra of graphene derivatives based on the nearest-neighbor tight-binding model, which has been successfully applied in studies of carbon nanotubes andnanoribbons. The Hamiltonian is defined by

H =

∑i

εi c†i ci +

∑〈i, j〉

ti j c†i c j , (2.1)

where εi is the site energy, ti j is the transfer energy and c†i is the creation operator of the π

electron at site i . The summation is taken over all nearest neighboring sites 〈i, j〉. Owing to theirhomogeneous geometrical configuration, we may take constant values for these energies, εi = εF

and ti j = t ≈ 2.70 eV. Then, the diagonal term in (2.1) yields just a constant, εF NC, where NC

is the number of carbon atoms in the system. The Hamiltonian (2.1) yields Dirac electronsfor graphene [1]–[3]. There exists one electron per carbon, and the band-filling factor is 1/2.It is customary to choose the zero-energy level of the tight-binding Hamiltonian (2.1) at thispoint so that the energy spectrum is symmetric between the positive and negative energy states.Therefore, the system is metallic, provided that there exist zero-energy states in the spectrum.A comment is in order. It is known that carbon atoms at edges are terminated by hydrogenatoms. We carry out the calculation [5, 12] together with this condition.

It is straightforward to derive the energy spectrum Ei together with its degeneracy gi foreach nanodisc by diagonalizing the Hamiltonian (2.1). The density of states is given by

D(ε)=

NC∑i=1

giδ(ε− Ei). (2.2)

We have found that the emergence of zero-energy states is surprisingly rare. Among typicalnanodiscs, only trigonal zigzag nanodiscs have degenerate zero-energy states and show metallicferromagnetism. As an example, we display the density of states (2.2) of trigonal zigzagnanodiscs in figure 2. We have classified them by the size parameter N as defined in figure 2(a),in terms of which the number of carbons is given by NC = N 2 + 6N + 6.

The size-N nanodisc has N -fold degenerated zero-energy states, where the gap energy isas large as a few eV. Hence, it is a good approximation to investigate the electron–electroninteraction physics only in the zero-energy sector, by projecting the system to the subspacemade of those zero-energy states. The zero-energy sector consists of N orthonormal states | fα〉,α = 1, 2, . . . , N , together with the SU(N ) symmetry. We can expand the wave function of thestate | fα〉 as

fα(x)=

∑i

ωαi ϕi(x), (2.3)

where ϕi(x) is the Wannier function localized at site i . The amplitude ωαi can be calculated bydiagonalizing the Hamiltonian (2.1). All values of it are found to be real. It is intriguing that oneof the wave functions is entirely localized on edge sites for the nanodisc with N = odd, as infigure 3, where the solid (open) circles denote that the amplitude ωi is positive (negative). Theamplitude is proportional to the radius of the circle. Such a wave function does not exist for thenanodisc with N = even.

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

5

0 1 3 5 70 2 4 6

–3

–2

–1

1

2

3(b)(a) ε(N)

N

N=0N=1

N=2N=3

N=4

Figure 2. (a) Geometric configuration of trigonal zigzag nanodiscs. It isconvenient to introduce the size parameter N in this way. The 0-trigonal nanodiscconsists of a single benzene, and so on. The number of carbon atoms is relatedas NC = N 2 + 6N + 6. (b) Density of states of the N -trigonal nanodisc for N =

0, 1, 2, . . . , 7. The horizontal axis is the size N and the vertical axis is the energyε(N ) in units of t = 2.7 eV. Dots on colored bars indicate the degeneracy ofenergy levels.

Figure 3. Zero-energy states of the trigonal nanodisc with size N = 5. Thereare five degenerate states. The solid (open) circles denote that the amplitude ωi

is positive (negative). The amplitude is proportional to the radius of the circle.Electrons are entirely localized on edges in one of the states.

2.2. Coulomb interactions

We include the Coulomb interaction between electrons in the zero-energy sector [15]. It isstraightforward to rewrite the Coulomb Hamiltonian as HD = HS + HU with

HS = −2∑α>β

JαβS(α) ·S(β), (2.4a)

HU =

∑α>β

(Uαβ −

1

2Jαβ

)n (α) n (β)+

∑α

Uααn (α) , (2.4b)

where Uαβ and Jαβ are the Coulomb energy and the exchange energy between electrons in thestates | fα〉 and | fβ〉. Here, n(α) is the number operator and S(α) is the spin operator,

n (α)=

∑σ

d†σ (α)dσ (α), (2.5a)

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

6

S(α)=1

2

∑σσ ′

d†σ (α)τσσ ′dσ ′(α), (2.5b)

where dσ (α) is the annihilation operator of the electron with spin σ =↑,↓ in the state | fα〉, andτ is the Pauli matrix.

A remarkable feature is that there exists a large overlap between the wave functions fα(x)and fβ(x), α 6= β, since the state | fα〉 is an ensemble of sites as in (2.3) and identical sitesare included in | fα〉 and | fβ〉. Consequently, the dominant contributions come from the on-siteCoulomb terms not only for the Coulomb energy but also for the exchange energy. Indeed, itfollows that Uαβ = Jαβ in the on-site approximation. We thus obtain

Uαβ ' Jαβ ' U∑

i

(ωαi ωβ

i )2, (2.6)

where

U ≡

∫d3x d3 y ϕ∗

i (x)ϕi(x)V (x−y)ϕ∗

i ( y)ϕi( y), (2.7)

with the Coulomb potential V (x − y). The Coulomb energy U is of the order of 1 eV becausethe lattice spacing of the carbon atoms is ∼1 Å in graphene. Coulomb blockade peaks appear atµ= εα and µ= εα + Uαβ , where new channels open [15]. We can determine experimentally theenergy εα and Uαβ by identifying Coulomb blockade peaks.

2.3. SU(N) approximation

Since the exchange energy Jαβ is as large as the Coulomb energy Uαβ , the spin stiffness Jαβ isquite large. Furthermore, we have checked [15] numerically that all Jαβ are of the same orderof magnitude for any pair of α and β, implying that the SU(N ) symmetry is broken but not sostrongly in the Hamiltonian (2.4a). It is a good approximation to start with the exact SU(N )symmetry. Then, the zero-energy sector is described by the Hamiltonian HD = HS + HU, with

HS = −J∑α 6=β

S(α) ·S(β), (2.8a)

HU =

(U

2−

J

4

)∑α 6=β

n(α)n(β)+ U∑α

n(α), (2.8b)

where J ≈ U . The term HS is known as the infinite-range Heisenberg model. We rewritethem as

HS = −JS2tot + 3

4 Jntot, (2.9a)

HU =

(U

2−

J

4

)(n2

tot + 1), (2.9b)

where Stot =∑

α S(α) is the total spin, and ntot =∑

α n(α) is the total electron number.The ground states of nanodiscs are half filled. We restrict the Hilbert space to the half-filling

sector,

n(α)= n↑(α)+ n↓(α)= 1. (2.10)

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

7

The Hamiltonian (2.9a) is reduced to the Heisenberg model,

HS = −JStot ·Stot, (2.11)

where we have neglected an irrelevant constant term, (3/4)J N . This is exactly diagonalizable,HS|9〉 = Es|9〉, with

Es = −Js(s + 1), (2.12)

where s takes values from N/2 down to 1/2 or 0, depending on whether N is odd or even,

s =N

2,

N

2− 1,

N

2− 2, . . . , s > 0. (2.13)

The Hilbert space is diagonalized,

H= ⊗NH1/2 = ⊕

gN (s)Hs, (2.14)

where Hs denotes the (2s + 1)-dimensional Hilbert space associated with an irreduciblerepresentation of SU(2). The multiplicities gN (s) satisfy the recursion relation coming fromthe spin synthesizing rule,

gN (s)= gN−1

(s −

12

)+ gN−1

(s + 1

2

). (2.15)

We solve this as

gN

(N

2− q

)= N Cq −N Cq−1. (2.16)

The total degeneracy of the energy level Es is (2s + 1)gN (s).At half filling, the eigenstate of the Hamiltonian HD is labeled as |9〉 = |ntot, s,m〉, where

s is the total spin and m is its z-component. We refer to the total spin Stot of a nanodisc as thenanodisc spin.

2.4. Thermodynamical properties

We have a complete set of eigenenergies together with their degeneracies. The partition functionof the nanodisc with size N is exactly calculable,

ZS =

∑s

(2s + 1)gN (s)e−βEs =

N/2∑q=0

(N − 2q + 1)(N Cq −N Cq−1)

× exp

[β J

(N

2− q

)(N

2− q + 1

)]. (2.17)

According to the standard procedure we can evaluate specific heat C(T ), entropy S(T ),magnetization 〈S2

tot〉 and susceptibility χ =1

kBT

(⟨S2

tot

⟩− 〈Stot〉

2)

from this partition function,where

Sg =

√N

2

(N

2+ 1

)(2.18)

is the ground-state value of the total spin. The entropy is given by

S (0)= kB log(N + 1) (2.19)

at zero temperature. We display them in figure 4 for size N = 1, 2, 22, . . . , 210.

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

8

(a) (c)

(d)(b)

14

1664

2561024

Spe

cific

heat

Mag

netiz

atio

nS

usce

ptib

ility

Ent

ropy

1.0

0.8

0.6

0.4

0.2

2

4

6

81.0

0.8

0.6

0.4

0.2

1.01.2

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0. .0

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.

8 1

8 1.0Tc

Figure 4. Thermodynamical properties of the nanodisc–spin system. (a) Specificheat C in units of kB N . (b) Entropy S in units of kB N log2. (c) Magnetization〈S2

tot〉 in units of S2g . (d) Susceptibility χ in units of Sg. The size is N =

1, 2, 22, . . . , 210. The horizontal axis stands for the temperature T in unitsof J N/kB. The arrow represents the phase transition point Tc in the limitN → ∞.

There appear singularities in thermodynamical quantities as N → ∞, which represent aphase transition at Tc between the ferromagnet and paramagnet states,

Tc =J N

2kB. (2.20)

For finite N , there are steep changes around Tc, although they are not singularities. It is not aphase transition. However, it would be reasonable to call it a quasi-phase transition between thequasi-ferromagnet and quasi-paramagnet states. Such a quasi-phase transition is manifest evenin finite systems with N = 100–1000.

The specific heat and magnetization take nonzero-values for T > Tc (figures 4(a) and (c)),which is zero in the limit N → ∞. The entropy for T > Tc is lower than that of the paramagnet(figure 4(b)). These results indicate the existence of some correlations in the quasi-paramagnetstate. The maximum value of the susceptibility increases linearly as N becomes large. This isan indicator of the quasi-phase transition.

3. Many-spin Kondo effects

We proceed to investigate how thermodynamical properties of the nanodisc are affected by theattachment of the leads. The nanodisc is no longer in the half-filled state when charges transferbetween the nanodisc and the leads. However, the nanodisc remains half filled when a chargetransfers from the lead to the nanodisc and then transfers back from the nanodisc to the lead.Such a process is the second-order effect in the tunneling coupling constant t .We now show thatit is described by the many-spin Kondo Hamiltonian.

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

9

Nanodisc–lead system

L R

Figure 5. The nanodisc–lead system. The nanodisc with N = 7 is connected tothe right and left leads by tunneling coupling tR and tL.

3.1. The nanodisc–lead system

We analyze a system made of a nanodisc connected to two leads (figure 5). The modelHamiltonian is given by

H = HS + HU + HL + HTL + HTR. (3.1)

Here, HL is the lead Hamiltonian,

HL =

∑kσ

ε (k) (cL†kσcL

kσ + cR†kσ cR

kσ ), (3.2)

describing a non-interacting electron gas in the leads with the dispersion relation ε(k), wherecχkσ is the annihilation operator of the electron with the wave number k and the spin σ in the left(χ = L) or right (χ = R) lead. On the other hand, HTL and HTR are the transfer Hamiltoniansbetween the left (L) and right (R) leads and the nanodisc, respectively,

HTL = tL

∑kσ

∑α

[cL†kσdσ (α)+ d†

σ (α)cLkσ ], (3.3a)

HTR = tR

∑kσ

∑α

[cR†kσdσ (α)+ d†

σ (α)cRkσ ], (3.3b)

with tχ being the tunneling coupling constant. We have assumed that the spin does not flip inthe tunneling process.

The nanodisc–lead system appears to be similar to the N -dot system [19]. However, thereexists a crucial difference. On the one hand, in the ordinary N -dot system, an electron hopsfrom one dot to another dot. On the other hand, in our nanodisc system, the index α of theHamiltonian runs over the N -fold degenerate states and not over the sites. According to theHamiltonian (3.3), an electron does not hop from one state to another state. Hence, it is moreappropriate to regard our nanodisc as a one-dot system with an internal degree of freedom.

It is convenient to make the transformation(ce

kσ

cokσ

)=

1

t

(t∗

L t∗

R

−tR tL

)(cL

kσ

cRkσ

)(3.4)

with

t =

√|tL|

2 + |tR|2, (3.5)

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

10

so that the right and left leads are combined into the ‘even’ and ‘odd’ leads. The leadHamiltonian HL is invariant under the above transformation,

HL =

∑kσ

ε(k)(

ce†kσce

kσ + co†kσco

kσ

), (3.6)

but the transfer Hamiltonian is considerably simplified,

HT = t∑kσ

∑α

(ce†

kσdσ (α)+ d†σ (α)c

ekσ

). (3.7)

It looks as if the tunneling occurs only between the ‘even’ lead and the nanodisc. Thus, we mayneglect the ‘odd’ lead term in the lead Hamiltonian (3.6).

3.2. Many-spin Kondo Hamiltonian

The total Hamiltonian is H = HS + HU + HL + HT. We analyze the Hamiltonian H = H0 + HT

by taking H0 = HU + HL as the unperturbed term and HT as the perturbation term. We makea canonical transformation known as the Schrieffer–Wolff transformation [20], H → H =

eiG H e−iG , with G the generator satisfying

HT +i

2[G, H0] = 0. (3.8)

We may solve this condition explicitly for the generator,

G =1

Ni

[∑αkσ

{t

εd − ε(k)+ U ′αα

d†ασce†

kσnασ +t

ε(k ′)− εdd†ασce†

kσ (1 − nασ )

}

+∑αkσ

∑βσ ′

{t

εd − ε (k)+ U ′

αβ

d†ασce†

kσnβσ ′ +t

ε(k ′)− εdd†ασce†

kσ

(1 − nβσ ′

)}− h.c.

, (3.9)

where σ =↓↑ for σ =↑↓, and

U ′

αβ =

(Uαβ −

Jαβ2

), εd = Uαα. (3.10)

The leading term is the second-order term, and given by

H (2)eff =

i

2[G, HT] =

−t2

2N

∑αβkk′σ

(1

εd − ε(k)+ U ′

αβ

+1

ε(k ′)− εd

)ce†

kσcek′σ

+2t2

N

∑αβkk′σσ ′

(1

εd − ε(k)+ U ′

αβ

−1

ε(k ′)− εd

)ce†

kστσσ ′cek′σ ′ ·S(α). (3.11)

The dominant contribution comes from the Fermi surface, ε(k)= εF. We now assume the SU(N )symmetry U ′

αβ = U ′ and the symmetric condition εF = εd + U ′

2 with respect to the Fermi level.Then, the second-order term becomes the many-spin Kondo Hamiltonian,

HK ≡ H (2)eff = JK

∑kk′σσ ′

ce†kστσσ ′ce

k′σ ′ ·Stot, (3.12)

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

11

with the Kondo coupling constant

JK = 2t2

(1

εd − εF + U ′−

1

εF − εd

)=

8t2

U ′. (3.13)

The difference between the above many-spin Kondo Hamiltonian and the ordinary KondoHamiltonian is whether the local spin is given by the summation over many spins Stot or asingle spin S. Note that S2

tot is a dynamical variable but S2 is not, S2= 3/4.

3.3. The Kondo–Heisenberg system

We have derived the Kondo Hamiltonian from the Coulomb and transfer terms by way ofthe Schrieffer–Wolff transformation. The resultant system is the Kondo–Heisenberg model,which comprises the Heisenberg Hamiltonian, the lead-electron Hamiltonian and the KondoHamiltonian,

Heff = HS + HL + HK, (3.14)

where we have ignored HU since it is just a constant at half filling. We note that the order ofthe Heisenberg Hamiltonian (∼U ) is much larger than the order of the Kondo Hamiltonian(∼4t2/U ) because U � t .

Our goal is to analyze the partition function of the coupled system (3.14). We define thespinor ψ = (ce

↑, ce

↓)t . The partition function in Matsubara form is given by

Z = TrS

∫DψDψ† exp

[−

∫ β

0dτ∫

dx(ψ†∂τψ +Heff

)], (3.15)

with Heff being the Hamiltonian density, where the lead electron’s degree of freedom isintegrated out by a functional integral, and the nanodisc spin is summed up. Since theHeisenberg term does not contain lead electrons, we can separate it as

Z = TrS

[exp (−βHS) ZK

], (3.16)

where ZK =∫DψDψ† exp[−SK], with

SK =

∫ β

0dτ∫

dx(ψ†∂τψ +HL +HK

). (3.17)

First, we evaluate the functional integration ZK in section 3.4. We obtain the effective spinHamiltonian, where the only active degree of freedom is the nanodisc spin. Then, we sum upover the nanodisc spin to obtain the total partition function Z . We do this for the graphene leadand for the metallic lead in sections 3.5 and 3.6, respectively.

3.4. Functional integration

Because an electron in the lead is constrained within a very narrow region, it is a goodapproximation to neglect momentum scatterings,

HK ' JK

∑kσσ ′

ce†kστσσ ′ce

kσ ′ ·Stot. (3.18)

The action (3.17) is summarized as

SK =

∫dω

2π

∑k

ψ†(k)M(k)ψ(k), (3.19)

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

12

with

M(k)= −[iω− ε(k)] + JKτ ·Stot. (3.20)

Performing the integration, we find that

ZK = Det[M] = exp[−βFK], (3.21)

where

FK = −1

2β

∑k

ln [coshβ JK |Stot| + coshβε(k)] (3.22)

is the Helmholtz free energy FK. This formula is reduced to the well-known one for freeelectrons with the dispersion relation ε(k) for JK = 0. We evaluate the momentum integral forthe graphene lead and for the metallic lead, separately, in sections 3.5 and 3.6.

3.5. Zigzag graphene nanoribbon leads

We consider the system where the leads are made of zigzag graphene nanoribbons. Owing tothe flat band at the zero energy, ε(k)= 0, the result of the functional integration (3.22) is quitesimple,

FK = −1

βln cosh

β

2JK|Stot|. (3.23)

The effective Hamiltonian for the nanodisc spin is HS + FK. The lead effect is to make theeffective spin stiffness larger and the ferromagnet more rigid.

The partition function (3.16) is

Z = TrS

[exp

[β JS2

tot

]cosh

β

2JK |Stot|

], (3.24)

which implies that the eigenstates of the total Hamiltonian are given by JS2tot ±

12 JK |Stot|.

Namely, the ground states are split into two by the existence of the lead-electron spin.Accordingly, the ground-state multiplicity is reduced to (N + 1)/2, which is just one half ofthat of the nanodisc without leads.

The trace over the total spin is carried out in (3.24),

Z =

N/2∑q=0

(N − 2q + 1) (N Cq −N Cq−1) exp

[β J

(N

2− q

)(N

2− q + 1

)]

× cosh

[β JK

2

√(N

2− q

)(N

2− q + 1

)]. (3.25)

In figure 6, we show specific heat CG(T ), entropy SG(T ), magnetization 〈S2tot〉 and susceptibility

χ for various sizes N connected with graphene leads.We compare the thermodynamical properties of nanodiscs without leads (figure 4) and

with leads (figure 6). A significant feature is the appearance of a new peak in the specific heatat TK = (JK/2J )Tc, although it disappears for large N . We examine the internal energy EG(T ),which is found to decrease around TK (figure 7). Near zero temperature, it reads as

EG(T )' −J S2g −

JK

2Sg + JKSg e−β JK Sg . (3.26)

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13

(c)

Mag

netiz

atio

n

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.00.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

(d)

Sus

cept

ibili

ty

2

4

6

8

(a)

(b)

Spe

cific

heat

Ent

ropy

1.0

0.8

0.6

0.4

0.2

1.01.2

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0

With graphene leads

TcTK

Figure 6. Thermodynamical properties of the nanodisc–spin system withgraphene leads. (a) Specific heat C in units of kB N . (b) Entropy S in units of(kB N log 2). (c) Magnetization 〈S2

tot〉 in units of S2g . (d) The susceptibility χ

in units of Sg. The size is N = 1, 2, 22, . . . , 210. We have set JK/J = 0.2. Thehorizontal axis stands for the temperature T in units of J N/kB. The arrowsrepresent the points corresponding to Tc and TK.

N = 8

N = 4

N = 2

N = 1 N = 1

N = 2

N = 4

N = 8

0.2

–0.2

–0.4

–0.6

–0.8

–0.2

–0.4

–0.6

–0.8

0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

(a) Without leads (b) With graphene leadsTcTK

Figure 7. Internal energy E/NkB for the system (a) without leads and (b) withgraphene leads, for size N = 1, 2, 4, . . . , 1024. The horizontal axis stands fortemperature. (a) The energy decreases except for N = 1 as the temperaturedecreases, which represents ferromagnetic order. (b) There exists an additionalenergy decrease around TK, which is prominent for N = 1, attributed to theKondo effect. The arrows represent the points corresponding to Tc and TK.

The first term represents energy stabilization due to the ferromagnetic order present in thenanodisc system without leads, while the second term represents that due to the Kondo couplingJK between spins in the nanodisc and in the leads. Furthermore, it follows that the entropy isreduced at zero temperature as

SG(0)− S(0)= −kB log 2 (3.27)

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14

(c)

Mag

netiz

atio

n

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.00.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

(d)

Sus

cept

ibili

ty

2

4

6

8

(a)

(b)

Spe

cific

heat

Ent

ropy

1.0

0.8

0.6

0.4

0.2

1.01.2

0.8

0.60.4

0.2

0.2 0.4 0.6 0.8 1.0

With metallic leads

TK

Tc

Figure 8. Thermodynamical properties of the nanodisc–spin system withmetallic leads. (a) Specific heat C in units of kB N . (b) Entropy S in units ofkB N log 2. (c) Magnetization 〈S2

tot〉 in units of S2g . (d) The susceptibility χ in units

of Sg. The size is N = 1, 2, 22, . . . , 210. We have set JK/J = 0.2 and D = 2kBTc.The horizontal axis stands for the temperature T in units of J N/kB. The arrowsrepresent the points corresponding to Tc and TK.

with (2.19), and this implies that the ground-state multiplicity at zero temperature is just onehalf of that of the system without leads, in accordance with the observation made below (3.24).These features indicate the occurrence of the Kondo effect due to the coupling between spins inthe nanodisc and in the leads.

3.6. Metallic leads

Next, we consider a system comprising metallic leads with a constant energy density,

ρ (ε)=

{ρ, |ε|< D,0, |ε|> D.

(3.28)

We change the momentum integration into the energy integration in (3.22),

FK = −ρ

2β

∫ D

−Ddε ln [coshβ JK |Stot| + coshβε] . (3.29)

The integration is carried out as

FK = −ρ

β

[βD JK|Stot| − 2D log 2 +

1

β

{Li2

(−eβ(D−JK|Stot|)

)− Li2

(−e−β(D+JK|Stot|)

)}], (3.30)

where Li2[x] is the dilogarithm function [21],

Li2 [x] =

∞∑n=1

xn

n2. (3.31)

It can be easily seen that the free energy is reduced to that of the nanodisc–spin system withgraphene leads in the limit D → 0 with ρ = 1/2D.

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

15

In figure 8, we show specific heat CM(T ), entropy SM(T ), magnetization 〈S2tot〉 and

susceptibility χ for various sizes N connected with metallic leads.The behaviors of magnetization and susceptibility are the same as those in the case of the

nanodisc–spin system with graphene leads. On the other hand, overall behaviors are the samewith respect to specific heat and entropy. In particular, the same relation as (3.27) holds forentropy,

SM(0)− S(0)= −kB log 2. (3.32)

There are some new features in the low-temperature regime. Using the asymptoticbehaviors [21]

limx→∞

Li2[−x] = −π 2

6−

1

2log2 1

x, lim

x→0Li2[−x] = 0, (3.33)

we obtain the free energy, entropy, specific heat and internal energy up to terms of the order ofe−βD as follows:

FK ' −ρ

β

[βD JK|Stot| − 2D log 2 +

π2

6β+β

2(D − JK|Stot|)

2

], (3.34)

SM(T )' kB logN + 1

2+π 2

3ρk2

BT, (3.35)

CM(T )'π2

3ρk2

BT (3.36)

and

EM(T )= EG(T )+1E(T ). (3.37)

The specific heat CM(T ) is identical to the specific heat of free electrons in the metallic lead.The internal energy EM(T ) consists of two terms: EG(T ) is identical to the energy (3.26) forthe nanodisc with graphene leads, and

1E(T )' −ρ

2

(D − JKSg

)2+π2

6ρ (kBT )2 (3.38)

is the energy for the metallic lead. Here, the first term shows that the bandwidth of free electronsin the lead (3.28) becomes narrower due to Kondo coupling. We may interpret that n freeelectrons in the lead with

n = ρ JKSg (3.39)

are consumed to produce spin coupling with electrons in the nanodisc. The second term is thethermal energy of free electrons in the metallic lead.

4. Spintronic devices and applications

We propose some applications of graphene nanodisc–lead systems in spintronic devices [22].The nanodisc–spin system is a quasi-ferromagnet, which is an interpolating system between asingle spin and a ferromagnet. It is easy to control a single spin by a tiny current but it doesnot hold the spin direction for a long time. On the other hand, a ferromagnet is very stable,but it is hard to control the spin direction by a tiny current. A nanodisc quasi-ferromagnet has

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

16

an intermediate nature. It can be controlled by a relatively tiny current and yet holds the spindirection for quite a long time. Indeed, its lifetime τferro is given by[12]

τferro ∝ exp

[J N 2

2kT

], (4.1)

which is quite long compared to the size. The important point is that its size is of the order ofnanometers, and it is suitable as a nanodevice.

The coupling of the nanodisc spin and the injected electron spin 12ψ

†τψ is described bythe Landau–Lifschitz–Gelbert equation [22],

∂n

∂t= γBeff ×n−αn×

∂n

∂t, (4.2)

where n = Stot/|Stot| is the normalized nanodisc spin, γ is the gyromagnetic ratio, α is theGilbert damping constant (α ≈ 0.01) and Beff is the effective magnetic field produced by theinjected electron spin,

Beff = −U

hγ |Stot|

⟨ψ†τψ

⟩. (4.3)

It is proportional to the injected current I in. A spin-polarized current rotates the nanodisc spinto the same direction as the current with the relaxation time

τfilter =1 +α2

2αγ |Beff|∝ N . (4.4)

We use these properties to design spintronic devices.

4.1. Basic components of spintronic devices

4.1.1. Spin filter. We consider a lead–nanodisc–lead system (figure 5), where an electronproduces a tunneling from the left lead to the nanodisc and then to the right lead. Leadelectrons with the same spin direction as the nanodisc spin can pass through the nanodiscfreely. However, those with opposite directions feel a large Coulomb barrier and are blocked(Pauli blockade) [22]. As a result, when we apply a spin-unpolarized current to the nanodisc,the outgoing current is spin polarized to the direction of the nanodisc spin. Consequently, thissystem acts as a spin filter.

4.1.2. Spin memory. A nanodisc can be used as a spin memory, where the spin direction is theinformation. We can read out the information by applying a spin-unpolarized current becausethe outgoing current from a nanodisc is spin polarized to the direction of the nanodisc spin.Furthermore, the direction of the nanodisc spin itself can be controlled by applying a spin-polarized current into the nanodisc.

4.1.3. Spin amplifier. A nanodisc can be used as a spin amplifier. We take the incoming currentto be partially polarized, whose average direction is assumed to be upward, I in

↑> I in

↓> 0. On the

other hand, the direction of the nanodisc spin is arbitrary. Since spins in the nanodisc experiencean effective magnetic field proportional to I in

↑− I in

↓, they are forced to align with that of the

partially polarized spin current after making damped precession. After a sufficient period oftime (τ � τfilter), all spins in the nanodisc take the upward direction and, hence, the outgoing

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

17

(a)

(b)

Spin valve

RL

RL

Figure 9. Illustration of a spin valve. (a) The spin valve is made of two nanodiscsof the same size, which are connected to leads. (b) Applying an external magneticfield, we control the spin direction of the first nanodisc to be |θ〉 and that of thesecond nanodisc to be |0〉 = |↑〉. The incoming current is unpolarized, but theoutgoing current is polarized, I out

↑= I cos θ

2 , I out↓

= 0.

(a)

(b)

Spin-field-effect transistor

RL

RL

Spin rotator

Figure 10. Illustration of a spin-field-effect transistor. (a) It is made up of twonanodiscs of the same size, which are connected to a rotator. (b) We set thespin direction of the two nanodiscs to be upward by a magnetic field. Theincoming current is unpolarized, but the outgoing current is polarized and givenby I out

↑= I cos θ

2 , I out↓

= 0. The up-spin current is rotated by the angle θ withinthe central lead acting as a rotator.

current is the perfectly up-polarized one, I out↑

= I in↑

, I out↓

= 0. Consequently, the small differenceI in↑

− I in↓

is amplified to the large current I in↑

. The amplification ratio is given by I in↑/(I in

↑− I in

↓),

which can be very large. This effect is very important because the signal of the spin will easilysuffer from damping by the disturbing noise in leads. By amplifying the signal we can makecircuits that are strong against noises.

4.1.4. Spin rotator. We can arrange a lead so that it has Rashba-type interaction [23],

HR =λ

h

(pxτ

y− pyτ

x). (4.5)

Spins precess while they pass through the lead. The spin-rotation angle is given ([24] andreferences therein) by θ = 2λm∗L/h, where m∗ is the electron effective mass in the lead andL is the length of the lead. We can control θ by changing the coupling strength λ externally by

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18

(a)

(b)

Spin diode

RL

RL

Spin rotator

(c)

RL

Figure 11. Illustration of a spin diode. (a) It is made up of two nanodiscs ofdifferent sizes, which are connected to a rotator. (b) We initially set the spindirection of the two nanodiscs to be upward by a weak magnetic field. Bycontrolling the bias voltage, the current flows from the left lead to the rightlead, or in the opposite direction. (c) We inject an unpolarized current from theleft (right), which is made up-polarized by the left (right) nanodisc, and thenrotated by the central lead. The spin of the right (left) nanodisc is rotated afterthe relaxation time τR

filter (τLfilter). When the incoming current is an unpolarized

pulse, the charge transported by the current is different whether it is injectedfrom the left or right. This acts as a spin diode.

applying an electric field [25]. In this way, we can rotate the direction of the spin current by anydegree θ . We call such a lead a spin rotator.

4.2. Some spintronic devices

4.2.1. Spin valve. A nanodisc can be used as a spin valve, inducing the giant magneto-resistance effect [26]–[28]. We set up a system composed of two nanodiscs sequentiallyconnected with leads (figure 9). We apply external magnetic field, and control the spin directionof the first nanodisc to be |θ〉 = cos θ

2 |↑〉 + sin θ

2 |↓〉 and that of the second nanodisc to be|0〉 = |↑〉. We inject an unpolarized spin current into the first nanodisc. The spin of the leadbetween the two nanodiscs is polarized into the direction of |θ〉. Subsequently, the currentis filtered to the up-spin one by the second nanodisc. The outgoing current from the secondnanodisc is I out

↑= I cos θ

2 . We can control the magnitude of the up-polarized current from 0 toI by rotating the external magnetic field. The system acts as a spin valve.

4.2.2. Spin-field-effect transistor. We again set up a system composed of two nanodiscssequentially connected with leads (figure 10). We now apply the same external magnetic fieldto both these nanodiscs, and fix their spin direction to be upward, |0〉 = |↑〉. As an additionalsetting, we use a lead acting as a spin rotator with the spin-rotation angle θ . The outgoingcurrent from the second nanodisc is I out

↑= I cos θ

2 . It is possible to tune the angle θ by applyingan electric field. Hence, we can control the magnitude of the up-polarized current. The systemacts as a spin-field-effect transistor [29].

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19

(a) Spin-XNOR gate

(b) Parallel external fields (spin current flows)

(c) Anti-parallel external fields (spin current is blocked)

RL

RL

RL

UL

DL

UR

DR

Figure 12. (a) Illustration of a spin-XNOR gate with unpolarized current comingfrom the left. The unpolarized current is filtered by the left nanodisc, and onlyup-spin electrons go through the central lead. (b) When we apply vertical spincurrents with parallel spin direction, the outgoing current exists. (c) When weapply vertical spin currents with antiparallel spin direction, the outgoing currentdoes not exist.

4.2.3. Spin diode. We set up a system composed of two nanodiscs sequentially connected withleads, where the two nanodiscs have different sizes (figure 11). The left nanodisc is assumed tobe larger than the right nanodisc. Then the relaxation time of the left nanodisc τL

filter is largerthan that of the right nanodisc τR

filter, i.e. τLfilter > τ

Rfilter. Secondly, we apply the same magnetic

field to the two nanodiscs, but we make it so small that the nanodisc spin can be controlled by apolarized current. For clarity we take the direction of the magnetic field to be upward. Thirdly,the central lead is taken to be a spin rotator with θ ≈ π . When no currents enter the nanodisc,the directions of the two nanodisc spins are identical due to the tiny external magnetic field,which is upward. This is the ‘off’ state of the spin diode. When we inject an unpolarized currentinto this system, the outgoing current is initially very small, I out

= I cos θ ' 0 for 1θ ≈ π .However, after the relaxation time τfilter, the outgoing current becomes large since the polarized

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20

(a) spin-NOR gate

(b) parallel external fields (spin current is blocked)

(c) anti-parallel external fields (spin current flows)

RL

RL

RL

UL

DL

UR

DR

Figure 13. (a) Illustration of a spin-XOR gate with unpolarized current comingfrom the left. The unpolarized current is filtered by the left nanodisc, and onlyup-spin electrons go through the central lead. The electron spin in the centrallead rotates θ = π by Rashba-type interaction. (b) When we apply verticalspin currents with parallel spin direction, the outgoing current does not exist.(c) When we apply vertical spin currents with antiparallel spin direction, theoutgoing current exists.

current rotates the spin of the second nanodisc by the angle θ . It takes a longer time when thenanodisc is larger. Now, let us inject an unpolarized pulse current either from the left or from theright. It follows that QL→R > QR→L since τL

filter > τRfilter, where QL→R and QR→L are the charges

transported by the current injected to the right and left leads, respectively. Hence, the systemacts as a spin diode.

4.3. Spin logic gates

We can construct spin logic gates in which the spin direction takes logic values: truth (false)identified with up (down) spin, by controlling a spin current by another spin current accordingto the following setups.

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21

4.3.1. Spin inverter (spin NOT gate). We take a spin rotator with the rotation angle θ = π .This rotator is used as a spin NOT gate, by regarding up spin as ‘true’ and down spin as ‘false’,because it interchanges up spin with down spin.

4.3.2. Spin XNOR. We may construct a spin-XNOR gate. We set up a system composed oftwo nanodiscs connected with seven leads, three horizontal leads and four vertical leads, asillustrated in figure 12. We control the nanodisc spins by vertically applied spin currents insteadof an external magnetic field. We inject an unpolarized current from the left lead to the rightlead. When the spin directions of the two vertical spin currents are parallel, the directions ofthe two nanodisc spins become parallel, and the horizontal current can pass through. However,when the spin directions of the two vertical spin currents are antiparallel, the horizontal currentcannot pass through. This system acts as a spin-XNOR gate by regarding up spin as ‘true’ anddown spin as ‘false’.

4.3.3. Spin XOR. We may construct a spin-XOR gate by changing the central lead to a spinrotator with the rotation angle 1θ = π . The horizontal current passes through if the spindirections of the two vertical spin currents are antiparallel, and cannot pass through if theyare parallel, as illustrated in figure 13.

5. Conclusions

The trigonal zigzag nanodisc has a remarkable property that it has N -fold degenerate zero-energy states with SU(N ) symmetry when its size is N . The SU(N ) symmetry is broken butnot so strongly by Coulomb interactions. The system is well approximated by the infinite-rangeHeisenberg model, where the site index runs over these N -fold degenerate states. We may regardit as a quasi-ferromagnet characterized by exchange energy as large as the Coulomb energy. Therelaxation time is finite but quite large even if the size is very small.

In this paper, we have investigated the thermodynamical properties of a nanodisc. We havefound the emergence of a quasi-phase transition between the quasi-ferromagnet and the quasi-paramagnet as a function of temperature even for samples with N ≈ 100. The transition pointTc is signaled by a sharp peak in the specific heat and in the susceptibility.

We have also examined how they are modified when the external leads are attached to thenanodisc. The lead effects are summarized by the many-spin Kondo Hamiltonian. One effectis to enhance the ferromagnetic order. This result is important to make spintronic circuits byconnecting leads in nanodevices. It is also significant that a new peak appears in the specificheat but not in the susceptibility for small N . The peak position is TK = (JK/2J )Tc for thezigzag graphene nanoribbon lead. The energy is found to decrease around the peak position,and the entropy is lowered by a factor k B log 2 in the zero-temperature limit, indicating theKondo effect due to a Kondo interaction between electrons in the lead and the nanodisc.

We have proposed some applications of nanodiscs in nanodevices. Being a ferromagnet, itcan be used as a spin filter. Namely, only electrons with spin parallel to the spin of the nanodisccan pass through it. Additionally, it has a novel feature in that it is not a rigid ferromagnet. Theincoming spin-polarized current can rotate the nanodisc spin itself. Combining the advantagesof both these properties, we have proposed a rich variety of spintronic devices, such as spinmemory, spin amplifier, spin diode, spin valve and spin-field-effect transistor. Furthermore,

New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)

22

we have proposed some spin logic gates such as spin XNOR gate and XOR gate. Graphenenanodiscs could well be the basic components of future nanoelectronic and spintronic devices.

Acknowledgments

I am grateful to N Nagaosa for many fruitful discussions on the subject. This work was partiallysupported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science,Sports and Culture of Japan (no. 20840011).

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New Journal of Physics 11 (2009) 095005 (http://www.njp.org/)