Quantum magnetocapacitance in ferromagnetic nanocontacts

Physics Letters A 342 (2005) 478–483 www.elsevier.com/locate/pla

Quantum magnetocapacitance in ferromagnetic nanocontacts Yu-Xian Li a,c,∗ , Yong Guo b , Bo-Zang Li c a College of Physics, Hebei Normal University, Shijiazhuang 050016, People’s Republic of China b Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China c Institute of Physics and Center for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China

Received 26 April 2005; accepted 31 May 2005 Available online 14 June 2005 Communicated by J. Flouquet

Abstract Using the scattering matrix theory, magnetocapacitance properties are investigated in magnetic nanocontacts. In the ferromagnetic alignment, the effective potential of the exchange splitting energy acts as a quantum well for spin-up electrons and as a quantum barrier for spin-down ones, which results in spin-dependent capacitance. When the system switches from the ferromagnetic alignment to the antiferromagnetic alignment induced by an external magnetic field, the capacitance changes correspondingly. This magnetocapacitance effect can be expected to be useful in designing a magnetocapacitor.  2005 Elsevier B.V. All rights reserved. PACS: 75.75.+a; 72.25.-b; 73.23.Ad; 73.63.Rt Keywords: Magnetocapcitance; Nanocontacts; Scattering matrix

1. Introduction Spin-dependent electronic transport in mesoscopic systems has received extensive attention [1–4]. When the constriction width of the mesoscopic system is comparable to the electron Fermi wavelength, the transverse energy is discrete, each transverse energy subband defines a conduction channel, and the conductance through the narrow constriction is quan* Corresponding author.

E-mail address: [email protected] (Y.-X. Li). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.091

tized [5–7]. Capacitance, another important property, has also been studied [8–13]. In a quantum point contact, Christen and Büttiker [8,9] indicated that capacitance decreases in a steplike manner when the channels open. Yu, Yeung, and Shangguan [10] calculated the capacitance and emittance in a quantum wire structure with reservoirs. They found there are resonant peaks in the capacitance spectrum due to the opening of an additional channel, and the oscillations are related to the longitudinal state of the wire. Because the scale of mesoscopic devices discussed above is nanosize, it is difficult to modify the structure

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once the devices are made, which naturally is a shortcoming for some applications. The capacitance of the capacitor is difficult to be tuned, thus the corresponding current in electrocircuit is difficult to be changed. Kinaret et al. reported that bending the nanotube to modify the capacitance [14]. Here, we present another way of tuning the capacitance of nanostructure. The capacitance of the system could be modified when the magnetic configuration of the contacts is changed from ferromagnetic to antiferromagnetic alignments induced by the external magnetic field.

2. Model and formulae Consider wide-narrow-wide ferromagnetic nanocontacts as shown in Fig. 1, where the magnetizations in the left (x
−d and |y|
W/2) and right (x  d and |y|
W/2) electrodes are fixed and parallel, while the direction of the magnetization in the middle part (|x|
d and |y|
L/2) can be tuned by the external magnetic field [15,16]. In the two electrodes the wave functions can be read as [10,17] Ψσ (x
−d, y) σ

= eiKN x ΦN (y) + Ψσ (x  d, y) =







rNσ  N e−iKN  x ΦN  (y), σ

(1)

N σ

tNσ  N eiKN  x ΦN  (y),

(2)

N

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and the wave function in the middle part can be expressed as
  σ σ −iknσ x AσnN eikn x + BnN φn (y), (3) e Ψσ (x, y) = n

where ΦN and φn are transverse eigenfunctions in the wide and narrow parts of the wire, respectively. The wave numbers KNσ and knσ are  KNσ = 2m(E − EN + h1 · σ )/h¯ and  knσ = 2m(E − En + h2 · σ )/h¯ , where h1 and h2 are the molecular fields in the wide part and the narrow part, respectively. σ is the Pauli operator with eigenvalue +1 for the spin-up state and −1 for the spin-down state. EN (En ) is the transverse energy in the wide (narrow) part. Using the matching condition of the wave function and its derivative, the coefficients in Eqs. (1)–(3) can be obtained. The detailed derivation can be seen in Refs. [10,17]. According to Landauer–Büttiker formula, the conductance through the nanowire is [18,19]   −∂f (E)  † e2
Tr S12 (σ, E)S12 (σ, E) , dE G= h σ ∂E (4) where f (E) is the Fermi distribution function, µ is the electrochemical potential, β (= 1/kB T ) is the inverse of the temperature, and h is the Planck constant, respectively. S12 is the scattering matrix, which can be calculated by the incident (emergent) velocity of the electrons and the coefficients of Eqs. (1)–(3) [10]. In the limit of T → 0, G depends only on the states near the Fermi energy. Once the wave function is obtained, the injectivity can be calculated by the following [20,21] dnL (x, σ ) dE  


2 −∂f 1

Ψσ (x, y) . = dE dy ∂E hvN

(5)

N

Fig. 1. Schematic diagram of magnetic nanocontacts. The direction of the moment in the middle part can be tuned by the external magnetic field.

Here the subscript L indicates the left contact. For the symmetric structure, dnR (x, σ )/dE = dnL /dE, where the subscript R indicates the right contact. The local density of states (LDOS) is then given by dn(x, σ ) dnL (x, σ ) dnR (x, σ ) = + . dE dE dE

(6)

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In the presence of a small ac voltage vac applied to the reservoir α (α = 1 or 2), the internal potential U (x, σ ) is given by U (x, σ ) = uα (x, σ )vac , where uα is called the characteristic potential. Using the Thomas–Fermi approximation, uα satisfies the Poisson equation −∇ 2 uα (x, σ ) + α = 1, 2.

e2 dn(r, σ ) e2 dnα (r, σ ) uα (x, σ ) = , ε0 dE ε0 dE (7)

We neglect the variation of the potential in the transverse direction, thus the characteristic function is a function of x only. Integrating over the transverse direction, the Poisson equation can be read as e2 dn(x, σ ) d 2 uα (x, σ ) uα (x, σ ) + − ε0 A(x) dE dx 2 e2 dnα (x, σ ) = (8) , α = 1, 2, ε0 A(x) dE where A(x) is the cross-sectional area of the system. To solve Eq. (8) for uα (x, σ ), we use the neutrality condition to determine the boundary val(xL ,σ ) dn(xL ,σ ) / dE and ues of uα (x, σ ): uα (xL , σ ) = dnαdE dnα (xR ,σ ) dn(xR ,σ ) / dE , where xL and xR are uα (xR , σ ) = dE left and right boundary lines of the system we have calculated. If the applied voltage is small, the distribution of the charge density in the system is given by

dq(x, σ ) = ρ(x, σ )dV (x) and ρ(x, σ ) can be written as d 2 uα (x, σ ) dq(x, σ ) = −ε0 A(x) . (9) dV (x) dx 2 The capacitance of the nanocontact can be calculated as ρ(x, σ ) =

0 Cσ =

ρ(x, σ ) dx xL

0  =e

2 xL

dnL (x, σ ) dn(x, σ ) − uL (x, σ ) dx. dE dE (10)

3. Results and discussion Throughout the Letter, we use ∆ (= π 2 h¯ 2 /2m∗ L20 ) as the unit of energy and L0 (= 50 nm) as the unit of length. In calculation, we set the width of the wide part D = 10, the thickness of the structure b = L0 /2, molecular fields h1 = 0.54, h2 = 0.36, and the Fermi energy Ef = 1.8. Fig. 2 shows the spin-dependent u(x) and ρ(x) in the ferromagnetic alignment. Fig. 2(a) and (b) correspond to the case of L = 1, and Fig. 2(c) and (d)

Fig. 2. Distribution of spin-dependent internal potential u(x) (top panels) and charge density ρ(x) (bottom panels) for (a), (b) L = 1 and (c), (d) L = 8. µ = 1.8, d = 2, W = 10, and b = 0.5.

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Fig. 3. (a) Conductance and (b) capacitance for nanocontacts in ferromagnetic alignment. µ = 1.8, h1 = 0.54, h2 = 0.36, d = 2, and W = 10.

correspond to that of L = 8, respectively. The contributions from spin-up and spin-down electrons are different because of the exchange interaction. When the width of the middle part is very small (see Fig. 2(a) and (b)), there is only one channel, and the difference can hardly be distinguished. With L increasing (see Fig. 2(c) and (d)), the difference between spin-up and spin-down electrons becomes larger. The screening length becomes shorter with a larger L, and as a result, the penetration depth becomes longer, the charge appears not only in the contacts but also in the electrodes. In Fig. 3, we plot the spin-dependent conductance and capacitance in the ferromagnetic alignment as functions of L. Because the molecular field is a quantum well for spin-up electrons and becomes a quantum barrier for spin-down ones, electrons with different spin orientations make different contributions to the conductance and capacitance. Here, we take the conductance and the capacitance as a unit contributed by one channel that electron tunnels. Based on the Landauer–Büttiker theory, the conductance and the capacitance can increase one unit once another channel opens for electrons. Therefore, the conductance dis-

plays a step-like structure with L increasing and the capacitance have discontinuous features [8]. The capacitance shows periodical resonant peaks around certain values of L, which correspond to the step jump of the dc conductance. This is caused by the opening of additional quantum channels in the contacts. When the width of the middle part is very small, the number of channels in the middle part is less than that in the electrode, the incoming electrons from the electrode experience multiple reflections at the interface between the narrow and wide parts. Thus, capacitance has small oscillations between the two neighboring resonances. When the width of the middle part almost equals that of the electrode, although the channels in the two parts are approximately equal, the screening length becomes very short, and as results, oscillations between the two neighboring resonant peaks become weak and the curve tends to smooth. The conductance and capacitance in the antiferromagnetic alignment are presented in Fig. 4 (see the dashed lines). The results of the ferromagnetic configuration are also shown in solid lines for comparison. In the antiferromagnetic case, the oscillations of the capacitance are less frequent than that in the ferromag-

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Fig. 4. (a) Conductance and (b) capacitance for nanocontacts in ferromagnetic and antiferromagnetic alignments, respectively. The parameters are the same as in Fig. 3.

netic one. When the width of the contacts is less than 5L0 , on the whole, the capacitance in the ferromagnetic alignment is bigger than in the antiferromagnetic one. However, when L  5L0 , the capacitance in the two alignments displays irregular oscillations. If we fix the width of the contacts at a certain value, whenever the structure changes from antiferromagnetic to ferromagnetic alignments, the capacitance will change correspondingly. One can see that the magnetocapacitance effect is complicated and sensitive to the width of the contacts.

4. Conclusions We have studied magnetocapacitance in ferromagnetic nanocontacts. Under the influence of the molecular field, spin-up and spin-down electrons make different contributions to conductance and capacitance. The capacitance decreases when the width of the contacts increases. When the system switches from the ferromagnetic alignment to the antiferromagnetic alignment, the capacitance changes correspondingly. The effect of magnetocapacitance revealed in this work

provides a feasible way to control the current in a nanoscale electrocircuit.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10074075) and Hebei province Natural Science Foundation of China (Grant No. A2005000147). Y. Guo was supported by the National Natural Science Foundation of China (Grant No. 10474052) and by Program for New Century Excellent Talents in University.

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