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Magnetic switches via electric field in BN nanoribbons
Liang Liu, Xue Ren, Jihao Xie, Bin Cheng, Weikang Liu, Taiyu An, Hongwei Qin, Jifan Hu*
* Corresponding author, email: hujf@sdu.edu.cn or hu-jf@vip.163.com
School of Physics, State Key Laboratory for Crystal Materials, Shandong University,
Jinan 250100, China
Abstract
The realization of strong magnetoelectric (ME) coupling for efficiently controlling of
magnetism is urgently needed but still a significant challenge. Based on first-principle
calculations and effective Hubbard models, we demonstrated that the polar charges and
locality of edge states in hexagonal BN nanoribbons (BNNRs) play key roles in the
behavior of edge magnetism M. By applying a transverse electric field, the magnetism in
both zigzag and armchair edges can be regulated sufficiently. In particular, the magnetic
on/off switches can be realized in armchair edges, it is robust against various edge
decorations including hydrogenation, fluoridation and hydroxylation. Furthermore,
uniaxial tensile strains can also produce significant magnetic modulation in Z-BNNRs,
which stem from the piezoelectricity. All these suggest that h-BNNRs are ideal platforms
for ME coupling researching and promising candidates for functional spintronics
application.
Keywords: Magnetoelectric, Magnetic switch, Boron nitride nanoribbons, Polar charge,
First-principle, Landau-Ginzburg model
Introduction
Controlling the magnetism via electric fields is highly desired both for fundamental
researching and for the ever-growing technical demanding and has attracted sharply
increasing number of researches[1-4]. Besides conventional magnetoelectric (ME)
coupling systems[5-11], the ground breaking of graphene especially the discovering of
magnetism in zigzag graphene nanoribbons (ZGNRs) have opened up a new era to
explore the ME effects in nanoribbons[12-16]. The major driving force for these
activities is that the intrinsic properties of one dimensional (1D) structures are mostly
contributed by boundary states amenable to efficient modulation by external fields or
strains[15, 17-24], in contrast to their two dimensional (2D) or three dimensional (3D)
counterparts.
Hexagonal-Boron nitride (h-BN) is a graphite-like layered material and has attracted
considerable interests owing to its novel properties and diverse applications[25-31].
Unlike other 2D materials including graphene and transition metal dichalcogenides
(TMDs), the intra-layer van der Waals interactions in h-BN is so weak that it influences
the electronic structure of each layer negligibly, and the high quality monolayer can be
readily exfoliated from h-BN single crystal[32]. Besides, 1D BNNRs have also been
experimentally synthesized by various routes including mechanical cutting, unzipping
and flatting of h-BN nanotubes[33-35]. There is a large volume of published studies
reporting the electronic and magnetic properties of BNNRs and it is now well established
that the electronic and magnetic structures of BNNRs can be tuned efficiently via
applying external electric field and edge passivation[36-48], e.g., full hydrogenation can
drive zigzag BNNRs (Z-BNNRs) into nonmagnetic states while half hydrogenation leads
to half metallic. However, most of the studies about magnetism are focused on Z-BNNRs,
there has been no quantitative analysis of ME effects in the armchair BNNRs (A-BNNRs)
or passivated edges, since they are usually thought to be nonmagnetic[49-52]. In addition,
previous studies have suggested that the spontaneous polarization P is strongly coupled to
mechanical strains, leading to remarkable band gap modulation and piezoelectric effect,
which can be attributed to the non-centrosymmetric structure[53, 54]. Therefore, it will
be very meaningful and interesting to investigate the ME coupling via applying strains,
too. Given all the facts above, hence, more systematic understanding of magnetic
modulations in each edge shape with or without edge-functionalization via applying field
and strains is urgently needed.
In this work, we report a comprehensive investigation on the ME effects in BNNRs
with various edge conditions and we demonstrate that the polar charges driven by P and
the locality of edge states are the two key factors that determine the magnetism M. To
verify our proposal, six models are examined utilizing density functional theory (DFT)
calculations. The first two is bare Z-BNNRs and A-BNNRs under external electric fields,
and we demonstrate the magnetic on/off switches in A-BNNRs and the effective
magnetic modulation in Z-BNNRs with respect to the biased voltage. Furthermore, we
construct the effective Hubbard model and phenomenological Landau-Ginsburg model
with extracted parameters from DFT results and show that the Curie temperature in A-
BNNRs can be well tuned by the biased voltage, too. Next, we consider three A-BNNRs
with edge-functionalized by H-, F- and HO-, all of them are capable to realize the
nonmagnetic-magnetic transition under critical biased voltage, revealing the magnetic
on/off switches are robust against edge-passivation, and their different magnetic
behaviors are also well understood via the analysis based on polar charges and locality.
The last model is bare Z-BNNRs with uniaxial tensile strains. In this case, although P is
dependent on the strains significantly due to the piezoelectric nature of Z-BNNRs, the
polar charges always keep constant. Therefore, edge locality dominates the evolution of
magnetism and makes M increase with respect to the increasing of P. All these findings
suggest BNNRs are promising candidates for future spintronic applications, as well as
provide an important advance in the understanding of ME effects in BNNRs.
Methods
Our calculations were performed within the DFT formalism using the generalized
gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) as exchange-
correlation functional, and the projected augmented wave (PAW) approach as
implemented in Vienna ab initio simulation package (VASP) code package[55, 56]. An
energy cutoff of 400 eV was used for the plane-wave expansion of the electronic wave
function. Brillouin zones (BZ) were sampled to 12×1×1 meshes for Z-BNNRs and 8×1×1
meshes for A-BNNRs using Monkhorst-Pack method. The energy convergence criteria
during full self-consistency in the electronic structure calculations were set at 10-5 eV per
unit cell. The lattice vectors and positions of the involved atoms were fully relaxed until
the max residual force was less than 10-2 eV/Å. To avoid spurious interplay interactions,
15 Å vacuum spaces were set in all non-periodic directions of the NRs.
Since the left and right edges of Z-BNNRs are different (see below for more details),
the work functions are also asymmetric in zero-field case, leading to an unphysical
electric field in the vacuum region between slabs and error in calculations of properties of
ground states. To offset this electric field, we place an artificial dipole sheet in the middle
of vacuum region, and the density of dipole moment can be determined self-consistently.
Furthermore, we add extra non-self-consistent dipole moments to the sheets to mimic the
effects of applied external electric fields. This has been proved an efficient way to study
the influences of external electric fields on systems with periodic boundary
conditions[57-60]. Note, for each periodic cell, the applied electric field cannot be too
large. If so, after self-consistent calculations, the artificial sheet may drag charge of
systems into the vacuum, giving rise to “field emission”.
Our Monte Carlo simulations were based on the standard Metropolis-Hasting
algorithm. In practice, we divide the continue part of Hamiltonian (see below for more
details) into twenty bins, each site thus can have at most twenty possible magnetic
moments (with sign representing the moment direction), and each state has its own
energy according to the Hamiltonian. In one Monte Carlo step, we first chose one site
then decide which moment it would take, stochastically. Based on the energy variation,
we accept or reject the operation with appropriate possibility. There are 500 unit cells
included in our simulations, using more than 108 sweeps (>1014 Monte Carlo steps) to get
converged results[61].
Results and discussion
1. Modulations of magnetic magnitude in bare Z-BNNRs and A-BNNRs
The natures of BNNRs are deeply inherited from their parent system: 2D h-BN
whose hexagonal lattice is composed of two distinct 2D sublattices of boron and
nitride atoms respectively. The inequality of two sublattices leads to an overall
point group of D
3v
thus open an energy gap at two inequivalent corners of the
Brillouin zone (BZ). The symmetry breaking also causes the bulk polarization[62,
63] which can be easily computed by representing the electronic structure in terms
of maximally localized Wannier functions (MLWF)[64]:
where
and
are the charge number and position of the nucleus in unit cell, and
is the position of the center of MLWFs, and
is the area of the unit cell, and
is one of the lattice vectors according to the modern theory of polarizations[65-
68]. In the case of h-BN, the computed MLWFs are three typical sp
2
-like Wannier
functions centred on the B-N bonds, together with one p
z
-like Wannier centred on
nitrogen and suspended on the h-BN plane (see Fig. 1(h)). Owing to the difference
in electron affinity, all the four Wannier functions are closer to nitrogen atom. A
particularly convenient way to calculate the polarization is thus to combine the
contribution of valence electrons and nitrogen ion core as one Wannier anion[68],
since the C
3v
symmetry fixes the dipole of Wannier anion to be zero and we can
only consider the contribution of boron ion and Wannier anion. Therefore, our
Wannier anion has the charge number
and
, and according to
equation (1) the bulk polarization of unit cell can be calculated straightforwardly:
here
are the two basic vectors of h-BN cell with lattice constant
a=2.51Å and
is a unit vector pointing along armchair direction. We have chosen
a specific representative element of polarization lattice with
in equation (1).
Hence, the polarization
we just obtained
points the armchair direction, and is
consistent with the C
3v
symmetry restriction[69, 70].
BNNRs only preserve one of the periodic directions which also define the name
of the NR, and the widths can be represented by the number of periodic atomic
chains N. Two such differently oriented NRs of Z-BNNRs with N=16 and A-
BNNRs with N=12 have shown in Fig 1(h). Let us first check the case of Z-
BNNRs with N=8. Since the orientation of symmetry-breaking is the same as
P
in
Z-BNNRs, an intrinsic electric field emerges and pushes the edge states to the
neighbours of Fermi level. Due to the strong locality of these edge states, they are
presented as high peaks in the density of states (DOS) diagram i.e. the von Hove
singularities. In Fig 1(b), we marked four such peaks including two for B-edge and
two for N-edge. On the other hand, the polar discontinuity induces polar charges to
screen the intrinsic electric field, avoiding the polar catastrophe[68, 71]:
where is the polar charges, is the length of boundary and we have in Z-BNNRs
case, is the unit vector orthogonal to the edge pointing to the vacuum, is the
difference of polarization across the edge. We can simply treat the polarization of
vacuum as zero, thus is identical to in equation (2) pointing from B- to N-edge, and
we get the maximal polar charges arise in the zigzag edges: . The positive
indicates hole-doping in N-edge and the negative value indicates the electron-doping in
B-edge. In terms of DOS, this kind of charge doping are represented by the hole
occupation in valence bands of N-edge (corresponding to N2 peak) and the electron
occupation in conducting bands of B-edge (corresponding to B1 peak). The partial filling
of van Hove singularities induce the Stoner instability and generate the edge magnetism.
Since both B1 and N2 peaks are totally spin polarized, we can get the relationship between
and magnetic moment:
here, the two integral terms correspond to the spin-polarized and localized
electrons in B- and holes in N-edges, respectively, and they are proportional to the
magnetic moment. The integral range is restricted between -2eV and 2eV because
all meaningful edge states in this study are located in this range.
and
correspond to the no spin-polarized charged contribution (electrons and holes,
respectively) which are delocalized and distributed in the inner part of Z-BNNRs.
Since the states in B-edge is more localized, revealing by the smaller width of
DOS peak and the spin density in Fig. 1(b), we can know that
,
therefore, the magnetic moment in B-edge is always larger than N-edge. Moreover,
noting that our Z-BNNRs with finite widths always preserve a residual intrinsic
electric field, the polar charges
can never reach the maximal value of
. And
the magnetic moments are well below
. When the width of ribbon increases,
more and more polar charges will be pushed to the edges, and
will grow and
approach 1e gradually. As well as, the total magnetic moments of Z-BNNRs
increase and asymptotically approach
, as shown in Fig. 2(a). To simply our
models, we mainly discuss the ferromagnetic case here. We stress that this
simplification does no limitations to our main conclusions.
When a transverse electric field is applied, extra polar charges will be generated
due to the dielectric effect:
here,
is the permittivity of our system. Therefore, the polar charges grow up or
drop down linearly with the magnitude of biased voltage (approximately suppose
that
is independent on the biased voltage), respectively, according to the
orientation of electric field. However, the evolution of magnetism is more complex
than linearity because of the contribution of no spin-polarized part. Let us first
examine the case in Fig.1 (c), in which the applied electric field takes the same
orientation as
. In terms of DOS, we can see that the states of B-edges (B
1
and B
2
peaks) get an extra electrostatic potential energy and move to lower energy region,
and the states of N-edges (N
1
and N
2
peaks) move reversely. The growing
(holes) in N-edge is mainly attributed to the growing holes occupation of N
1
peak,
i.e. the integral term in equation (4). Hence, the magnetic evolution shows a nearly
linear relationship. Meanwhile, the magnetic evolution in B-edge is much more
complex and we divide it into three stages. Firstly, in the weak field region,
corresponding to the 0 to -2V biased voltage in Fig. 1(a), the growing
(electrons) in B-edge mainly origins from the integral term, i.e. the occupation of
B
1
peak, thus the magnetic moment grows rapidly with the increasing of biased
voltage. In the second stage, corresponding to the -2 to -4 V biased voltage, whole
B
1
peak is occupied and Fermi level lays in the gap between B
1
and B
2
peaks, the
variation of
is attributed to the
term, which can be understood by the
collapse of wavefunction under strong electric field. This process also enlarges the
area of B
1
peak to generate the smoothly growing of magnetism. In the last stage,
the strong field pulls Fermi level across B
2
peak, therefore the filling of B
2
peak
becomes the main factor for the growing of
, and the growing of magnetic
moment comes back to the rapid and linear level.
As for the case in Fig. 1(d), the orientation of applied electric field is opposite
against
P
, and the polar charges decrease with respect to the growing of biased
voltage. In terms of DOS filling, the extra electrostatic potential pulls the states of
B-edge (N-edge) to the higher (lower) energy region. Since Fermi level always
cross the B
1
and N
2
peak, the decreasing of
mostly origins from the
occupation-reductions in these two spin-polarized peaks. Hence, the magnetic
moment in both edges decrease with an approximate linear relation.
As shown in Fig.2 (a), Z-BNNRs with different widths (including N=6-16) all
have the similar ME behaviour, that is, the magnetic moment decreases with the
positive electric field, and increases with the negative electric field. We thus may
extrapolate these conclusions to Z-BNNRs with arbitrary widths. Furthermore, it is
apparent from the Fig.2 (a) that the greater N corresponds to the steeper slope,
indicating that wider Z-BNNRs are more sensitive to
E
. This is not difficult to
understand since wider ones sustain greater electrostatic-potential differences
under the same
E
, the shifts of DOS are also more significant.
Next we would like to turn to the case of A-BNNRs. According to equation (2)
and (3), all polar charges and intrinsic electric field vanish since
is exactly
orthogonal to the edge-vector
. As a result, the edge states are deeply mixed with
bulk and the systems show nonmagnetic and insulate features. In the DOS diagram
of Fig. 1(f), the valence peaks of left edge (L
1u
and L
1d
) are spin-degenerated and
energetically degenerated with the right edge (R
1u
and R
1d
), and so are the
conducting peaks. Owing to the C
2
symmetry of A-BNNRs, only one direction of
external electric field is needed to consider. While such a transverse biased voltage
is applied, the degeneracy between two opposite sides is broken and the
electrostatic potential drags both L
1
and R
2
peaks to the Fermi level, such a Stark-
driven gap reduction is general in nanosystems and has already been reported both
theoretically and experimentally, and the gap can be totally closed under a critical
biased voltage[38, 72, 73]. What we want to stress here is that the gap-closing
effect depends on the biased voltage rather electric field intensity. If we neglect the
coupling between two opposite edges, the extra electrostatic potential needed to
close gap is just equal to the zero-field energy gap. In our case, calculated gap of
A-BNNRs is 4.6eV, approximately equal to the critical biased voltage 4.2V times
the charge of one electron. Our truly interest here is on the magnetic behaviour
under biased voltage. We want to ask that: are the L
1
and R
2
peaks meet in Fermi
level high enough to generate Stoner instability? Our spin-polarized calculations
give a positive result. Spontaneous spin-polarization emerges along with the
insulator-metal transition. As shown in Fig. 1(g), the spin degenerations in L
1
and
R
2
peaks are broken. Being similar to the case in Z-BNNRs, now the polar charges
are contributed from two parts, the inner part and the spin-polarized edge part:
where
and
correspond to the no spin-polarized charged contribution
(electrons and holes for right and left edges, respectively)
Beyond the critical biased voltage, the evolution of magnetism can be divided
into three stages. While the biased voltage is smaller than 5.2V, the growing of
polar charges mainly origin from the filling of
and
peaks, which are
totally spin-polarized, indicating the linearly growing of magnetic moments with
respect to the increasing of biased voltage. We call this stage as linear stage. The
second stage is saturation stage, corresponding to the biased voltage more than
5.2V and smaller than 6V. In this stage, R
2u
peak is completely occupied, and the
increasing of polar charges on the right edge are mostly supported by the no spin-
polarized term
. In the left edges, the filling of
peak starts to contribute to
polar charges. Noting that the spin-polarized holes in left edge is:
The contribution from
(which is negative valued) reduces the growing ratio of
magnetic moment. Therefore, in the saturation stage, the total magnetic moment
grows much smoothly. When the intensity of applied field gets stronger, the
magnetic evolution will start the third stage, i.e. reduction stage. Now the filling of
peak starts to contribute to the polar charges in right edge, and the holes
occupation of
dominate the variation of polar charges in left edge, the
magnetic moments of A-BNNRs thus decrease with respect to the growing of
biased voltage.
In Fig. 2(b), we show the magnetic evolutions for a series of A-BNNRs with
different widths. Since the magnetic transition emerges along with the gap-closing,
which is decided by the biased voltage, wider ribbons can switch to magnetic
states under lower external electric field. Beyond the critical
E
, their ME behaviors
are very similar as the one we discussed before. Note, while the ribbon become too
wide, a relatively high
E
may lead to the “field emission” effects as described in
method part. That is the reason we only give the low field results for wide systems.
To establish whether the magnetic state is energetically stable, we calculated the
magnetic energy for both kinds of BNNRs under biased voltage, which is defined
as the energy difference between spin-polarized and no spin-polarized solutions.
As shown in Fig. 3 (a) and (b), the magnetic states of Z-BNNRs under each biased
voltage and the magnetic states of A-BNNRs under biased voltage beyond critical
value of 4.2V are energetically favoured. Furthermore, the magnetic energies are
relied on the magnitude of biased voltage significantly. In Z-BNNRs, the most
stable magnetic state is accessed under -3V biased voltage, indicating that the
magnetic contribution from the filling of B
2
peak would destabilize the magnetic
state. In A-BNNRs, the lowest magnetic energy was found at 5.8 V biased voltage,
near to the end of saturation stage. It can be well understood via the Stoner criteria:
, where
represents the strength of electron-electron interaction and
is
DOS on Fermi level. Since the Fermi level crosses the largest DOS while A-
BNNRs are under 5.8 V biased voltage, the Stoner criteria is largely fulfilled, thus
leading to the most stable magnetic state.
2. Modulations of Curie temperature in A-BNNRs
Since the magnetic energy reveals some aspects of magnetic coupling in the Stoner
magnetic systems, the strong voltage dependent magnetic energy in A-BNNR indicates
that the Curie temperature may be well tuned by electric field, too. To verify this
assumption and examine the ME effect in A-BNNRs under finite temperature, we
construct an effective Hubbard model:
where the first part is the standard particle Hamiltonian and we consider the 2
nd
nearest hopping,
create (annihilate) one electron at i
th
site. The extra
electrostatic potential induced by external electric field was added to the diagonal
elements and was represented by the middle term:
, here
is the
electric field and
is the position vector and
, in which
is the occupation number operator of up-spin (down-spin). The
last part is a crucial two-body term, describing the onsite interaction:
, where
denotes the strength of electron-electron interaction. To
simplify our model,
was approximated by mean field:
, and
is assumed to be isotropic, our Hamiltonian thus regress
back into single-body case and
only induce the reoccupation of states while
dispersion relation keeps unchanged as given by the non-interacting case. The
reoccupation leads to the feedback to mean field
, the model thus can be
resolved iteratively. All parameters including hopping factors
and interaction U
are determined by fitting the electronic structure to the result of DFT calculations.
In the framework of mean-field, we can calculate the Curie temperature directly.
However, mean-field solution thoroughly neglects collective excitations such as spin-
density-waves which are the essential features for magnetic disorder at low temperature,
thus the Curie point would be considerably overestimated. Hence, we alternatively chose
the phenomenological Landau theory to describe our system. The double-well shape of
energy versus magnetic moment in the inset of Fig. 3(c) suggests the discrete model
which has been widely used to study the phase transition under finite temperature[74, 75].
In practice, we found that the Taylor series up to eighth order can well fit the energy
curve under each external field, and our Landau-Ginzburg expansion is:
where the first four terms are related to the free energy contribution from magnetization
of in each cell. The last term captures the magnetic coupling between nearest cells
and the coupling constant was extracted from the DFT calculations via comparing the
energy of ferromagnetism and anti-ferromagnetism solutions: .
The other four parameters (A-D) were decided by fitting the Landau-Ginzburg model
with ferromagnetic solution to the result of mean-field theory. As seen from the red line
in the inset of Fig. 3(c), the double-well anharmonic potential was well described.
Based on this effective Hamiltonian, we employ a series of Monte Carlo
simulations to investigate the magnetic phase transition under finite temperature.
As discussed in method part, each lattice site can takes one of the twenty different
discrete magnetic moment values during the simulations, ranging from -2 to
.
One can see the abrupt transition of magnetic moment in Fig. 3(d), indicating the
behaviour of first-order phase transition. The field-dependent Curie temperature
can be observed in the pattern of susceptibility more clearly. The divergent of
susceptibility, corresponding to the abrupt colour transitions in Fig. 3(e), reveals
the magnetic order-disorder transition and thus gives the information of Curie
temperature. There is no surprise that the Curie temperature can be sufficiently
tuned by the biased voltage, since both magnetic moment and magnetic energy are
relied on the biased voltage as discussed before.
All the three properties including magnetic moment, magnetic energy and Curie
temperature have a similar evolution including the on/off switches with respect to
the biased voltage, and can be well tuned in a large range, implying that the A-
BNNRs are excellent platform to realize the ME coupling. More importantly, as
discussed above, since the magnetic on/off switches emerges simultaneously with
the gap-closing which only depends on critical biased voltage rather than huge
field intensity, its experimental realization is truly feasible and promising.
3. Modulations of magnetism in functionalized A-BNNRs
The edge chemical functionalization has proved to be a powerful way to engineer the
electronic structures of NRs, and because of the existence of dangling bonds in bare
BNNRs, edge-passivation often stabilize the nanostructures thus is favorable for the
experimental realization. Former studies have reported the significant influence of edge
decoration on the ME coupling in Z-BNNRs[45]. Therefore, we would focus on the case
of A-BNNRs here. In what follows, we investigate three kinds of edge decorations
including hydrogenation (H-A-BNNRs), hydroxylation (HO-A-BNNRs) and fluorination
(F-A-BNNRs).
The energy gaps are not influenced by hydrogenation and fluorination but reduced by
hydroxylation. In Fig. 4(a), we can see that the HO-A-BNNRs first overcome the
nonmagnetic insulator state and become magnetic. However, the magnetism in
hydroxylated edges is not stable. As seen in Fig. 4(b), only in a small range of biased
voltage with magnitude less than 4.8 V dose the HO-A-BNNRs show magnetic ground
states. The reason is that the hydroxylation has eliminated the von Hove singularities to a
large extent. One can see from Fig. 4(e) that the DOS intensity for edge states are
comparatively low, indicating the states are so iterative and spread into the inner part of
NRs. Therefore, there are no enough electron-electron interactions to generate Stoner
magnetic instability.
As for H- and F-A-BNNRs, edge passivation dose not reduce edge localization too
much, the DOS peaks are still high enough to drive stable magnetic ground states. The
electronic structure of H-A-BNNR is more like the bared case that the magnetism in right
edge depends on the filling of two DOS peaks Ru1 and Ru2. Hence, the growing rates of
magnetic moment with respect to biased voltage have changed three times before
saturation stage. For F-A-BNNRs, these two DOS peaks are combined into single peak,
and the growing of magnetic moment is much faster. Also owing to the high DOS, the
magnetic energy of F-A-BNNRs is larger than that of hydrogenated counterparts.
Comparing with bare A-BNNRs, the magnetic energies of all these three cases are lower,
demonstrating that the edge passivation instincts to the formation of magnetism. However,
the important magnetic on/off switch properties are still preserved for all passivated
systems. Thus, we can confirm that the nonmagnetic-magnetic transitions under critical
biased voltage are robust against edge chemical pollutions and stable for applications in
real condition.
4. Modulations of magnetism in Z-BNNRs via strains
Considering the piezoelectric nature in Z-BNNRs, it is reasonable to realize the
ME coupling via applying strains. Here, we only discuss the in-plane uniaxial
tensile strains which are along the periodic directions since the compressive strains
are hard to realize in experiments. In the view on parent 2D BN systems, the
tensile strains of this orientation can be seen as the transformations which alter the
length and relative orientation of the two basic crystal vectors
.
Due to the
symmetry, the length of
should be equal to each other during the
transformations. We suppose that the length of basic crystal vectors are
transformed to
, and the angle between them is
. Then the area of stretched cell
is:
, and according to equation (2), we have
And since the length of periodic unit of Z-BNNRs , the
polar charges thus can be computed according to equation (e): . This is such a
surprising result that it tells us the polar charges are independent of applying strains.
Besides, is indeed reduced thus the intrinsic electric field is reduced under tensile
strains, and the DOS of stretched Z-BNNRs (Fig. 5(b)) is similar to the case of applying
counter electric field (Fig. 1(d)), in which the filling of B1 and N2 peaks are lowered. To
keep the polar charges unchanged, the capacity of these edge DOS peaks have to be
enlarged, leading to the increasing of edge magnetism, as shown in Fig. 5(a). And the
increased DOS intensity on Fermi level also stabilizes the magnetic states more,
representing by the lowering of magnetic energy in Fig. 5(a). These discussions are based
on the intrinsic bulk properties and edge states which are not related to the ribbon width,
we thus can expect the generality of these findings for different sized Z-BNNRs.
Hence, applying of uniaxial tensile strains is truly an effective way to control the
magnetism in Z-BNNRs, with the intrinsic electric field and edge locality as bridges.
Conclusions
In summary, we have examined the ME coupling in BNNRs with bare zigzag edges or
armchair edges. It was demonstrated that the edge magnetism M is dependent on polar
charges and the localization of edge states. Since the polar charges are sensitive to the
biased voltage, M thus can be efficiently modulated by E. In particular, the polar charges
in A-BNNRs are tuned from no spin-polarized to spin-polarized spontaneously with the
gap-closing effect driven by biased voltage, thus a novel magnetic on/off switch can be
realized in armchair edges. And the calculations suggest that the switch is robust against
edge-passivation including hydrogenation, fluorination and hydroxylation. Based on an
effective phenomenological Landau-Ginzburg model and a series of Monte Carlo
simulations, we show that the Curie temperature in A-BNNRs can also be well tuned by
E. In the final, we also discussed the realization of ME effects in Z-BNNRs through the
applying of tensile strains. We found that the polar charges are independent on the strains
and the effect of localization of edge states leads to an abnormal magnetic enhancement
with respect to the increasing of P. All these findings reveal the promising applications of
BNNRs in manifold spintronics, as well as give a comprehensive understanding of the
ME mechanism in BNNRs.
Acknowledgements
We acknowledge the inspiring discussions with X. B. Liu and the support of the National
Natural Science Foundation of China (Grant Nos. 51472150, 51472145 and 51272133),
Shandong Natural Science Foundation (Grant No. ZR2013EMM016), National 111
Project (B13029), and the Fundamental Research Funds of Shandong University (No.
2018GN037).
Declaration of interest: None
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Figures
Figure 1 The biased voltage dependent magnetic moments for Z-BNNRs (a) and A-
BNNRs (e), respectively, in which the red and blue squares represent the magnetic
moment of B- and N-edges. DOS and spin density for Z-BNNRs under the biased voltage
of zero (b), -7.8 V (c) and 7.8 V (d), respectively. The green and grey peaks represent the
DOS of B- and N-edges, respectively. (f) and (g) are the DOS and spin density for A-
BNNRs under biased voltage zero and 5.5 V, respectively. The red and blue peaks
represent the DOS of left and right edges, respectively. Up panel of each DOS diagram
represents the DOS of up-spin, and down panel draws down-spin. The Fermi level in
each DOS diagram is set to zero and the isosurface of spin density is set to 0.006 . (h)
give the full relaxed geometry structures for h-BN Z-BNNRs and A-BNNRs, respectively.
The isosurfaces represent computed Wannier functions. a1 and a2 are the two basic
crystal vectors of h-BN. The blue, grey and white balls represent boron, nitrogen atoms
and the centres of Wannier functions. The black circle represents the Wannier anion and
the arrow points to the direction of intrinsic P.
Figure 2 The external electric field dependent total magnetic moments for Z-BNNRs (a)
and A-BNNRs (b), respectively. Each colour corresponds to a specific ribbon width,
represented by N (defined in main text).
Figure 3 The magnetic energies (defined in main text) of Z-BNNRs (a) and A-BNNRs
(b). (c) give the free energy pattern of A-BNNRs versus magnetic moment and biased
voltages, with reference of energy of nonmagnetic state. The inset shows the two-well
energy curve of A-BNNRs under special biased voltage of 4.8 V. (d) and (e) are the
magnetic moment and susceptibility patterns of A-BNNRs versus biased voltages and
temperatures, respectively
Figure 4 Evolution of magnetic moment (a) and magnetic energy (b) of edge decorated
A-BNNRs with respect to the biased voltage. Blue, green and red lines represent
hydrogenated, fluorinated and hydorxylated cases, respectively. (c)-(e) are DOS for
different edge decorated A-BNNRs, blue and red peaks represents the DOS of right and
left edges, respectively. The Fermi energy is set to zero.
Figure 5 The magnetic dependence on uniaxial tensile strain. (a) gives the evolutions of
magnetic moments and magnetic energy of Z-BNNRs under tensile strains. The green
line represents the magnetic moment of B-edge, and the grey line represents the N-edge.
Black line represents the magnetic energy. (b) shows DOS of Z-BNNRs under 9% tensile
strain. Green peaks represent B-edge and grey peaks represent N-edge and dashed lines
represent total DOS. Fermi level was set to zero. And the inset shows spin density with
isosurface set to 0.006
