Using transfer-matrix and stationary phase methods, we study the tunneling time (group delay time) in a ferromagnetic monolayer graphene superlattice. The system we peruse consists of a sequence of rectangular barriers and wells, which can be realized by putting a series of electronic gates on the top of ferromagnetic graphene. The magnetization in the two ferromagnetic layers is aligned parallel. We find out that the tunneling time for normal incident is independent of spin state of electron as well as the barrier height and electron Fermi energy while for the oblique incident angles the tunneling time depends on the spin state of electron and has an oscillatory behavior. Also the effect of barrier width on tunneling time is also investigated and shown that, for normal incident, the Hartman effect disappears in a ferromagnetic graphene superlattice but it appears for oblique incident angles when the x component of the electron wave vector in the barrier is imaginary.

Graphene is a flat monolayer of carbon atoms densely packed in to a two-dimensional (2D) honeycomb lattice of sp2-bonded, with two nonequivalent triangular sublattices.1 Graphene sheet was first fabricated by Novoselov et al. in 2004.2–4 Sutter et al. using the new method, epitaxial of graphene on ruthenium, produced arrays of macroscopic single-crystalline graphene domains.5 The low energy electron states near the edges of Brillourin zone can be described by the conical energy spectrum. As a result, charge carriers i.e., electrons and holes close to the Dirac points K and K′ are described by the Dirac equation, at these points the Hamiltonian is given by |$\hat H = \hbar v_F \hat \sigma.\vec k$|Ĥ=vFσ̂.k,6 where |$\vec k$|k is the quasiparticles wave vector, |$\hat \sigma$|σ̂ is the 2D Pauli matrix and vF ≈ 106m/s is the Fermi velocity in graphene which is almost 100 times larger than in normal metal and thus it is sure to neglect the coulomb interaction comparing to kinetic energy in graphene. Also Fermi velocity plays the role of the speed of light in relativistic Dirac equation. The eigenvalues of the Hamiltonian are given by E = ±ℏvFkF. As a result, charge carriers near the Dirac points K and K′ exhibit linear dispersion relation where signs + and – denote the conical conduction and valence bands, respectively. From a basic research point of view, graphene-based systems, due to their lower “light speed” can be quite useful for studying relativistic effects such as the Klein paradox.6–8 It predicts that the electron can pass through a high potential barrier with the probability of one, regardless of height and width of the barrier at normal incident. In contrast to the conventional nonrelativistic problems, transmission probability exponentially decays with the increasing of the barrier height and width.9,10

Several graphene- based structures have been designed and many interesting results have been achieved.11,12 When a bias is applied to a graphene sheet, a potential barrier is created therefore a graphene superlattice can be realized by applying a periodical gate voltage to graphene layer. Transport of electron in a ferromagnetic monolayer graphene barrier,7,13 a double ferromagnetic monolayer graphene barrier,14 and in a clean,15 disordered,9 ferromagnetic,16 or magnetic17 graphene superlattice has been studied. It is shown that transmission though a graphene superlattice is strongly dependent on the well width, Fermi energy, barrier width and height.

The dwell time characterizes the time a particle spends in a region of space, averaged overall scattering channels.18 The tunneling time through one potential barrier19–25 double-barrier structures26–29 and semiconductor superlattice30,31 has been studied. Therefore, it would be worthwhile to investigate tunneling time and Hartman effect in a ferromagnetic graphene superlattice. The Hartman effect in quantum tunneling basically explains on the basis that the tunneling time for a particle tunneling through a barrier is independent of the barrier thickness.22 

In this work, the tunneling time (group delay time) in a ferromagnetic monolayer graphene superlattice with rectangular barrier is studied by solving the Dirac equation, using transfer-matrix and stationary phase methods. We also investigated the transmission probability and the tunneling time of the Dirac fermions through the superlattice when electron wave vector in the barrier, qx, is imaginary.

To study spin dependent transport, we consider a ferromagnetic graphene superlattice with rectangular potential barrier which is obtained by applying a series of magnetic insulator bar with metallic gate, such as EuO, on the top of monolayer graphene sheet which is located on a substrate such as |${\textit SiO}_2$|SiO2. The grown direction is taken to be the x axis. In order to neglect the strip edges, we focus here on the case where the width of the graphene strip is much larger than the width of barriers in graphene superlattice.9,15 The schematic of potential profiles for the ferromagnetic graphene superlattice is shown in Fig. 1.

FIG. 1.

The schematic view of the ferromagnetic graphene superlattice (a), the electronic energy profile (b), and N electrostatic barriers of width D, and (N-1) wells of width L (c).

FIG. 1.

The schematic view of the ferromagnetic graphene superlattice (a), the electronic energy profile (b), and N electrostatic barriers of width D, and (N-1) wells of width L (c).

Close modal

The rectangular potential profile along the grown direction is given by:

\begin{equation}V(x) = \left\{ {\begin{array}{c@{\qquad}c}{V_0 \pm \Delta } & {\textit for\; the\; barrier\; areas,} \\[2pt]0 & {\textit for\; the\; well\; areas.} \\\end{array}} \right.\end{equation}
V(x)=V0±Δforthebarrierareas,0forthewellareas.
(1)

Where V0 is the electronic potential which is controlled by the metallic gate and Δ is the exchange field, the signs + and − denotes spins anti parallel and parallel to the exchange field, respectively. A rough estimate suggests that Δ can be ±5 meV.7,16,32

The charge carriers in graphene superlattice are described by the Dirac equation in which the Hamiltonian of carriers is written as |$\hat H = \hat H_0 + V(x)$|Ĥ=Ĥ0+V(x), where |$\hat H_0 = \hbar v_F \hat \sigma.\vec k.$|Ĥ0=vFσ̂.k. In order to study the transport problem in graphene superlattice, we shall solve the Dirac equation. To solve this equation, we suppose that incident electron propagates at an angle φ along the x axis. The general solution to the Hamiltonian |$\hat H = \hat H_0 + V(x)$|Ĥ=Ĥ0+V(x) in the ith strip can be expressed in the following form.9,15–17

\begin{eqnarray}\psi _1^i &=& (a_i e^{ik_{ix} x} + b_i e^{ - ik_{ix} x})e^{ik_y y},\nonumber \\\psi _2^i &=& s_i (a_i e^{ik_{ix} x + i\varphi _i } - b_i e^{ - ik_{ix} x - i\varphi _i })e^{ik_y y}.\end{eqnarray}
ψ1i=(aieikixx+bieikixx)eikyy,ψ2i=si(aieikixx+iϕibieikixxiϕi)eikyy.
(2)

Here |$\psi _1^i$|ψ1i and |$\psi _2^i$|ψ2i are the components of the Dirac spinor, ai and bi are the transmission and reflection coefficients, respectively.

and

\begin{equation}s_i = {\rm sgn} (E - V(x)),\end{equation}
si= sgn (EV(x)),
(3)
\begin{equation}k_{ix} = \left\{ {\begin{array}{c@{\qquad}c}\displaystyle{k_x = \frac{{E\cos \varphi }}{{\hbar v_F }},} & {\textit for\; the\; well\; areas,} \\[7pt]\displaystyle{q_x = \frac{{E - V_0 \mp \Delta }}{{\hbar v_F }}\cos \theta,} & {\textit for\; the\; barrier\; areas.} \\\end{array}} \right.\end{equation}
kix=kx=EcosϕvF,forthewellareas,qx=EV0ΔvFcosθ,forthebarrierareas.
(4)
\begin{equation}k_y = \left\{ {\begin{array}{c@{\qquad}c}\displaystyle{k_y = \frac{{E\sin \varphi }}{{\hbar v_F }},} & {\textit for\; the\; well\; areas,} \\[7pt]\displaystyle{q_y = \frac{{E - V_0 \mp \Delta }}{{\hbar v_F }}\sin \theta,} & {\textit for\; the\; barrier\; areas.} \\\end{array}} \right.\end{equation}
ky=ky=EsinϕvF,forthewellareas,qy=EV0ΔvFsinθ,forthebarrierareas.
(5)

where E is energy of incident electrons and θ = tan −1(ky/qx) is the angle of refraction, i.e., the corresponding angle inside the barriers. Because the system in our model is homogeneous in the y-direction, the momentum parallel to the y-axis is conserved.7 

\begin{equation}k_y = q_y = \frac{E}{{\hbar v_F }}\sin \varphi.\end{equation}
ky=qy=EvFsinϕ.
(6)

By applying the continuity of wave function at the boundaries for the system which consist of N barriers, we obtain the rs and tsN where rs and tsN are spin reflection and spin transmission coefficient, respectively. The angle dependent transmission probability can be evaluated by Ts(φ) = |tsN|2.

The phase time or group delay time, τg is the time it takes for the peak of the transmitted wave packet to appear.22 It is calculated by the method of stationary phase which is defined as

\begin{equation}\tau _g = \left| {t_s } \right|^2 \tau _{gt} + \left| {r_s } \right|^2 \tau _{gr},\end{equation}
τg=ts2τgt+rs2τgr,
(7)

where |ts|2 and |rs|2 are the transmission and reflection probabilities, respectively. The group delay time in transmission τgt and reflection τgr can be written as

\begin{equation}\tau _{gt} = \frac{\hbar }{{dE}}(k_x l_{(2N)} + \Theta _t),\end{equation}
τgt=dE(kxl(2N)+Θt),
(8)
\begin{equation}\tau _{gr} = \hbar \frac{{d \Theta}_r }{{dE}},\end{equation}
τgr=dΘrdE,
(9)

where l(2N) is the length of the system |$\Theta _t = \arg t_s,$|Θt=argts, and |$\Theta _r = \arg r_s$|Θr=argrs. It can be shown that for a symmetric system τg = τgt = τgr.23 The first term in Eq. (8) represents the time it take for a particle to travel the distance l(2N) in the absence of the barriers. The dwell time is the time spent by a particle in the barrier regardless of whether it is ultimately transmitted or reflected.19,23 This quantity is defined as the total probability of the particle within the barrier divided by the incident probability current.33 

\begin{equation}\tau _d = \frac{1}{{j_{in} }}\int\limits_0^D {\left| \psi \right|^2 dx}.\end{equation}
τd=1jin0Dψ2dx.
(10)

Here, jin = vFcos φ and ψ is the steady-state scattering solution of the Dirac equation. According to Eq. (21) in Ref. 19 for gapless graphene structure the group delay time equals the dwell time i.e.,τd = τg.

In this section we present a numerical study based on equation derived in previous section. At first, we calculate the transmission probabilities of charge carriers through a ferromagnetic graphene superlattice. The transmission probabilities as a function of the barrier height V0, are shown in Figs. 2(a) and 2(b) for one barrier and five barriers, respectively. In these figures the Fermi energy E of the incident electrons is E=83 meV and the incident angles are set to be φ = 0, π/6, respectively. We present the transmission probability as a function of the barrier thickness D for one barrier and five barriers in Figs. 3(a) and 3(b) respectively. In case of V0=200 meV, all other parameters are set to be the same as those in Fig. 2.

FIG. 2.

Transmission probability as a function of the barrier height V0 for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier thickness is D=50nm and the well thickness is L=10nm.

FIG. 2.

Transmission probability as a function of the barrier height V0 for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier thickness is D=50nm and the well thickness is L=10nm.

Close modal

According to Figs. 2 and 3 perfect transmission with Ts = 1 at normal incident (φ = 0) is observed for spin-up and spin-down. This is due to massless the Dirac fermions and directly related to Klein tunneling.6,8 while for incident angle φ ≠ 0 the transmission probability depends on spin state of electron and has an oscillatory behavior. This is because of the fact that the transmission probability Ts for incident angles φ ≠ 0 is an oscillating function of D and qx, which qx is determined by V0. For example, in the limit of high barrier |V0| > >E, Ts = cos 2φ/[1 − cos 2(qxD)sin 2φ] for graphene superlattice with single barrier.6 

FIG. 3.

Transmission probability as a function of the barrier thickness D for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier height isV0 = 200meVand the well thickness is L=10nm.

FIG. 3.

Transmission probability as a function of the barrier thickness D for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier height isV0 = 200meVand the well thickness is L=10nm.

Close modal

We plot the τg0 as function of the barrier height V0 for one barrier and five barriers in Figs. 4(a) and 4(b), respectively, where τ0 is the time it takes for a particle would take to travel the distance l(2N) in the absence of the barriers. Fermi energyE, incident angle φ, barrier thickness D and well thickness L are taken the same as in Fig. 2. We also plot the τg0 as function of the barrier thickness D for one barrier and five barriers in Figs. 5(a) and 5(b), respectively.

As can be seen from Figs. 4 and 5, the τg0 to the normal incident (φ = 0) is always 1. Because in this situation Θt and τg in Eq. (9) for graphene superlattice with N barriers can be written as:

\begin{eqnarray}\Theta _t &=& - (k_x + q_x){\textit ND}, \nonumber\\\tau _g &=& \hbar \frac{1}{{dE}}(k_x l_{(2N)} - (k_x + q_x){\textit ND}),\end{eqnarray}
Θt=(kx+qx)ND,τg=1dE(kxl(2N)(kx+qx)ND),
(11)

so,

\begin{equation}\tau _g = \frac{{(N - 1)L + {\textit ND}}}{{v_F }} = \frac{{l_{(2N)} }}{{v_F }} = \tau _0.\end{equation}
τg=(N1)L+NDvF=l(2N)vF=τ0.
(12)

This is directly related to Klein paradox. While for incident angles φ ≠ 0 the τg0 depends on the spin state of electron and it has an oscillatory behavior. For example, in the limit of high barrier |V0| > >E, Θt = −tan −1(sin 2qxD[1 + cos φ]/2cos 2qxD[1 − cos φ] − 1) for graphene superlattice with single barrier, but according to Eq. (3)qx is determined by V0, so that Θt and τg are also oscillating functions of V0 and D.

FIG. 4.

The ratio of group delay time to the equal time τ0 as a function of the barrier height V0 for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier thickness is D=50nm and he well thickness is L=10nm.

FIG. 4.

The ratio of group delay time to the equal time τ0 as a function of the barrier height V0 for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier thickness is D=50nm and he well thickness is L=10nm.

Close modal
FIG. 5.

The ratio of group delay time to the equal time τ0 as a function of the barrier thickness barrier D for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier height isV0=200 meV and the well thickness is L=10nm.

FIG. 5.

The ratio of group delay time to the equal time τ0 as a function of the barrier thickness barrier D for spin-up and spin-down through the ferromagnetic graphene superlattice with one barrier (a) and five barriers (b). The barrier height isV0=200 meV and the well thickness is L=10nm.

Close modal

The most interesting behavior is found for 2E > |V(x)|, where the x component of the electron wave vector in the barrier is defined as |$q_x = k_F \sqrt {(E - V(x))^2 /E^2 - \sin ^2 \varphi }$|qx=kF(EV(x))2/E2sin2ϕ that can be imaginary. On the other hand, the Hartman effect occurs when qx is imaginary, because imaginary qx corresponds to evanescent wave which decay exponentially with distance across the barrier region. Consequently, the integrated probability density of the particle in large thickness limit becomes insensitive to the barrier thickness D, i.e., the Hartman effect.22,23,34 Figs. 6(a) and 6(b) show the transmission probability as a function of the barrier thickness D when qx is imaginary through the system for one barrier and five barriers, respectively. Fig. 7 shows theτg0as a function of barrier thickness D. In Figs. 6 and 7 the Fermi energy E, of incident electrons is taken to be E=200 meV and the incident angles are supposed to be φ = π/6, π/4, π/3. According to Figs. 6 and 7, the shape of transmission probability and τg0 decay exponentially with the barrier width and for the wide barrier, τg0 is independent of the barrier thickness (Hartman effect). It should also be mentioned that this condition, spin-up and spin-down behave very similarly.

FIG. 6.

Transmission probability as a function of the barrier thickness D through the graphene superlattice with one barrier (a) nd five barriers (b). The barrier height is V0=250 meV and the well thickness is L=5nm.

FIG. 6.

Transmission probability as a function of the barrier thickness D through the graphene superlattice with one barrier (a) nd five barriers (b). The barrier height is V0=250 meV and the well thickness is L=5nm.

Close modal
FIG. 7.

The ratio group delay time to the equal time τ0 as a function of the barrier thickness barrier D through the graphene superlattice with one barrier (a) and five barriers (b). The barrier height is V0=250 meV. The well thickness is L=5nm.

FIG. 7.

The ratio group delay time to the equal time τ0 as a function of the barrier thickness barrier D through the graphene superlattice with one barrier (a) and five barriers (b). The barrier height is V0=250 meV. The well thickness is L=5nm.

Close modal

In summary, we have evaluated numerically the tunneling time of the Dirac fermions in a ferromagnetic graphene superlattice. Our results show that the Hartman effect exits in ferromagnetic graphene superlattice when the wave vector in the barriers is imaginary. Moreover, we showed that in ferromagnetic graphene superlattice unlike the semiconductor superlattice, the Hartman effect does not exist for normal incident and the tunneling time for normal incident is independent of the spin state of electron. In addition, when qx is real and for incident angles φ ≠ 0 the tunneling time, as well as, transmission probability, exhibits strong dependence not only on the Fermi energy and barrier width, barrier height but also on the spin state of electron and it has an oscillatory behavior.

Finally the authors are grateful to the Iran University of Science and Technology for financial support.

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