You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We propose a graphene device that can generate spin-dependent negative differential resistance (NDR). The device is composed of a sufficiently wide and short graphene and two gated EuO strips deposited on top of it. This scheme avoids graphene edge tailors required by previous proposals. More importantly, we find clear significant of a spin selectivity and a [region tunability]{} in the spin-dependent NDR: by changing the top gates of the device, NDR for spin up only, spin down only, or both spins (occurring sequentially) can be respectively realized; meanwhile, the central position of the NDR [region]{} in each case can be monotonously tuned in a wide range of bias voltage. These remarkable features are attributed to a gate controllability of the spin-dependent resonant levels in the device hence their deviations from the Fermi energy and Dirac point in the source electrode, respectively. They add a spin and a bias degree of freedom to [conventional]{} NDR-based devices, which paves a way for designing a whole new class of NDR circuits.'
author:
- Yu Song
- Yang Liu
- Xiaolong Feng
- Fei Yan
- 'Wei-Zhi Zhang'
title: 'Spin-selectable, [region-tunable]{} negative differential resistance in graphene double ferromagnetic barriers'
---

[^1]

Introduction
============

The intersection of spintronics and nonlinear transport can lead to phenomena of interest and use. For example, the realization of a spin-dependent negative differential resistance (NDR), in which [fermions of one spin display NDR]{} while the current of the other spin increases monotonically as the bias increases, can be applied as spin-resolved [oscillators,[@alekseev2000large] amplifiers,[@laskar1989gate] switchings,[@liu1997novel] and memories]{}.[@van1999tunneling] Till now, several schemes have been proposed based on graphene nanoribbons. For example, spin-dependent [NDRs]{} have been demonstrated in edge-doped zigzag graphene nanoribbons (ZGNRs),[@wu2012negative; @jiang2014spin; @yan2014effects] B(N)-doped ZGNRs,[@yang2015half] ZGNRs with nitrogen-vacancy defects,[@xu2014negative] FeN$_4$-embedded ZGNRs,[@li2017strong] and zigzag-edged trigonal graphenes linked to ZGNRs electrodes.[@hong2015axisymmetric] In these devices, spin-polarized edge states[@son2006half] play an important role. Other proposals considered armchair graphene nanoribbons (AGNRs), including [a]{} FeN$_4$ embedded and N-doped AGNR[@liu2016efficient] and a compound system comprising an AGNR and a set of ferromagnetic insulator strips deposited on top of it.[@munarriz2013strong; @diaz2014spin] [Very recently, experiments on nanoribbons for graphene and other two-dimensional materials have been reported.[@bai2010very; @Llinas2017short; @Chhowalla2016two] However, proper introductions of magnetic doping’s, defects, or superlattices are still required for these devices. That’s partly why till now none of the proposals has been experimentally implemented. ]{}

In this work, we propose a bulk graphene based, spin-dependent double-barrier resonant tunneling diode (DB-RTD), which requires only depositing two *gated* EuO strips[@haugen2008spin] on top of a sufficiently wide and short graphene (see upper panel in Fig. \[fig:setup\]). [In this device, no specific control of the graphene edge type is needed because the transport is dominated by bulk states; no doping’s or defects are required because the ferromagnetism is induced by magnetic insulators; and two ferromagnetic barriers instead of a ferromagnetic superlattice is enough because large energy gaps are induced.]{} We find clear spin-dependent resonant states in the device, which dominate the biased transport when the Fermi energy is setting around [the resonant levels]{}. The resonant levels decrease with an increasing bias, resulting in a fewer transversal mode number hence the occurrence of the desired NDR feature. Remarkably, we find a clear significant of a spin selectivity and a [region tunability]{} in the spin-dependent NDR: by changing the top gates hence the spin-dependent resonant levels in the device, the spin index of the NDR can be tuned as spin up only, spin down only, or both spins; meanwhile, the central position of the NDR [region]{} in each case can be monotonously tuned in a wide bias range. Together with the relative ease in fabrication, these remarkable features make the proposed device an important building block for future spintronic or NDR circuits.

![(color online) Schematic diagrams of the proposed device (upper panel) and its spin-dependent voltage profiles under zero or finite biases (middle or bottom panel). Red for spin up and blue for spin down. The (spin-dependent) dispersion in each region is also shown.[]{data-label="fig:setup"}](spinNDR-device "fig:"){width="0.75\linewidth"}\
![(color online) Schematic diagrams of the proposed device (upper panel) and its spin-dependent voltage profiles under zero or finite biases (middle or bottom panel). Red for spin up and blue for spin down. The (spin-dependent) dispersion in each region is also shown.[]{data-label="fig:setup"}](voltage-profile "fig:"){width="0.65\linewidth"}

The structure of the paper is as follows. In Sec. \[sec:II\] we present the setup of the device and the formula to calculate the stationary envelop function as well as the transmission coefficient for a given spin, energy, incident angle, and bias voltage across the sample. An ab-initio dispersion of graphene on EuO[@yang2013proximity; @hallal2017tailoring] and a linear voltage drop through the device are considered. The resulting current-voltage (I-V) characteristics of the device, [comprised of]{} a spin-dependent NDR with a spin selectivity and a [region]{} tunability, are discussed in the subsequent Sec. \[sec:III\]. The mechanism for the arising and tunability of the spin-dependent NDR, as well as possible applications are emphasized. Finally, Sec. \[sec:IV\] concludes the paper.

Setup and Formula {#sec:II}
=================

The proposed system is composed of a sufficiently wide ($W$=250 nm) and short rectangular graphene and two rectangular EuO strips on top of it (see the upper panel of Fig. \[fig:setup\]). The length of [each strip]{} is $a =$ 21.75 nm and the spacing between them is $b =$ 8.7 nm. The source and drain electrodes, which may induce a charge doping ($U$), are separated [from the EuO strips by two buffer regions of $a/10$ length]{}. The total length $l=2.2a+b$ is several times smaller than $W$, hence the transport is dominated by bulk states[@tworzydlo2006sub] and no specific control of the graphene edge type is needed. Experiments[@swartz2012integration; @wang2015proximity; @wei2016strong; @zhang2017realization] and ab-initio calculations[@yang2013proximity; @hallal2017tailoring] have shown that, through a magnetic proximity effect, the two EuO strips induce ferromagnetism as well as [heavy electron doping’s]{} (with a Dirac point of $E_D=-1.37$eV) in graphene just underlying them (see the middle panel of Fig. \[fig:setup\]). On top of the EuO strips, two top gates ($V_t^{(i)},i=1,2$) are applied. They are used to tune the local carrier concentrations and [to lift up]{} the Dirac points of the two ferromagnetic graphene’s to align with that of the prinstine graphene (see the middle panel of Fig. \[fig:setup\]). Therefore, the array of two gated EuO strips creates the desired [ferromagnetic double barriers]{} or spin-dependent DB-RTD. Below the substrate, a back gate ($V_g$) is applied to tune the Fermi energy ($E_F$) through the whole sample.

When a bias voltage ($V_b$) is applied between the source and drain, a net current will be produced by the electrons or holes in the source within an energy range of [$(0,E_F)$ or $(E_F-eV_b,0)$]{} at zero temperature, see the bottom panel in Fig. \[fig:setup\]. Due to the spin difference in the ferromagnetic regions, the current will be spin dependent. Unlike other works which [considered]{} a step-like voltage drop through the device, in this work we also consider a linear voltage drop, $V(x)=f x$ with $f=V_b/l$, between the source and drain along the $x$-direction.

The magnetic proximity effect in graphene is usually described by a simple Zeeman splitting model,[@haugen2008spin; @munarriz2013strong; @diaz2014spin; @lu2013tunable; @dell2009wave; @niu2008spin; @faizabadi2012spin; @zou2009negative] based on which [a]{} spin wavevector filtering [effect]{},[@lu2013tunable] [a]{} spin conductance filtering [effect]{},[@dell2009wave] and [a]{} tunneling magnetoresistance [effect]{} [@niu2008spin; @faizabadi2012spin; @zou2009negative] have been proposed. (These effects have also been found in other two-dimensional systems subjected to magnetic modulations; just to cite a few.[@wang2002spin; @papp2005spin; @wu2011spin; @shakouri2015tunable; @shakouri2014spin]) The Zeeman splitting model is roughly estimated from an analogy with a EuO/Al interface and considers only opposite energy shifts for opposite spins added to the linear dispersion of a pristine graphene.[@haugen2008spin] However, very recent ab-initio calculations[@yang2013proximity; @hallal2017tailoring] and experiments[@swartz2012integration; @wang2015proximity; @wei2016strong; @zhang2017realization] demonstrate a totally different picture: the [gapless and linear]{} Dirac cone for each spin shifts down to a negative Dirac point ($D_s$), opens a large energy gap ($\Delta_s$), and changes its Fermi velocity ($v_s$) ($s=\pm1$ for spin up and [spin down]{}), [all depending on the spin index]{}; see middle and bottom panels in Fig. \[fig:setup\]. For graphene on six bi-layer EuO, the parameters read $D_{+(-)}=$42 (-24) meV, $\Delta_{+(-)}=$67 (49) meV, and $v_{+(-)}=$ 1.337 (1.628) $v_F$, respectively.[@yang2013proximity; @song2015spin] Others[@zollner2016theory; @su2017effect; @hallal2017tailoring] and we[@song2015spin; @song2017electric] have developed effective Hamiltonians in a sublattice-spin direct produce space or a sublattice space, based on which nontrivial effects, such as quantum anomalous Hall effect,[@su2017effect] simultaneous spin filter and spin valve effect,[@song2015spin] electric-field-induced extremely large [change in resistance]{},[@song2017electric] and pure crossed Andreev reflection[@ang2016nonlocal] have been explored.

In a sublattice space[,]{} the effective Hamiltonian reads[@beenakker2008colloquium; @song2015spin; @song2017electric]

$$\mathcal{H}^f_{\bm{k},s,\xi}= \bm\sigma \cdot \hbar v_s \bm{k}+ \xi \sigma_z \Delta_s + \mathcal{I} (D_s +e V_t^{(i)}+e V(x)).\label{Hamiltonian:fm}$$

Here $\bm{\sigma}=(\sigma_{x},\sigma _{y})$ is a pseudospin Pauli matrices, $\bm{k}=(k_x, k_y)$ is a momentum operator, $\xi =\pm 1$ is an index for valley $K$ and $K^\prime$, and $\mathcal{I}$ is an [identity]{} matrix. For the pristine graphene in the well and [the]{} buffer regions, the Hamiltonian is well known $$\mathcal{H}^p_{\bm{k}}= \bm\sigma \cdot \hbar v_F \bm{k} + \mathcal{I} e V(x).\label{Hamiltonian:pr}$$ For the contacted graphene in the source and drain[,]{} the Hamiltonian reads $$\mathcal{H}^c_{\bm{k}}= \bm\sigma \cdot \hbar v_F \bm{k} + \mathcal{I} U.\label{Hamiltonian:con}$$

[For convenience, below we will express all physical quantities in their dimensionless form,]{} in terms of a characteristic length $l_0=$56.55nm and a corresponding energy unit $E_0=\hbar v_F/l_0=$10meV. The transversal momentum $k_y=(E-U)\sin\alpha$ ($E$ and $\alpha$ are the energy and incident angle of a fermion, respectively) is conserved, and it suffices to [solve]{} the longitudinal wave function $\bm{\Psi}_j(x)$, where $ j=c,f,p$ for the contacted, ferromagnetic, and pristine graphene, respectively. However, straightforward decoupling of $\mathcal{H}_j\bm\Psi_j(x) = E \bm\Psi_j(x)$ in Eqs. (\[Hamiltonian:fm\]) and (\[Hamiltonian:pr\]), which contain a linear term of $x$, results in an unsolvable two-order differential equation.[@sonin2009effect] To solve this problem, we perform a rotation of the Dirac equation by $\pi/2$ around the $y$-axis in the pseudospin space (see the upper pannel of Fig. \[fig:setup\]). The envelope functions become resolvable and the result for the ferromagnetic graphene reads

\[wavefunction\] $$\label{wavefunction1}
\bm{\Psi}^s_f=
f^+\left(\begin{array}{c}
F_s(x)\\
G_s^*(x)\end{array}\right)e^{ik_{y}y}
+f^-\left(\begin{array}{c}
G_s(x)\\
F^{*}_s(x)\end{array}\right)e^{ik_{y}y},$$ where $F_s(x)=D[-1/2+iq_s^2/2a_s,(1+i)(E-D_s-eV_t^{(i)}+fx)/\sqrt{a_s}]$ and $G_s(x)=D[-1/2-iq_s^2/2a_s,(-1+i)(E-D_s-eV_t^{(i)}+fx)/\sqrt{a_s}]$. [Here $D[,]$ is]{} the Weber parabolic cylinder function, $q_s^2=\Delta_s^2+k_y^2$, and $a_s=f v_s$. Note, $F_s(x)$ and $G^{*}_s(x)$ \[$F^{*}_s(x)$ and $G_s(x)$\] are spin resolved and have properties of a right (left)-going wave function.[@sonin2009effect] For the pristine graphene the result becomes $$\label{wavefunction2}
\bm{\Psi}_p=p^+\left(\begin{array}{c}
F(x)\\
G^{*}(x)\end{array}\right)e^{ik_{y}y}
+p^-\left(\begin{array}{c}
G(x)\\
F^{*}(x)\end{array}\right)e^{ik_{y}y},$$ where $F=D[-1/2+ik_y^2/2f,(1+i)(E+fx)/\sqrt{f}]$ and $G=D[-1/2-ik_y^2/2f,(-1+i)(E+fx)/\sqrt{f}]$. In the rotated pseudospin space, the envelope functions for the contacted graphene become $$\label{wavefunction3}
\bm\Phi_c^\pm=
c^\pm\left(\begin{array}{c}
%\frac{E_j\pm k_j+iq_j}{2E_j}
1\pm e^{\pm i \alpha} \\
-1 \pm e^{\pm i \alpha}
\end{array}\right)
%[E_j\pm k_j+iq_j,-E_j\pm k_j+iq_j]^T
e^{\pm ik_cx+ik_yy},$$

where $k_c^{s,d}=\textmd{sign} [E-U+eV_b^{s,d}]\sqrt{(E-U+eV_b^{s,d})^2-k_y^2}$ with $V_b^{s,d}=0,V_b$. In the [device]{} source, $c^+=1$ and $c^-=r_s$, while in the [device]{} drain $c^+=t_s$ and $c^-=0$. $r_s$ and $t_s$ are spin-dependent reflection and transmission coefficients, respectively.

![(color online) (a,b) Linear conductances for spin up (a) and spin down (b) as a function of the Fermi energy. Dashed: $l=2$ and $eV_t^{(i)}=0$, solid: $l=1$ and $eV_t^{(i)}=-2,0,2$. (c) The corresponding spin polarizations as a function of the Fermi energy. [An ideal electrode doping $U=0$ is considered.]{} []{data-label="fig:unbias"}](G-unbiased "fig:"){width="0.95\linewidth"}\
![(color online) (a,b) Linear conductances for spin up (a) and spin down (b) as a function of the Fermi energy. Dashed: $l=2$ and $eV_t^{(i)}=0$, solid: $l=1$ and $eV_t^{(i)}=-2,0,2$. (c) The corresponding spin polarizations as a function of the Fermi energy. [An ideal electrode doping $U=0$ is considered.]{} []{data-label="fig:unbias"}](P-unbiased "fig:"){width="0.95\linewidth"}

With the standard transfer-matrix method,[@born2013principles] $t_s$ can be obtained by matching the spinor envelope functions in [Eqs. (\[wavefunction1\]), (\[wavefunction2\]), and (\[wavefunction3\])]{} at the potential boundaries. The [spin-dependent]{} transmission probability reads $
T_s(E,\alpha,V_b)=|t_s|^{2}
(k_c^d/E-U+eV_b)(k_c^s/E-U)^{-1}
$ for $(k_c^s)^2>0$ and $(k_c^d)^2>0$, and $T_s=0$ otherwise. The spin-dependent net current at zero temperature can be calculated by the Landauer-Büttiker formalism[@buttiker1985generalized] $$\label{eq:current}
I_s(E_F,V_b)=I_0\int_{E_{F}-eV_b}^{E_{F}}\int_{-\pi/2}^{\pi/2}T_s (E,\alpha,V_b)
|{E-U}|
\cos\alpha
d\alpha dE,$$ where $I_0=2ev_{F}W/(2\pi l_0)^2$ is a current unit with the factor 2 coming from the valley degeneracy. Considering the values of $W$ and $l_0$, $I_0=60\mu A$, which is one or two orders of magnitudes larger than those in [devices based on]{} graphene nanoribbons.[@wu2012negative; @jiang2014spin; @yan2014effects; @yang2015half; @xu2014negative; @hong2015axisymmetric; @son2006half; @liu2016efficient; @munarriz2013strong] $(E_F-eV_b,E_F)$ is an energy [integral interval]{}, in which a spin-dependent differential conductance at a given energy can be defined as $g_s(E,V_b)\equiv\int_{-\pi/2}^{\pi/2}T_s(E,\alpha,V_b) \cos\alpha d\alpha$. It is also seen that, $g_s(E,V_b)$ contributes to $I_s(E_F,V_b)$ by a weight of $M(E,W)=2|{E-U}|W/hv_F$, which is the transversal modes number (TMN) at $E$.[@tworzydlo2006sub]

Results and Discussion {#sec:III}
======================

Spin-dependent linear conductance at zero bias {#sec:III.A}
----------------------------------------------

Figure \[fig:unbias\] (a,b) [show]{} the spin-dependent linear conductance through the unbiased device as a function of the Fermi energy: $G_\pm(E_F,V_b=0)=M G_0\int_{-\pi/2}^{\pi/2}T_s(E_F,\alpha,V_b=0) \cos\alpha d\alpha$, where $G_0=e^2/h$ is the quantum conductance. The calculations can be made by following Ref. [@song2017electric]. [Here an ideal contacting ($U=0$) is considered.]{} It is seen that, already for a relatively short device ($l=2$ with $a\rightarrow2a$ and $b\rightarrow2b$, the dashed curve), its linear conductance for each spin shows a good accordance to the corresponding band gap, whose conduction band minimum (CBM) and valence band maximum (VBM) are indicated by the vertical bars [in Fig. \[fig:unbias\] (a,b)]{}. Inside each [energy]{} gap, two resonant peaks $E_+^\pm$ and $E_-^\pm$ arise, [where the subscript $\pm$ stand for spin up and spin down and the superscript $\pm$ stand for the CBM or VBM part]{}. This is a clear significant of resonant tunneling in the device. For a shorter device ($l=$1, the solid [curves]{}) the resonant peaks shift outside and become stronger (indicated by the arrows). The spin-dependent linear conductance at different top gates are also shown. It is seen that, the resonant levels become larger (smaller) as a positive (negative) top gate is applied. This will play an important role in the gate control of the spin-dependent NDR. [Compared]{} with spin up, the resonant tunneling is much weaker for spin down. This is a result of a smaller effective spin Dirac gap, i.e., $\Delta_\textmd{eff}^-=\Delta_-/v_-\approx 3
<\Delta_\textmd{eff}^+=\Delta_+/v_+\approx 5$.[@song2017electric]

The total conductance is spin dependent, which manifests itself clearly in the polarization, which is defined as $P_G=(G_+-G_-)/(G_++G_-) \times 100\%$ and plotted [in Fig. \[fig:unbias\](c)]{}. [In Fig. \[fig:unbias\](c)]{}, near $-100\%$ ($100\%$) polarization can be found in the [energy]{} ranges between CBMs (VBMs) of different spins. They are much higher than spin polarizations obtained in Refs. [@dell2009wave; @zou2009negative] and can be applied as a highly efficient spin conductance filtering. On the other hand, the spin filtering effect is found to be weaken by the resonant tunneling and can be exactly controlled by the top gates.

![(color online) I-V characteristics for (a) spin up and (b) spin down at various Fermi energies with a step of 1. [In (a), $E_F$ changes from 12 to 10, from 9 to 4, and from 3 to 0 for curves along the red, black, and blue arrows, respectively. In (b), $E_F$ decreases from 12 to 0 for curves along the arrow. In both (a) and (b), the curves with no NDR features are plotted in dashed. The other device parameters are $l=1$, $eV_t^{(i)}=0$, and $U=0$.]{} []{data-label="fig:biased"}](spin-up "fig:"){width="0.95\linewidth"}\
![(color online) I-V characteristics for (a) spin up and (b) spin down at various Fermi energies with a step of 1. [In (a), $E_F$ changes from 12 to 10, from 9 to 4, and from 3 to 0 for curves along the red, black, and blue arrows, respectively. In (b), $E_F$ decreases from 12 to 0 for curves along the arrow. In both (a) and (b), the curves with no NDR features are plotted in dashed. The other device parameters are $l=1$, $eV_t^{(i)}=0$, and $U=0$.]{} []{data-label="fig:biased"}](spin-dn "fig:"){width="0.95\linewidth"}

Spin-dependent NDR at finite bias {#sec:III.B}
---------------------------------

The spin-dependent I-V [characteristics]{} of the device [with]{} various Fermi energies are plotted in Fig. \[fig:biased\](a,b). It is seen in Fig. \[fig:biased\](a) that, for Fermi energies around the resonant level $E_+^+=7.37$ (black solid), the [spin-up]{} current displays obvious NDR. However, for Fermi energies much higher ($>9$, red dashed) or smaller ($<3.5$, blue dashed) than the resonant [level]{}, the NDR features disappear. The case is the same for spin down as shown in Fig. \[fig:biased\](b), where NDR features [are found only for]{} Fermi energies around the resonant level $E_-^+ = 3.77$.

![(color online) Differential conductance as a function of the energy for (a) spin up and (b) spin down at several typical biases. [Curves with biases in the NDR region are plotted in solid: red for current peak, blue for current valley, and black for between. Curves with biases out of the NDR region are plotted in dashed: red for zero bias, black for bias before the current peak, and blue for bias after the current valley. The other device parameters are $l=1$, $eV_t^{(i)}=0$, and $U=0$.]{} []{data-label="fig:dc"}](resonant-peak "fig:"){width="0.95\linewidth"}\
![(color online) Differential conductance as a function of the energy for (a) spin up and (b) spin down at several typical biases. [Curves with biases in the NDR region are plotted in solid: red for current peak, blue for current valley, and black for between. Curves with biases out of the NDR region are plotted in dashed: red for zero bias, black for bias before the current peak, and blue for bias after the current valley. The other device parameters are $l=1$, $eV_t^{(i)}=0$, and $U=0$.]{} []{data-label="fig:dc"}](resonant-peak-dn "fig:"){width="0.95\linewidth"}

![image](control){width="0.49\linewidth"} ![image](both){width="0.49\linewidth"}

To understand these interesting behaviors, we plot the [spin-up]{} differential conductance as a function of energy in Fig. \[fig:dc\](a). Several typical biases in the strongest NDR case ($E_F=5.5$) are considered. It is seen that, for Fermi energies around the resonant level, the transport under the considered biases are dominated by the resonant state. As the bias increases from zero (the $eV_b=$ 0 and 5.4 [curves]{}), the resonant peak shifts into the [integral interval]{} (the origin of which, $E_F$, is indicated by the vertical bar) with an increasing peak value. This [results in]{} an increase behavior of the spin-up current with an increasing bias in the device. The [spin-up]{} current reaches its maximum at a bias of $eV_P=8.8$, where the resonant peak has totally inserted into the [integral interval]{}. As indicated above, the spin-up resonant state contributes to the spin-up current by a weight of TMN$\propto|E|$ [(where $U=0$)]{}, see Eq. (\[eq:current\]). As the bias increases further (the $eV_b=$11.2 [curve]{}), the TMN decreases, which results in a decrease behavior of the [spin-up]{} current with an increasing bias, i.e., the desired NDR feature. The TMN decreases [with the bias]{} because, as the bias increases the *positive* resonant level [is pulled down]{} and gets closer to the Dirac point in the source, i.e., $|E|=E$ becomes smaller as $E$ becomes smaller. The spin-up current achieves its minimum at a larger bias ([the]{} $eV_V=14.1$ [curve]{}), where the [pull-down]{} resonant level almost aligns to the Dirac point in the source and the TMN approaches [near]{} zero. However, as the bias increases further (the $eV_b=$16.5 [curve]{}), the [pulling down]{} of a *negative* resonant level leads to an increase of the TMN, i.e., $|E|=-E$ becomes bigger as $E$ becomes smaller. As a result, the [spin-up]{} current increases again.

On the other hand, for Fermi energies far from the [spin-up]{} resonant level, the biased transport is also contributed by [a]{} quasi-ballistic state outside the spin Dirac gap, see Fig. \[fig:dc\](a). As a result, the bias-induced decrease of the TMN ([integral weight]{}) is counteracted by the bias-induced increase of the [integral interval]{}, and the NDR feature disappears. For spin down, the NDR is much less pronounced. This stems from a more expanding resonant peak, which is shown in Fig. \[fig:dc\](b), where several typical biases for the strongest NDR case ($E_F=4.0$) are considered.

Top-gate control of the spin-dependent NDR
------------------------------------------

The spin-dependent I-V characteristics at various top gates are shown in Fig. \[fig:control\](a,b). Here $E_F=5.5$ is focused, for which both spins display obvious NDRs. It is seen that, as the top gates increase in the considered range, spin up displays NDR (the -6 to 2 [curves]{}), no NDR (the 4 to 8 [curves]{}), and NDR (the 10 to 14 [curves]{}), respectively. In contrast, spin down shows no NDR (the -6 to -2 [curves]{}), NDR (the 0 to 10 [curves]{}), and no NDR (the 12 to 14 [curves]{}), respectively. This is according to that the resonant levels $E_+^+$, $E_-^+$, and $E_+^-$ are sequentially [lifted up]{} around the Fermi energy and [dominate]{} the biased transport. For each spin, the peak-to-valley currents ratio (PVR) are summarized in Fig. \[fig:control\] (c) as a function of the top gates. It is seen that, according to the occurrence of the spin-dependent NDR only at certain top gates, the PVR shows a non-monotonous dependence on the top gates, increasing from 1 (no NDR) to certain value and then decreasing to 1 again. The maximal PVR for each spin is found at certain top gates that the lifted resonant level is slightly lower than the Fermi energy. For spin up, the value reads 2.5, which is comparable with those obtained in experiments on spin-independent NDRs.[@britnell2013resonant; @sharma2015room]

Although the NDR happens at different top gates for spin up and spin down, NDRs for both spins can be achieved for certain top gates, [see the green windows in Fig. \[fig:control\] (c) and the $eV_t^{(i)}=0,2$ curves in Fig. \[fig:control\](a,b). In Fig. \[fig:control\] (a,b),]{} it is also observed that the NDR [regions]{} for different spins do not overlap. As a result, the proposed device can be applied as *a [dual-spin]{} Esaki diode*: at low [biases]{} spin down displays NDR and at high [biases]{} spin up displays NDR. The [dual-spin]{} NDR occurs because the resonant levels are lifted up that the Fermi energy lies between $E_+^+$ and $E_-^+$ ($E_-^+$ and $E_+^-$). For top gates higher (lower) than the left green [window in Fig. \[fig:control\](c)]{}, i.e., the red (blue) windows, only spin down (spin up) shows NDR while the current for the other spin increases monotonously with the bias. This can be applied as *a [single-spin]{} Esaki diode*. Hence, as the top gates increases from -7 to 15, the proposed device works consecutively as a spin-up only, dual-spin, spin-down only, dual-spin, and spin-up only Esaki diode. In other words, spin-dependent NDR with spin selectivity can be realized in the proposed device, by simply [changing]{} the top gates.

![image](contacting){width="0.49\linewidth"} ![image](pvr-contacting){width="0.49\linewidth"}\

The NDR [regions]{} ($eV_P^\pm,eV_V^\pm$) for each spin [are summarized in Fig. \[fig:control\] (d) as a function of the top gates]{}. It is seen that, when the top gates increase from -7.5 to -1.2 (the left blue window in Fig. \[fig:control\] [(d)]{}), the center of the [spin-up NDR region]{} shifts monotonously from $\sim$3.3 to $\sim$9.7. This implies that, the operation [region]{} of the spin-up Esaki diode is variable in a large range of $\sim$ 64meV. When $eV_t^{(i)}$ increase from -1.2 to 3.7 (the left green window in Fig. \[fig:control\] [(d)]{}), both the [spin-up and spin-down NDR regions]{} of the dual-spin Esaki diode are tunable. The variable ranges are $\sim53$ meV and $\sim33$ meV, respectively. For top gates changed between 3.8 and 9 (the red window in Fig. \[fig:control\] (d)), the operation [region]{} of the spin-down Esaki diode is monotonously tunable in a range of $\sim$45 meV.

The spin selectivity and [region]{} tunability of the spin-dependent NDR can be understood as follows. As the top gates increase, the spin-dependent resonant levels formed between the [ferromagnetic double barriers]{} are lift. As a result, $E_+^+$, $E_-^+$, and $E_+^-$ are consecutively closed to the fixed Fermi energy hence [dominate]{} the biased transport, solely or simultaneously. The former leads to a [single-spin]{} Esaki diode while the latter results in a [dual-spin]{} Esaki diode. On the other hand, for each resonant level dominating the [biased]{} transport, its [deviation]{} from the Dirac point in the source enlarges with the increasing top gates, see Fig. \[fig:unbias\](a,b). From the NDR mechanism discussed in Sec. \[sec:III.B\], the decrease of the TMN ([integral]{} weight) hence the spin-dependent NDR should be happen at a higher bias. In Fig. \[fig:biased\] the spin-dependent NDR [region is]{} also found to be controlled by the Fermi energy. However, the change is non monotonous.

Spin oscillator, spin amplifier, spin switching, and spin memory can be realized based on a spin-dependent NDR. The gate-induced spin selectivity in the proposed device adds a spin degree of freedom to these spin devices: by simply [changing]{} the top gates as low, high, and medium values, spin devices for spin up only, spin down only, or both spins (occurring sequentially) can be respectively realized. On the other hand, the gate-induced [region]{} tunability in the proposed device further adds an output degree of freedom to these spin devices. This may be especially important in [spin switchinges and spin memories]{}, whose lowest output voltage and two stored states are found to be exactly determined by the position of the NDR [regions]{} (see Ref. [@mizuta2006physics] and relevant references therein).

Electrode doping effect
-----------------------

[ We at last investigate the electrode doping effect on the spin-dependent NDR. Fig. \[fig:electrode\](a,b) show the spin-dependent I-V characteristics at various light contact doping’s. The spin-dependent PVR and NDR region ($eV_P^\pm,eV_V^\pm$) are summarized in Fig. \[fig:electrode\](c,d) as a function of the contact doping. A dual-spin NDR ($E_F=5.5$ and $V_t^{(i)}=2$) is considered. As can be seen, the dual-spin NDR only maintains for -23meV$<U<$44meV. For $U$ smaller than -23meV or larger than 44meV, the NDR becomes spin-down only or spin-up only. The single-spin NDR feature even disappears for $U$ smaller than -37meV or larger than 50meV. However, for familiar metal electrodes such as Ag, Ti, Cu, Au, and Pt at their *equilibrium distances* to graphene, the contact doping’s are -320meV, -230meV, -170meV, 250meV, and 320meV, respectively.[@lee2008contact; @giovannetti2008doping] These values are much larger than the doping range ($-23$meV$<U<$44meV) the dual-spin NDR can survive. To maintain the proposed NDR, one possible way is to tune the metal-graphene distances to special values (e.g., Au/Cu/Ag at 3.2/3.4/3.7 $\AA$) to obtain ideal doping’s.[@giovannetti2008doping] Another possible way is to adopt improved contacts such as Ti/Au. [@lee2008contact; @robinson2011contacting] ]{}

[ In Fig. \[fig:electrode\](a,b) it is also clear that, both the peak and valley currents decrease with an increasing doping. This is a result of that, the TMNs for the dominating resonant states ($|E_\pm^+-U|=E_\pm^+-U>0$) decrease with the doping $U$. However, the peak and valley currents decrease differently. As shown in Fig. \[fig:dc\], the peak current is dominated by the resonant peak away from the Dirac point, so as the Dirac point is lifted up it decreases first gently and then sharply, see Fig. \[fig:electrode\](a); in contrast, the valley current is dominated by the resonant peak around the Dirac point, so as the Dirac point is lifted up it decreases first sharply and then gently. As a result of this contrast, the PVR first increases and then decreases with an increasing doping, see Fig. \[fig:electrode\](c). (The spin-down PVR shows an exception for -10meV$<U<$0meV because the resonant peak for the peak current extends considerably to the Dirac point.) Importantly, the PVR for the spin-up NDR increases from 1.92 at an ideal doping to 2.89 at a light hole doping of 22meV, with a large enhancement factor of 51$\%$. For spin down, the PVR of 1.15 at an ideal contacting approaches its maximum of 1.27 at a light hole doping of 15meV, with an enhancement factor of 10$\%$. The above specific metal-graphene distances can be enlarged a little to achieve these light hole doping’s[@giovannetti2008doping] and larger PVR’s. ]{}

[ In Fig. \[fig:electrode\](d) it is seen that, the NDR regions are rather sensitive to the contact doping’s, with variable ranges of 84 meV and 110 meV for spin up and spin down, respectively. The larger the contact doping, the lower the NDR regions. This is because as $U$ increases, the Dirac node in the source gets closer to the resonant levels and the biases needed to align them become smaller. However, using the contact doping instead of the top gates to tune the NDR regions would be much harder.]{}

Conclusion {#sec:IV}
==========

In summary, we have proposed a bulk graphene based, spin-dependent NDR device, which is composed of a sufficiently wide and short graphene and two gated EuO strips deposited on top of it. [An ideal or light hole electrode doping is also essential.]{} The spin-dependent NDR arises because the energies (transversal mode numbers) of the spin-dependent resonant states decrease with an increasing bias. Compared with spin-dependent NDR devices based on zigzag or armchair GNRs, the proposed device holds the advantages of no need of edge tailors, generation of larger currents, and most importantly, features of a spin selectivity and a [region]{} tunability. These remarkable features stem from a top-gate control of the deviations of the spin-dependent resonant levels from the Fermi energy and [the]{} Dirac point in the source electrode, respectively. They add a spin and a bias degree of freedom to [conventional]{} NDR-based devices, which paves a way for design of a whole new class of NDR circuits such as spin-selectable and output-tunable spin switching and spin memory.

Acknowledgements {#acknowledgements .unnumbered}
================

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11404300 and the S$\&$T Innovation Fund of IEE, CAEP under Grant No. S20140807.

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[^1]: Corresponding author