⇦Chinese Physics Letters, 2017, Vol. 34, No. 11, Article code 118503 Fano Factor in Strained Graphene Nanoribbon Nanodevices Walid Soliman1, Mina D. Asham1**, Adel H. Phillips2 Affiliations 1Faculty of Engineering, Benha University, Benha, Egypt 2Faculty of Engineering, AinShams University, Cairo, Egypt Received 1 September 2017 **Corresponding author. Email: minadanial@yahoo.com Citation Text: Soliman W, Asham M D and Phillips A H 2017 Chin. Phys. Lett. 34 118503 Abstract We investigate the Fano factor in a strained armchair and zigzag graphene nanoribbon nanodevice under the effect of ac field in a wide range of frequencies at different temperatures (10 K–70 K). This nanodevice is modeled as follows: a graphene nanoribbon is connected to two metallic leads. These two metallic leads operate as a source and a drain. The conducting substance is the gate electrode in this three-terminal nanodevice. Another metallic gate is used to govern the electrostatics and the switching of the graphene nanoribbon channel. The substances at the graphene nanoribbon/metal contact are controlled by the back gate. The photon-assisted tunneling probability is deduced by solving the Dirac eigenvalue differential equation in which the Fano factor is expressed in terms of this tunneling probability. The results show that for the investigated nanodevice, the Fano factor decreases as the frequency of the induced ac field increases, while it increases as the temperature increases. In general, the Fano factors for both strained armchair and zigzag graphene nanoribbons are different. This is due to the effect of the uniaxial strain. It is shown that the band structure parameters of graphene nanoribbons at the energy gap, the C–C bond length, the hopping integral, the Fermi energy and the width are modulated by uniaxial strain. This research gives us a promise of the present nanodevice being used for digital nanoelectronics and sensors. DOI:10.1088/0256-307X/34/11/118503 PACS:85.30.De, 72.80.Vp, 85.40.Qx, 73.50.Pz © 2017 Chinese Physics Society Article Text Two-dimensional (2D) crystalline materials have recently been identified and analyzed.[1,2] The first material in this new class is graphene, a single atomic layer of carbon. This new material has a number of unique properties which make it interesting for both fundamental studies and future applications.[3] Graphene has a number of applications in the field of nanotechnology. It is an ultimately thin, mechanically very strong, transparent and flexible conductor.[3-6] Its conductivity can be modified over a large range either by chemical doping or by an electric field.[3-6] The mobility of graphene is very high,[7] which makes the material very interesting for nano-electronic high frequency applications.[8] Flexible electronics and gas, mass sensors[9-11] are the most promising potential applications. Graphene nanoribbons[12-14] are of increasing interest due to their promise of a band gap, overcoming the gapless band structure of truly 2D graphene.[2,3] In particular, their overall semiconducting behavior allows the fabrication of graphene field effect transistors,[15] tunneling barriers,[16] and quantum devices.[17] Graphene nanoribbons have one-dimensional structures with hexagonal two dimensional carbon lattices, which are stripes of graphene.[18] Their structures and their electronic and magnetic properties have been intensively studied both experimentally and theoretically. Due to their various edge structures (armchair and zigzag), graphene nanoribbons present different electronic properties ranging from normal semiconductors to spin-polarized half metals.[18,19] The effects of uniaxial strain on the electronic properties of graphene nanoribbons have been extensively studied.[20,21] These studies revealed the potential of uniaxial strain as a way of tuning the electronic properties of graphene nanoribbons. It is widely recognized that time-dependent current fluctuations due to the discreteness of the electrical charges, the shot noise power, provide further physical information of an electronic system than other transport quantities such as the conductance. For instance, the shot noise takes into account the Pauli principle, thus it can reveal electron correlations. Markus et al.[22] recognized the importance of the shot noise for understanding the problem of quantum electron transport and made major contributions to this topic. Effects of the presence of disorder on the shot noise in graphene sheets have been experimentally and theoretically investigated.[23-26] It has been found that the Fano factor (the ratio of noise power and mean current) is affected by the strength of the disorder.[27,28] We have investigated the quantum transport characteristics and thermoelectric effect of a graphene nanoribbon field effect transistor under the influence of both the ac field with different frequencies and magnetic field, and studied the effect of an external tensile strain on the quantum transport characteristics of graphene nanoribbons.[29,30] In this Letter, we investigate the noise Fano factor in strained armchair and zigzag graphene nanoribbon field effect transistors at different temperatures and in a wide range of frequencies of the induced ac field. Graphene nanoribbon nanodevices are modeled as follows:[29,30] a graphene nanoribbon is connected to two metallic leads (Fig. 1). These two metallic leads operate as a source and a drain. The conducting substance is the gate electrode in this three-terminal nanodevice. Another metallic gate is used to govern the electrostatics and the switching of the graphene nanoribbon channel. The substances at the graphene nanoribbon/metal contact are controlled by the back gate.
cpl-34-11-118503-fig1.png
Fig. 1. The schematic diagram of the modeled graphene nanoribbon nanodevice.
The Dirac fermion charge carrier tunneling through the present investigated nanodevice is induced by an external applied ac field given by $$ V=V_{\rm ac}\cos(\omega t),~~ \tag {1} $$ where $V_{\rm ac}$ is the amplitude of the applied ac field, and $\omega$ is its frequency. The Fano factor $F$ is defined as the ratio of noise power and mean current,[22] given by[31,32] $$\begin{align} &F=\Big[\int\limits_{E_{\rm F}}^{E_{\rm F} +n\hbar \omega} \!\!\!\!\!\!dE\int \!\!{\it \Gamma} _{\rm withphotons} (E)[1-{\it \Gamma} _{\rm withphotons} (E)]\\ &\cdot({f_{\rm L} -f_{\rm R}})^2\cos \varphi\cdot d\varphi\Big]\Big[\int\limits_{E_{\rm F}}^{E_{\rm F} +n\hbar \omega} dE\int {\it \Gamma} _{\rm withphotons} (E)\\ &\cdot(f_{\rm L} -f_{\rm R})\cos \varphi\cdot d\varphi\Big]^{-1},~~ \tag {2} \end{align} $$ where $\phi$ is the incident angle on the graphene nanoribbon, $f_{\rm L}$ and $f_{\rm R}$ are the left and right Fermi–Dirac distribution functions, $$\begin{align} f_{\rm L} =\,&\frac{1}{1+\exp ({\frac{E-E_{\rm F} +n\hbar \omega}{k_{\rm B} T}})},~~ \tag {3} \end{align} $$ $$\begin{align} f_{\rm R} =\,&\frac{1}{1+\exp ({\frac{E-E_{\rm F} +n\hbar \omega +eV_{\rm sd}}{k_{\rm B} T}})}.~~ \tag {4} \end{align} $$ In Eq. (2) ${\it \Gamma}_{\rm withphotons}(E)$ is the photon-assisted tunneling probability, which will be determined by solving the Dirac eigenvalue differential equation, is given by[29,30] $$\begin{align} H{\it \Psi} =E{\it \Psi},~~ \tag {5} \end{align} $$ where $E$ is the scattered energy of quasiparticle Dirac fermions, and the Hamiltonian $H$ is expressed as $$\begin{alignat}{1} H=H_0 +eV_{\rm sd} +eV_{\rm g} +eV_{\rm ac} \cos (\omega t)+\frac{\hbar eB}{2\,m^\ast},~~ \tag {6} \end{alignat} $$ where $V_{\rm sd}$ is the bias voltage, $h$ is the reduced Planck constant, and $m^{\ast}$ is the effective mass of quasiparticle Dirac fermions in graphene nanoribbons. Now, due to the transmission of these quasiparticles Dirac fermions through the present investigated nanodevice, a transition from the central band to side-bands of graphene nanoribbons is at energies equal to $E\pm n\hbar \omega$,[29,30,33] where $n$ is an integer with values $0,\pm 1,\pm 2,\ldots$. The transport of quasiparticle Dirac fermions in graphene nanoribbon nanodevices is described by the following Dirac Hamiltonian $H_{0}$, which is given as[29,30] $$\begin{align} H_0 =-i\hbar v_{\rm f}\sigma\nabla +V_{\rm b},~~ \tag {7} \end{align} $$ where $v_{\rm f}$ is the Fermi velocity, $\sigma=(\sigma_{x}, \sigma_{y})$ are the Pauli matrices, and $V_{\rm b}$ is the barrier height. Since the present graphene nanoribbon is connected to two metallic leads and applying a top gate with gate voltage $V_{\rm g}$, the transports of quasiparticle Dirac fermions are influenced by applying both an ac field and a magnetic field $B$. The solution to Eq. (5) gives the following eigenfunctions.[29,30] The eigenfunction of incident quasiparticle Dirac fermions is $$\begin{align} {\it \Psi} _{\rm in} (x,y;t)=\,&\sum\limits_{n=1}^\infty J_n \Big(\frac{eV_{\rm ac}}{n\hbar \omega}\Big) \Big[\Big(\frac{1}{se^{i\varphi}}\Big)\exp (i(k_x x\\ &+k_y y))+r\Big(\frac{1}{-se^{-i\varphi}}\Big)\exp (i(-k_x x\\ &+k_y y))\Big]e^{-in\omega t}.~~ \tag {8} \end{align} $$ The eigenfunction of the transmitted quasiparticle Dirac fermions is $$\begin{align} {\it \Psi} _{\rm tr} (x,y;t)=\,&\sum\limits_{n=1}^\infty J_n \Big({\frac{eV_{\rm ac}}{n\hbar \omega}}\Big) \Big[t\Big(\frac{1}{se^{i\varphi}}\Big)\\ &\cdot\exp (i(k_x x+k_y y))\Big]e^{-in\omega t},~~ \tag {9} \end{align} $$ where $r$ and $t$ are the reflection and transmission amplitudes, $s={\rm sgn}(E)$ is the signum function of $E$, and $\phi$ is the angle of incidence of the quasiparticle Dirac fermions (see Eqs. (8) and (9)). The wave vectors $k_{x}$ and $k_{y}$ are expressed in terms of $\phi$ as $$\begin{align} k_x =k_{\rm f} \cos \varphi,~~~ k_y =k_{\rm f} \sin \varphi,~~ \tag {10} \end{align} $$ where $k_{\rm f}$ is the Fermi wave vector. It is well known that electrons will be laterally confined in narrow graphene nanoribbons, similar to constrictions in conventional 2DEGs. The narrow width yields the quantization of the wave vector in the lateral direction: $k_y W\approx m'\pi$ (for $m'\geqslant 1$). Here a squarewell potential equal to the nanoribbon width $W$ is presumed.[29,30,34] The wave vector $k_{y}$ has the following form $$\begin{align} k_y ={{m}'\pi}/{W}.~~ \tag {11} \end{align} $$ The eigenfunction inside the region of the barrier is given by $$\begin{align} &{\it \Psi} _{\rm b} (x,y;t)=\sum\limits_{n=1}^\infty J_n \Big(\frac{eV_{\rm ac}}{n\hbar \omega}\Big) \Big[\Big(\frac{\alpha}{{s}'\beta e^{i\theta}}\Big)\exp (i(q_x x\\ &+k_y y))+ \Big(\frac{\alpha}{-{s}'\beta e^{-i\theta}}\Big)\exp (i(-q_x x +k_y y))\Big]e^{-in\omega t},~~ \tag {12} \end{align} $$ where ${s}'={\rm sgn}(E-V_{\rm b})$, $\theta =\tan ^{-1}({\frac{k_y}{q_x}})$, and $q_x =({k'}_{\rm f}^2 -k_y^2)^{\frac{1}{2}}$. The wave vector ${k'}_{\rm f}$ is given by $$\begin{align} {k'}_{\rm f} =\sqrt {\frac{(V_{\rm b} -{E}')^2-({E_{\rm g}^2}/{2})}{(\hbar v_{\rm f})^2}},~~ \tag {13} \end{align} $$ where $E_{\rm g}$ is the energy gap, and the parameter ${E}'$ is expressed as $$\begin{alignat}{1} {E}'=E-eV_{\rm sd} -eV_{\rm g} -V_{\rm b} -n\hbar \omega +\frac{\hbar eB}{2m^\ast}.~~ \tag {14} \end{alignat} $$ In Eqs. (8), (9) and (12), $J_{n}$ is the $n$th order Bessel function. This solution must be generated by the presence of different side-bands $n$, which come with phase factor $\exp(-in\omega t)$ that shifts the center energy of the transmitted quasiparticle Dirac fermions by integer multiples of $h\omega$.[35] The parameters $\alpha$ and $\beta$ in Eq. (12) are given by $$\begin{align} \alpha =\,&\sqrt {1+\frac{{s}'({E_{\rm g}}/{2\hbar v_{\rm f}})}{\sqrt {({E_{\rm g}^2}/{4(\hbar v_{\rm f})^2}+{k'}_{\rm f}^2)}}},~~ \tag {15} \end{align} $$ $$\begin{align} \beta =\,&\sqrt {{1-\frac{{s}'({E_{\rm g}}/{2\hbar v_{\rm f}})}{\sqrt {({E_{\rm g}^2}/{4(\hbar v_{\rm f})^2}+{k'}_{\rm f}^2)}}}}.~~ \tag {16} \end{align} $$ The parameters $\alpha$ and $\beta$ correspond to $K$ and ${K}'$ points, respectively. Now, applying the boundary conditions (8), (9) and (12) at the boundaries of the barrier, we obtain the tunneling probability ${\it \Gamma}_{withphotons}(E)$ as follows: $$\begin{align} &{\it \Gamma} _{\rm withphotons} (E)=\sum\limits_{n=1}^\infty J_n^2 \Big(\frac{eV_{\rm ac}}{n\hbar \omega}\Big)\big|[\cos (q_x d)\\ &-i(ss'\eta\sec (\varphi)\sec (\theta) +\tan (\varphi)\tan (\theta))\sin (q_x d)]\big|^{-2} ,~~ \tag {17} \end{align} $$ where $d$ is the barrier width. The parameter $\eta$ is given by $$ \eta =\frac{\sqrt{\big[E^2_{\rm g}/(4(\hbar v^2_{\rm f}))\big] +{k'_{\rm f}}^2}}{k'_{\rm f}}.~~ \tag {18} $$ Substituting Eq. (17) into Eq. (2) and the complete equation for the Fano factor $F$ will be solved numerically as it will be shown in the following. Since we intend to compute the conductance of the present nanodevice under the influence of strain, then the band gap of both armchair and zigzag graphene nanoribbons varies with strain. The relation between the energy gap of the armchair graphene nanoribbons and the external strain[36] is given by $$\begin{align} E_{\rm g} =2\Big| {\gamma _1 +2\gamma _2 \cos ({\frac{p\pi}{N+1}})}\Big|,~~ \tag {19} \end{align} $$ where $\gamma_{1}$ and $\gamma_{2}$ are the strained hopping parameters, $p$ is an integer, and $N$ is the number of dimmer lines across the ribbons. The strained hopping parameters $\gamma_{1}$ and $\gamma_{2}$ are related to the unstrained hopping parameter $\gamma_{\rm o}$ by[36] $$\begin{align} \gamma _1 =\frac{\gamma _{\rm o}}{({1+\varepsilon})^2},~~~\gamma _2 =\frac{\gamma _{\rm o}}{({1+\frac{\varepsilon}{4}})},~~ \tag {20} \end{align} $$ where $\varepsilon$ is the uniaxial strain. Also, the relation between the energy gap $E_{\rm g}$ of the zigzag graphene nanoribbon and the strain $\varepsilon$ is given by[37] $$\begin{align} E_{\rm g} =3\gamma _{\rm o} S_{\rm t} (1+\nu _{\rm z}) \varepsilon,~~ \tag {21} \end{align} $$ where $S_{\rm t}$ is a constant and its value is 1.29,[37] and $\nu_{\rm z}$ is Poisson's ratio and its value is 0.16. The variation of the effective mass, $m^{\ast}$, of the quasiparticle Dirac fermion with the energy gap of the graphene nanoribbon is given by[38] $$\begin{align} m^\ast =\frac{2\hbar ^2}{9\gamma _{\rm o}^2 a^2}E_{\rm g},~~ \tag {22} \end{align} $$ where $a$ is the equilibrium bond length of the graphene nanoribbon. Numerical calculations are performed for both strained armchair and zigzag graphene nanoribbons with certain widths. This can be achieved by computing the Fano factor $F$ (see Eq. (2)) under the effects of the induced ac field with different values of frequency and different temperatures. In the calculation, the barrier height $V_{\rm b}$ = 0.12 eV and its width $d=10$ nm, the unstrained nearest neighbor hopping parameter $\gamma _{0}$ = 2.7 eV, the unstrained C–C bond length equals 0.142 nm, and the angle of incidence of the quasiparticle Dirac fermions $\phi$ equals 1.335 rad.[29,30,33] The unstrained Fermi-energy $E_{\rm F}$ is determined in terms of $\gamma_{0}$ by the equation $E_{\rm F}=0.072\gamma$.[1] It must be noted that the hopping integral $\gamma_{0}$ between the $\pi$ orbitals of armchair graphene nanoribbon (AGNR) is altered upon strain. This causes the up and down shift of the $\sigma^{\ast}$ band to the Fermi level $E_{\rm F}$.[39] The value of the applied magnetic field $B$ is 0.5 T. The strained hopping integral $\gamma'$ is related to the unstrained hopping integral $\gamma_{0}$ through[40,41] $$\begin{align} {\gamma}'=\gamma _{0} (a/a')^2,~~ \tag {23} \end{align} $$ where $a$ is the unstrained C–C bond length, and ${a}'$ is the strained C–C bond length. An armchair ribbon is cut so that the edge looks as if it consists of repeated armchairs.[42] Each edge is terminated by atoms of the A- and B-sublattice. The width of an armchair ribbon can be defined in terms of the number of dimer lines $N_{\rm a}$, $$\begin{align} W_{\rm a} =({N_{\rm a} -1})\frac{\sqrt 3 {a}'}{2}.~~ \tag {24} \end{align} $$ Also, for zigzag graphene nanoribbons (ZGNR), the atoms at one edge are of the same sublattice, e.g., A atoms at the left edge and B atoms at the right edge. The width of a zigzag ribbon is now identified with the number of zigzag chains $N_{\rm z}$ as[42] $$\begin{align} W_{\rm z} =({N_{\rm z} -1})\frac{3{a}'}{2}.~~ \tag {25} \end{align} $$ Now, the effect of uniaxial strain is modeled as a modification to the tight-binding nearest neighbor hopping integral (see Eq. (23)) and the corresponding strained C–C bond length, and the corresponding widths $W_{\rm a}$ and $W_{\rm z}$ (see Eqs. (24) and (25)) to give optimum values.[42] Also, by applying moderate uniaxial strain, ($\varepsilon=15{\%}$), to estimate the energy band gaps of both armchair and zigzag graphene nanoribbons (see Eqs. (19) and (21)) and the effective mass $m^{\ast}$ (see Eq. (22)). The calculations are performed to simulate Eqs. (19) and (21) and using the density functional theory.[42] The features of the results for the Fano factor $F$ (see Eq. (2)) for both strained armchair and zigzag graphene nanoribbons are obtained. Figures 2(a)–2(d) show the variation of the Fano factor $F$ with the gate voltage $V_{\rm g}$ for strained armchair graphene nanoribbons at different frequencies of $f=10^{11}$ Hz and $f=10^{12}$ Hz and at different temperatures of 10 K, 30 K, 50 K and 70 K.
cpl-34-11-118503-fig2.png
Fig. 2. Fano factor $F$ with gate voltage at two different frequencies and at temperatures (a) 10 K, (b) 30 K, (c) 50 K, (d) 70 K, for strained armchair graphene nanoribbons.
As shown in Fig. 2(a) that for temperature $T=10$ K, the peaks of the Fano factor are $F_{\max}=0.331$ for frequency $f=10^{11}$ Hz and $V_{\rm g}=0.41$ V, while for $f=10^{12}$ Hz and $V_{\rm g}=-0.4$ V, $F_{\max}=4.1215\times10^{-3}$. The results in Fig. 2(b) show that for $T=30$ K, the peaks of the Fano factor are $F_{\max}=0.4406$ for frequency $f$=10$^{11}$ Hz and $V_{\rm g}=0.45$ V, while for $f=10^{12}$ Hz and $V_{\rm g}=-0.4$ V, $F_{\max}=0.1051$. The results in Fig. 2(c) show that for $T=50$ K, the peaks of the Fano factor are $F_{\max}=0.4642$ for frequency $f=10^{11}$ Hz and $V_{\rm g}=0.45$ V, while for $f=10^{12}$ Hz and $V_{\rm g}=-0.4$ V, $F_{\max}=0.2046$. The results in Fig. 2(d) show that for $T=70$ K, the peaks of the Fano factor are $F_{\max}=0.4743$ for frequency $f=10^{11}$ Hz and $V_{\rm g}=0.45$ V, while for $f=10^{12}$ Hz and $V_{\rm g} =-0.4$ V, $F_{\max}=0.2705$.
cpl-34-11-118503-fig3.png
Fig. 3. Fano factor $F$ with gate voltage at two different frequencies and at temperatures (a) 10 K, (b) 30 K, (c) 50 K, (d) 70 K, for strained zigzag graphene nanoribbons.
Figure 3 shows the variation of Fano factor $F$ with the gate voltage $V_{\rm g}$ for strained zigzag graphene nanoribbons at different frequencies $f=10^{10}$ Hz and $f=10^{11}$ Hz and at different temperatures of 10 K, 30 K, 50 K and 70 K. The results in Fig. 3(a) show that for $T=10$ K, the peaks of the Fano factor are $F_{\max}=0.481$ for frequency $f=10^{10}$ Hz and $V_{\rm g}=0.35$ V, while for $f=10^{11}$ Hz and $V_{\rm g}=-0.4$ V, $F_{\max}=0.331$. The results in Fig. 3(b) show that for $T=30$ K, the peaks of the Fano factors are $F_{\max}=0.494$ for frequency $f=10^{10}$ Hz and $V_{\rm g}=0.35$ V, while for $f=10^{11}$ Hz and $V_{\rm g}=0.4$ V, $F_{\max}=0.4406$. The results in Fig. 3(c) show that for $T=50$ K, the peaks of the Fano factor are $F_{\max}=0.4964$ for frequency $f=10^{10}$ Hz and $V_{\rm g}$=0.45 V, while for $f=10^{11}$ Hz and $V_{\rm g}=0.45$ V, $F_{\max}=0.4641$. The results in Fig. 3(d) show that for $T=70$ K, the peaks of the Fano factor are $F_{\max}=0.4972$ for frequency $f=10^{10}$ Hz and $V_{\rm g}=0.45$ V, while for $f=10^{11}$ Hz and $V_{\rm g}=0.45$ V, $F_{\max}=0.4741$. The results (Figs. 2 and 3) show that for both strained armchair and zigzag graphene nanoribbons, the Fano factor $F$ decreases as the frequency of the induced ac field increases, while $F$ increases as the temperatures of both investigated nanodevices increase. The decrease of Fano factor $F$ when the frequency of the induced ac field increases might be explained as follows: the photon energy is invoked as a useful energy measurement scale, and the side-bands are induced by the ac field in the THz range provide additional quantum channels for the Dirac fermion electrons to tunnel through the present investigated graphene nanoribbon nanodevice.[43,44] Also the values of the Fano factor are quite in agreement with those in Refs. [26,27,45–49]. We notice that the values of Fano factor for both strained armchair and zigzag graphene nanoribbons are different. This is due to the different uniaxial strain effects on the electronic structure of them.[29,30] In summary, we have investigated the Fano factor in strained armchair and zigzag graphene nanoribbons at different frequencies of the induced ac field and at different temperatures. The results show that the values of Fano factor for both graphene nanoribbons increase with the temperature. Also, the Fano factor decreases with the increase of the frequency of the induced ac field. Accordingly, both strained armchair and zigzag graphene nanoribbon nanodevices might be promised to nanotechnology applications, namely, photo-detectors in the THz region. This is a consequence of the moderate values of Fano factor in the range of frequencies. Our results present useful information on shot noise in strongly correlated nano-devices, and the results also provide additional information on coherent transport other than the conductance and tunneling current. 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