Strain effects on the quantum capacitance of graphene nanoribbon devices

Applied Surface Science 502 (2020) 144292

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Strain effects on the quantum capacitance of graphene nanoribbon devices George S. Kliros

T

Department of Aeronautical Sciences, Division of Electronics, Electric Power and Telecommunications, Hellenic Air-Force Academy, Dekeleia Air-Base, Dekeleia 13671, Attica, Greece

ARTICLE INFO

ABSTRACT

Keywords: Strained graphene Graphene nanoribbons Quantum capacitance Strain effect

One of the most important characteristics of graphene-based nanodevices is the quantum capacitance (QC) of the channel which plays a dominant role in governing the device performance. Moreover, QC can be utilized to realize various types of high performance sensors and energy storage devices. In the present work, the effect of in-plane uniaxial strain on the QC of GNR-based devices is investigated utilizing a compact analytical model which accounts for several finite-size and edge effects. Carrier density and QC are calculated for the three distinct families of armchair GNRs. The strong modulation of QC by uniaxial strain is related to strain-induced changes in both bandgap and effective mass of GNRs. Our study could be helpful for designing GNR-based nanodevices in the Quantum Capacitance Limit.

1. Introduction Graphene has emerged as a promising material for nanoelectronics due to its outstanding electrical, thermal, optical and mechanical properties [1,2]. However, large-area graphene is unable to function as a switch in transistor devices due to zero bandgap. Fortunately, if graphene is patterned into nanoribbons (GNR), a sizeable bandgap can be created due to quantum confinement and edge effects opening up the possibility of improving the switching performance of GNR-based transistors [3] and design logic circuits [4]. The bandgap of a GNR depends on its width and edge orientation. Zigzag edged nanoribbons have a very small gap due to the presence of localized edge states at the Fermi level. Such edge states are absent in an armchair graphene nanoribbon (AGNR). Significant experimental work has demonstrated the ability to fabricate narrow AGNRs [5–7]. Scanning tunneling microscope lithography is a method which is able to create GNRs with well defined edge orientation, having a width down to 2.5 nm [8]. Son et al. [9] and recently, Wang et al. [10] have shown that AGNRs can be categorized according to their bandgap variation with GNR’s width in three distinct families N = 3p 1, N = 3p and N = 3p + 1, where N is the number of carbon atoms in the transverse direction and p is an integer. The GNR’s width can be calculated using the relation W = 0.123(N 1) nm. In the presence of edge bond relaxation, all AGNRs are semiconducting with bandgaps Eg inversely proportional to GNR’s width W. Moreover, the bandgap family hierarchy of Eg (3p 1) < Eg (3p) < Eg (3p + 1) is maintained. One of the most important characteristics of graphene-based nanodevices is the quantum capacitance (QC) of the channel which arises

from the finiteness of the density of states within the channel and is related to the change in electron density with the chemical potential [11]. When the device operates in the quantum capacitance limit (QCL) where the small QC dominates the total device capacitance, QC plays a significant role in governing the device performance [12,13]. Moreover, in the QCL the channel potential is mainly controlled by the gate electrode with 1:1 ratio with the gate voltage. On the other hand, the negative local QC that is achieved in GNRs [14], could be used to design nanoelectronic gates. Existing studies have already established the fundamental science as well as device applications underlying QC for both monolayer and bilayer graphene [15–18]. However, the possibility of tuning the device QC via mechanical strain remains largely unexplored. Tailoring the QC of a nanodevice is important for several applications since (i) changes in QC can enhance sensing signals in graphene biosensors [19] (ii) for energy storage graphene nanodevices high specific QC is desirable [20] (iii) for interconnect application, low QC is preferable in order to reduce signal transmission delay inside the chip [21] (iv) QC plays significant role in governing the quality factor of graphene varactors for wireless sensing [22], (v) the modulation speed of a graphene based optical modulator can be controlled by the QC [23]. Strain engineering is shown to be an effective method for boosting device performance and could be used to design various elements for all-graphene electronics [24]. However, experimental research on strain engineering of GNR - based devices is rather limited. Uniaxial strain has recently been applied to individual GNRs by atomic force microscopy manipulation [25] and in-situ stretching patterned GNRs in a transmission electron microscope [26]. The possibility of boosting the performance of AGNR-based devices by applying tensile uniaxial strain

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.apsusc.2019.144292 Received 10 November 2018; Received in revised form 30 September 2019; Accepted 5 October 2019 Available online 18 October 2019 0169-4332/ © 2019 Elsevier B.V. All rights reserved.

Applied Surface Science 502 (2020) 144292

G.S. Kliros

after applying strain, respectively. Treating these changes as small perturbations, we can obtain the band structure of uniaxially strained AGNR in the armchair direction [27]

En± (k x ) = ± EC2, n + (

n kx )

2

(1)

with

c1 ( 1 +

EC, n =

3)

+ 2c2 1 scos (n )

+ 2c2 3 [c3 + (1

c3) cos (2n )]

(2)

and 1 s c cos (n 2 1 2

(

n)

2

)[c1 ( 1 +

+ 2c2 3 (c3 + (1

= (3acc )2 ×

3 [c1 1

+ (c1

+ 2c2 3 (c3 + (1

3)

c3) cos (2n ))] 1)

3

c3) cos (2n ))]

(3)

where = /(N + 1), ± indicates the conduction band and valence band respectively, N is the total number of carbon atoms in the ribbon, n denotes the subband index, and EC, n is the band edge energy of the nthsubband. The strain parameters are expressed as c1 = 1 + , c2 = 1 + , c3 = ( 3 c2 + 1)/ 3 c2 (N + 1) with = 2 + 3 2 and . The first set of conduction and valence bands have band index s = 1. Supposing symmetric band structure of electrons and holes, one obtains for the energy gap EG, n = 2EC, n . Also, 1 = 3.2 eV and 3 = 0.3 eV refer to the first and third-nearest neighbor hopping parameters and 1 = 0.2 eV is the correction to 1 accounting for the edge bond relaxation. Effective mass approximation (EMA) is quite accurate in narrow GNRs since their band dispersion curves are approximately parabolic. The electron effective mass of each conduction subband can be calculated using the relation

Fig. 1. (a) Schematic of double-gate GNR FET where a semiconducting AGNR is used as channel material. (b) The structure of a strained H-passivated AGNR. Uniaxial tensile strain is applied along the armchair direction.

has also been explored [27–29]. In the present paper, the influence of uniaxial strain on the QC of AGNR-devices is thoroughly investigated utilizing a compact analytical model for calculating both the carrier density and the quantum capacitance for the three AGNR families. Our results indicate the strong modulation of QC under the applied strain for all AGNR families which is related to strain-induced changes in both bandgap and effective mass of GNRs. The temperature dependence of the QC is also discussed.

mn =

1

2E (k ) n x k x2

2

(4)

and at the conduction band minimum is given by mn = EC , n/ the EMA the Density of States (DOS) is given by

D (E ) = 2. Theoretical study

2mn

1 E

n> 0

EC , n

(E

EC , n)

2 n.

Within

(5)

It is worth noting that line broadening due to impurity scattering and electron-hole puddle effect [33] are not considered here, assuming that such effects can be overcome by processing advancements in the future. Integrating the DOS over all possible energies, the one-dimensional carrier density inside the channel is obtained.

A schematic of the device structure used in our study is illustrated in Fig. 1(a). Double-gate configuration is considered with gate-insulator HfO2 of thickness tins = 1.5 nm and relative dielectric constant = 16. The channel consists of an intrinsic armchair-edged GNR with length LG = 30 nm. The power supply is set VDD = 0.5 V and room temperature (T = 300 K) is assumed. Fig. 1(b) shows the atomic structure of AGNR channel under a uniaxial tensile strain along the armchair direction (xdirection). Moreover, the carbon-carbon (C-C) bonds at the edges are bonded to hydrogen atoms to terminate dangling bonds. This allows keeping the CeC bond length constant for both interior and edge atoms. It should be noticed that, the hopping parameter for edge atoms can increase up to 12% in lack of edge bond relaxation [9]. It has been verified that a third nearest neighbor tight binding model (3NN) incorporating the edge-bond relaxation can accurately predict the band structure of GNRs [30]. On the other hand, the second nearest neighbor (2NN) interaction only shifts the dispersion relation in the energy axis and does not affect the GNR’s band structure. When uniaxial tensile strain ( ) is applied to the relaxed structure of GNR, in (1 + ) rix x-direction, the three CeC bond vectors are changed as rix (1 ) riy, i = 1, 2, 3 where and = 0.165 is the strength of and riy uniaxial strain and the Poissson’s ratio, respectively [31]. As a result, the hopping parameters in the Hamiltonian matrix of the unstrained GNR should be modified. Bonding length dependence of the hopping parameters can be described by the Harrison’s model [32] as ti = t 0 (di / d0 ) 2 , where di and d 0 are the CeC bond lengths before and

n1D =

kB T 2 2

mn [F

1/2 ( n, S )

+F

1/2 ( n, D )]

n>0

(6)

where Fj is the Fermi-Dirac integral of order j defined by

Fj ( )

1 (j + 1)

xj 0

1 + exp [x

( / kB T )]

dx

(7)

where (x ) represents the Gamma function and n, S = (EFS EC , n)/ kB T , EC , n)/ kB T . Moreover, from device electrostatics, the foln, D = (EFD lowing relation between the gate voltage and Fermi energy EF can be found [13]

VG (EF )

VFB =

qn1D (EF ) EF + q Cins

(8)

where q is the carrier charge, Cins is the gate-insulator capacitance per unit length of the GNR and VFB denotes the flat-band voltage. The value of VFB depends on the work function difference between the metal-gate electrode and the GNR and it can be set simply to zero [13]. The gateinsulator capacitance (classical capacitance) can be calculated by the expression [34] 2

Applied Surface Science 502 (2020) 144292

G.S. Kliros

Cins = NG

0

W + tins

(9)

where NG is the number of gates, is the relative dielectric constant of the gate insulator, tins is the gate-insulator thickness and 1 is a dimensionless fitting parameter that takes into account the electrostatic edge effects [34]. It should be pointed out that, in the QCL, the potential distribution within the channel is determined by the gate potential rather than the channel charge and thus, short channel effects are suppressed [13]. The bias-dependent quantum capacitance per unit length is defined as CQ = q2 n1D / EF where q is the electron charge and n1D is the one-dimensional carrier density. By using Eq. (6) the QC can be written in terms of Fermi-Derac integrals of order ( 3/2) as follows:

CQ =

q2 (2

2k T )1/2 B n> 0

mn [F

3/2 ( n, S )

+F

3/2 ( n, D )]

(10)

3. Results and discussion Firstly, the strain dependence of both bandgap and effective mass is presented. We have chosen GNRs with N = 23, N = 24 and N = 25 in our calculations to represent the three distinct families of AGNR - channels N = 3p 1, N = 3p and N = 3p + 1 respectively. In order to verify the validity of our results based on the analytical model, we have calculated numerically both bandgaps and effective masses for different strains employing the 1D -supercell method of Xie et al. [36], after taking into account both bond relaxation and 3NN interactions. This method has the advantage to obtain accurate results without heavy computations because it only needs to deal with small matrices. As shown in Fig. 2(a) and (b), a good agreement between analytical and numerical results is observed. As shown both bandgap and effective mass show a zig-zag pattern versus tensile strain in the range 0–15% with minima that occur at the same values of strain. The bandgap changes in a linear fashion between turning points and the slope is almost identical for the three GNR families [37]. It can be also seen that the amplitude of the zig-zag pattern decreases as the index of GNR family increases. The strong modulation of both bandgap and effective mass can be correlated to the alternative movements of the sub-bands to lower and higher energies with applied strain [35]. It is worth noting that, if there is no change in the time scale for quasiparticle scattering under strain, the carrier mobility would be inversely proportional to the effective mass [38]. Therefore, changes in the effective mass can affect both the DOS and the carrier transport properties. It is instructive to explore the effect of strain on the band structure by plotting the DOS for the three GNR families. Fig. 3(a)–(c) depict the DOS in the energy range ± 1 eV around the Fermi energy EF = 0 , for three increasing values of strain percentage. As seen, the positions of the first van Hove singularities change with applied strain according to the bandgap modulation and the corresponding peak heights follow opposite trends in N = 3p and N = 3p + 1 families. Moreover, Fig. 4(a)–(c) display the carrier density at room temperature as a function of Fermi energy for increasing values of tensile strain. As you can see, there is a difference in trends related to the effect of strain on the carrier densities for the three GNR-families. In the strain range [0–6%], the carrier density in (3p 1) -GNR channels increases monotonically with strain whereas the density of 3p -GNR channels decreases. On the other hand, the carrier density of (3p + 1) -GNR channels changes non-monotonically and the density curves for strains 2% and 4% almost coincide. We now focus on the effect of strain on the QC characteristics CQ VG of the device under study. Fig. 5(a)–(c) shows the CQ VG characteristics, at drain bias V, for the three families N = 3p 1, N = 3p and N = 3p + 1 respectively. The QC shows significant non-linear variation with gate bias including a well defined peak. These variations follow the effect of gate bias on the conduction band edge in the GNR channel. Actually, the peak of QC occurs at the gate bias corresponding to the onset of barrier collapse, that is, at the point where incremental change of gate voltage is unable to introduce new carriers into the channel. As is depicted

Fig. 2. Bandgap (a) and effective mass (b) versus strain for the three distinct AGNR families N = 3p 1, N = 3p and N = 3p + 1. Lines represent the analytic results while symbols indicate results based on numerical supercell method of Ref. [36].

in Fig. 5(a)–(c) the CQ VG characteristics are strongly modulated by uniaxial strain following the non-monotonic variations of the bandgaps of each family. More specifically, the peaks in CQ are decreased and moved toward lower values of VG as tensile strain is increased before the turning point of bandgap variations and are increased and moved toward higher values of VG as strain is increased after the turning point. Fig. 6 displays the peak values of QC extracted from Fig. 5(a)–(c), versus applied tensile strain, for the three distinct AGNR. In addition, Fig. 7 plots the QC in the ’on-state’ corresponding to VGS = VDS = 0.5 V, versus applied tensile strain. As seen, in the same range of applied strain the GNR-23 and GNR-24 operating in the ’on-state’ have larger values of QC than the GNR-25 with pronounced peaks around = 4% and = 8%. Therefore, the application of strain in specific GNR-families can increase the QC above threshold and thus, can drastically affect the transconductance and cut-off frequency of the GNR-device [13]. It should be noticed that the classical capacitance Cins in the DGGNR device under study, has values several times larger than the calculated values of QC as shown in Figs. 5 and 6. This is well evident from Fig. 8 where the classical capacitance versus the AGNR channel width at different HfO2-insulator thickness tins , is plotted. As a result, for insulator thickness tins = 1.5 nm and GNR-width W 3 nm, the QC is dominant compared to classical capacitance over the entire gate-voltage range. In general, the design window where the QCL is relevant Cins which is usually fulfilled can be determined by the inequality CQ for DG-FETs with ultrathin and high- gate insulators. However, since the QC is gate-voltage dependent, its comparability to classical 3

Applied Surface Science 502 (2020) 144292

G.S. Kliros

Fig. 3. Density of states D (E ) under different strain percentages for the families N = 3p 1 (a), N = 3p (b) and N = 3p + 1 (c) respectively.

Fig. 4. Carrier density versus Fermi energy under increasing strain percentages for the families N = 3p 1 (a), N = 3p (b) and N = 3p + 1 (c) respectively.

capacitance depends on the gate voltage and thus, the QCL can be less accurate when the Fermi level reaches the subband energies. Finally, we investigate the temperature dependence of CQ VG characteristics. Fig. 9(a) and (b) show, as an example, the temperature dependence of the CQ VG characteristics for the unstrained and strained GNRs with N = 24 . A decrease of QC with increasing temperature is observed within a finite voltage range around maximum. The maximum values of QC follows the relation CQ, max 1/ 4kB T independently of the applied strain. Actually, the thermal broadening of the Fermi-Dirac distribution prevents QC from diverging once the conduction band aligns with the Fermi level of the source. Similar results for the temperature dependent QC are also obtained for the other GNR families.

4. Concluding Remarks An analytical model is presented to investigate the effects of uniaxial strain on the quantum capacitance (QC) of AGNR devices. As a first step, the high sensitivity of both bandgaps and effective masses to the applied uniaxial strain is explored. This sensitivity makes AGNRs promising candidates for sensing applications, such as strain gauges [39]. Next, numerical results for the DOS, carrier density and QC for a double-gate AGNR-FET operating in the QCL, are presented. The strong modulation of CQ VG characteristics due to the changes in applied tensile strain is directly related to strain-induced changes in both bandgap and effective mass of the three distinct families of AGNRchannel. Our results indicate that, a suitable choice of strain applied to

4

Applied Surface Science 502 (2020) 144292

G.S. Kliros

Fig. 6. Peak values of quantum capacitance CQ as a function of tensile strain for the three GNR families N = 3p 1 (a), N = 3p (b) and N = 3p + 1(c) respectively.

Fig. 7. On-state quantum capacitance CQ as a function of tensile strain for the three GNR families N = 3p 1 (a), N = 3p (b) and N = 3p + 1(c) respectively.

Fig. 5. Quantum Capacitance CQ versus gate voltage VGS under increasing strain percentages for the families N = 3p 1 (a), N = 3p (b) and N = 3p + 1(c) respectively.

a specific family of AGNR channel, can be used to reduce or enhance the QC. Thus, strain could be an effective way to control the high frequency response and switching performance of a GNR-based device [13] or the modulation speed of a GNR-based optical modulator [23]. Our study has restricted to the application of tensile strain on the GNR-channel since it has been demonstrated that narrow GNRs have critical compressive strain for bucking several orders of magnitude smaller than the corresponding tensile strain for fracture. Such a large asymmetry implies that strain engineering of GNR-devices is only viable with application of tensile strain [40]. It should be also noted that our approach may underestimate the actual carrier concentration in the channel when parabolic band structure misses to match the exact dispersion relation. However, we believe that the present analytical model provides an easy way for technology benchmarking. More specifically,

Fig. 8. Classical capacitance versus channel width at different HfO2 -insulator thickness for the doubled-gated GNR-device of Fig. 1.

5

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G.S. Kliros

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

Fig. 9. Quantum Capacitance CQ versus gate voltage VGS for increasing temperature for a) unstrained GNR with N = 24 and b) strained by 4% GNR with N = 24 .

[28] [29]

our study can provide some insight and guidance for controlling the quantum capacitance of GNR-based transistors, sensors, optical modulators as well as GNR-based energy storage devices.

[30] [31]

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