Vacancies in Epitaxial Graphene - Springer Link

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Emphasis is placed on the effect of vacancies on the. DOS of EG. In addition, we discuss the charge trans fer between graphene and the substrate. The consider.
ISSN 10637826, Semiconductors, 2015, Vol. 49, No. 8, pp. 1069–1078. © Pleiades Publishing, Ltd., 2015. Original Russian Text © S.Yu. Davydov, 2015, published in Fizika i Tekhnika Poluprovodnikov, 2015, Vol. 49, No. 8, pp. 1095–1103.

CARBON SYSTEMS

Vacancies in Epitaxial Graphene S. Yu. Davydov Ioffe Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia email: [email protected] Submitted November 12, 2014; accepted for publication December 23, 2014

Abstract—The coherentpotential method is used to consider the problem of the influence of a finite con centration of randomly arranged vacancies on the density of states of epitaxial graphene. To describe the den sity of states of the substrate, simple models (the Anderson model, Haldane–Anderson model, and parabolic model) are used. The electronic spectrum of free singlesheet graphene is considered in the lowenergy approximation. Charge transfer in the graphene–substrate system is discussed. It is shown that, in all cases, the density of states of epitaxial graphene decreases proportionally to the vacancy concentration. At the same time, the average charge transferred from graphene to the substrate increases. DOI: 10.1134/S1063782615080072

1. INTRODUCTION Studies of epitaxial graphene (EG) [1–4] are, in a sense, no less important than studies of free single sheet graphene (SSG), since it is EG in particular that serves as an element of the planar technology of device structures. Therefore, it is important to understand exactly how the density of states (DOS) of SSG changes under the influence of the substrate. First of all, it is necessary to establish whether an energy gap is formed in the spectrum of the initial zerogap SSG, whether or not the linear behavior of the electron dis persion relation ε(k) (k is the twodimensional wave vector) is altered in the vicinity of the Dirac point εD, and to where and how the position of the Dirac point εD is shifted with respect to the chemicalpotential level of the system. Since graphene contains defects inherent in any real structure, interest in disordered SSG is quite understandable. For example, a number of studies (see [5, 6] and references therein) are concerned with the influence of a finite concentration of carbon vacancies on the DOS of SSG. The same problem for EG was considered in [7]; in [7], the coefficients involved in the dependence of the DOS on the vacancy concen tration were calculated for ideal SSG. In this study, we calculate these coefficients for defectfree EG, and as a result, we are able to avoid a singularity at the Dirac point. The major part of the study is concerned with EG on a semiconductor substrate, since the electron spec trum of EG formed on such a substrate undergoes the most radical changes in the energy range correspond ing to the band gap. However, the case of a metal sub strate is considered as well. Emphasis is placed on the effect of vacancies on the DOS of EG. In addition, we discuss the charge trans

fer between graphene and the substrate. The consider ation is of the model character. 2. GENERAL RELATIONS Let the Green’s function for a carbon atom bound to a solid substrate be –1

g C ( ω ) = ω – ε C – Λ ( ω ) + iΓ ( ω ).

(1)

Here, ω is the energy variable; εC is the energy of the free |p〉 orbital of a carbon atom (εC is further taken to be equal to the energy at the Dirac point εD); Γ(ω) = 2

πV CS ρ s ( ω ) is the function of broadening of the quasi level of carbon (VCS) is the matrix element of interac tion of the |p〉 orbital of a carbon atom with the sub strate and ρs(ω) is the DOS of the substrate, and Λ(ω) is the function of the shift of the quasilevel of carbon. The function Λ is related to the function Γ(ω) by the relation ∞

Γ ( ω' )dω' , Λ ( ω ) = 1P  π ω – ω'



(2)

–∞

where P is the symbol of the principal integral value. Making use of the results of [7], where the coher entpotential method [8, 9] is used to solve the prob lem of vacancydisordered EG, we write the averaged Green’s function of a graphene layer as

1069

˜ eg ( ω )〉 layer = F˜ 1 ( ω ) – iF˜ 2 ( ω ). 〈G

(3)

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Here, F˜ 1 ( ω ) =

Ω – ε ( k ) + αA 1

, ∑  ( Ω – ε ( k ) + αA ) + ( Γ ( ω ) + αA ) 2

k

F˜ 2 ( ω ) =

In order to simplify the problem, we use the func tions A1, 2(ω) corresponding to defectfree EG instead of the functions A 1, 2 ( ω ) :

∑ k

2

1

2

(4)

Γ ( ω ) + αA 2  2 2 ( Ω – ε ( – k ) + αA 1 ) + ( Γ ( ω ) + αA 2 )

and F˜ 1, 2 ( ω ) , A 1, 2 ( ω ) =  F˜ 1 ( ω ) + F˜ 2 ( ω )

(5)

Ω = ω – ε D – Λ ( ω ). Thus, expressions (4) present a set of two selfcon sistent nonlinear equations. Here, we are interested in the averaged DOS of the graphene layer 〈 ρ˜ eg ( ω )〉 layer = –1 ˜ eg ( ω )〉 layer , wherefrom – π Im 〈 G 〈 ρ˜ eg ( ω )〉 layer = 1F˜ 2 ( ω ). π

(6)

We here use the lowenergy approximation, in which the real electron spectrum in the vicinity of the Dirac point is replaced by the linearized expression ε±(|q|) = εD ± 3ta|q|/2, where the wave vector q is reck oned from the wave vector at the Dirac point, t is the energy of electron transfer between nearest neighbors spaced by a distance of a = 1.42 Å in SSG, the plus sign refers to the conduction band of SSG, and the minus sign refers to the valence band of SSG. In this approx imation, in accordance with [7], we obtain ( Ωα − + ξ ) + Γα ( ω ) 2 1   F˜ 1 ( ω ) = − +  – 2 Ω α ln  2 2 ξ ξ Ωα + Γα ( ω ) 2

2

(8)

Here, F 1, 2 ( ω ) are determined by Eqs. (7) at α = 0, so that Ωα transforms into Ω and Γα(ω) into Γ(ω). In the bandgap region of the crystalline substrate, its DOS ρs is identically equal to zero, whence it fol lows that Γ(ω) = 0. It can be easily shown that, in this case, from (7) it follows that Ω, − 2 + 2Ω F1 ( ω ) = +  ln  ξ ξ2 Ω− +ξ ⎧ 2π Ω ⎪ , Ω ≤ ξ, F2 ( ω ) = ⎨ ξ2 ⎪ ⎩ 0, Ω > ξ.

(9)

Here, the minus sign refers to the conduction band of EG (Ω > 0) and the plus sign to the valence band (Ω < 0) of defectfree EG. Let us consider some features of expressions (9). The energy at the Dirac point of defectfree EG, ωD, is defined from the equation Ω = 0. If |ωD| ≤ Eg/2, − 2 /ξ), F 2 (ωD = 0, from (9) it follows that F 1 (ωD = +

− ξ /2, and A 2 (ωD) = 0. A 1 (ωD) = + The condition for the appearance of band gaps in the DOS ρeg(ω) = F 2 (ω)/π of defectfree EG formed on a semiconductor substrate is the inequality |Ω| > ξ (see, e.g., [7] and below). The edges of the band gaps −ξ . are defined by the equation Ω* ≡ ω* – εD – Λ(ω*) = + − ∞ and F 2 (ω*) = It can be easily seen that F 1 (ω*) +

2π/ξ, so that A 1 (ω*) = 0 and A 2 (ω*) = 0. To conduct analysis further, we must choose a model for the DOS of the substrate ρs(ω).

− ξ⎞ Ωα Ωα + – 2Γ α ( ω ) ⎛ arctan   – arctan   , ⎝ Γα ( ω ) Γα ( ω ) ⎠ (7) − ξ) + ( Ωα + 1   F˜ 2 ( ω ) = 2 Γ α ln  2 2 Ωα + Γα ( ω ) ξ 2

F 1, 2 ( ω ) 2 . A 1, 2 ( ω ) =  2 ( F1 ( ω ) ) + ( F2 ( ω ) )

2 Γα ( ω )

Ωα Ωα − + ξ⎞ +2Ω α ⎛ arctan   – arctan   . ⎝ Γα ( ω ) Γα ( ω ) ⎠ Here, Ωα = Ω + αA1(ω), Γα(ω) = Γ(ω) + αA2(ω), the minus sign refers to the conduction band of EG (Ωα > 0), and the plus sign refers to the valence band of EG (Ωα < 0). The Dirac point ωαD of imperfect EG is defined by the equation Ωα = 0.

3. SEMICONDUCTOR SUBSTRATE: THE HALDANE–ANDERSON MODEL We start from the Haldane–Anderson model [10], in which the DOS ρs(ω) is a constant at |ω| ≥ Eg/2 and zero at |ω| < Eg/2, where Eg is the band gap. Then Γ(ω) = Γ = const at |ω| ≥ Eg/2 and Γ(ω) = 0 at |ω| < Eg/2. The function ω – E g /2 Λ(ω) = Γ  ln   π ω + E g /2

(10)

is specified on the entire energy axis. Here and below, we assume that zero energy corresponds to the middle of the band gap of the substrate. At |ω| > Eg/2 (the SEMICONDUCTORS

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8

2 (a)

(a)

εD = 0 4

2 1 4

2 ×10

0 – ai

– ai

1

0 1

–1

3

Haldane–Anderson DOS –4

εD = 0 γ=1

–2

–3 –1.0

–0.9

–0.8

×10

Haldane–Anderson DOS γ = 0.2 γ=5

–0.7

–0.6

0.5

–8 –1.0

–0.5 x

–0.9

–0.8

–0.7

–0.6

–0.5 x

–0.1

0 x

0.6

(b)

(b) Haldane–Anderson DOS 0.4

0 – ai

1 2

0.2 – ai

–0.5

4

0 Haldane–Anderson DOS εD = 0 γ=1

–1.0

–0.2

εD = 0 3

γ = 0.2 γ=5 –1.5 –0.5

–0.4

–0.3

–0.2

–0.1

0 x

Fig. 1. The Haldane–Anderson model: dependences of the reduced coefficients a 1, 2 = A 1, 2 /ξ on the dimension less energy x = ω/ξ at εD = 0 and Eg/ξ = 1 for γ ≡ Γ/ξ = 1. Only the lefthand portions of (curve 1) the odd function a 1

–0.4 –0.5

–0.4

–0.3

–0.2

Fig. 2. The same as in Fig. 1 for γ = (circles) 0.2 and (squares) 5. γ = 0.2: (1) (solid circles) a 1 × 10 and (2) (open circles) a 2 × 10. γ = 5: (3) (solid squares) a 1 and (4) (open squares) a 2 .

and (curve 2) the even function a 2 are shown.

valence band and conduction band) and ω ±Eg/2, − ±∞. Then we have Λ(ω) + ∞ and Ω F 1 (±Eg/2)

1/Ω, F 2 (±Eg/2)

±∞, and A 2 (±Eg/2) band gap) and ω 1/Ω, F 2 (±Eg/2) A2 ( − + E g /2) = 0.

0, A 1 (±Eg/2)

0. However, at |ω| < Eg/2 (the ±Eg/2, we obtain F 1 (±Eg/2) − 0, A 1 (±Eg/2) + ∞ , and

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Figures 1 and 2 show the dependences of the reduced coefficients a 1, 2 = A 1, 2 /ξ on the dimension less energy x = ω/ξ at εD = 0, Eg/ξ = 1, γ ≡ Γ/ξ = 1 (Fig. 1), 0.2, and 5 (Fig. 2). In Figs. 1 and 2, we show only the regions of negative energies, since at εD, the function a 1 is odd and the function a 2 is even with respect to zero energy. The same is true for the func tions F 1, 2 (ω). When choosing the parameters, we assumed ξ ~ t ~ Eg ~ 3 eV [1, 11–13], so that the ratio Eg/ξ = 1 approximately corresponds to a 6HSiC sub

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strate. It should be noted that the inequality γ Ⰶ 1 cor responds to weak graphene–substrate coupling and the inequality γ Ⰷ 1 to strong graphene–substrate cou pling; the case of γ = 1 is intermediate. All of the abovelisted cases exist in reality. For example, the condition of weak coupling corresponds to a socalled quasifree graphene sheet, whereas the condition of strong coupling to a buffer layer (see [11–13] and ref erences therein for details). Let us first consider the case of γ = 1. From Fig. 1a it follows that the coefficients a 1, 2 vary only slightly up to the top of the valence band of the substrate (| a 1 | ~ 0.5, | a 2 | ~ 1). At the top of the valence band, the coeffi cients start sharply decreasing, and the coefficient a 2 decreases to zero almost stepwise. Such behavior of the coefficients is defined by the specific features of the Haldane–Anderson model, in which the density of allowed states is assumed to in the form of two semi infinite steps. Figure 1b demonstrates much more complex dependences a 1, 2 on the reduced energy x. At the Dirac point (xD = ωD/ξ = 0 in the dimensionless form), we have a 1 = 0.5 and a 2 = 0. In the range (–0.5 x *1 ), where x *1 = ω *1 /ξ ≈ –0.377, the coefficient a 2 is identically equal to zero. This region is the energy gap in the DOS of defectfree EG ρeg(ω) = F 2 /π, and the dimensionless energy x *1 is the upper edge of the band gap (see below); as shown above, a 1 ( x 1* ) = 0 and a 2 ( x *1 ) = 0. The coefficient a 1 is zero also at the point x' = ω'/ξ ≈ –0.315, where the energy ω' is determined from the equation Ω Ω = + − 1.  ln  ξ Ω− +ξ

(11)

In (11), lower signs must be taken for the valence band region (see (9)). Let us now turn to Fig. 2. Upon weak coupling (γ = 0.2), the coefficients | a 1 | (curve 1) and a 2 (curve 2) in the valenceband region (Fig. 2a) are smaller than those at γ = 1 (Fig. 1), whereas in the condition of strong coupling (γ = 5), in contrast, the coefficients a 1 (curve 3) and a 2 (curve 4) are larger. In the bandgap region (Fig. 2b), at γ = 0.2, the energy gap is narrowed so that it is not evident at the scale of Fig. 2 (curve 2 for the coefficient a 2 ). It can be shown that, at γ Ⰶ 1, we have the energy ω 1* –Eg/2, so that the band gap is ξ – E g /2⎞ Δ = ω *1 – E g /2 ≈ E g exp ⎛ – π   . ⎝ Γ ⎠

(12)

Consequently, at ξ > Eg/2, the reduced energy gap is small: Δ/ξ Ⰶ 1. If γ Ⰷ 1, we obtain ω *1 ≈ –πξEg/Γ, so that Δ ~ Eg/2. This lastpresented result means that the band gap at γ = 5 (curve 4) is widened compared to the band gap at γ = 1 (Fig. 1b, curve 2). The depen dences a 1 (x) are changed in the corresponding man ner: in Fig. 2b, the features of curve 1 are not evident (at the scale of Fig. 2b), and the portion in curve 3 (compared to curve 1 in Fig. 1b) corresponds only to the small vicinity of the point x *1 ≈ –0.133. We now consider how the coefficients a 1 and a 2 vary as the Dirac point is shifted from the position εD. It can be easily shown that, in this case, all of the aboveconsidered dependences a 2 (ω) and a 1 (ω) lose their symmetry (correspondingly, parity and imparity) with respect to ω = 0. In fact, when turning to the energy variable ε = ω – εD in expression (5) for Ω, we can rewrite expression (10) as Λ(ε) = (Γ/π)ln|(ε – EC)(ε – EV)|, where the energies of the edges of the conduction band and valence band are EC, V = ±(Eg/2) – εD. The primary effect of asymmetry mani fests itself as a difference between the energy gaps adja cent to the edges of the valence band and conduction band (see [13] for details). Knowing the coefficients a 1, 2 , we can turn to cal culations of the DOS of imperfect EG. Figure 3 shows the results of calculation of the reduced DOS f2 = πξ〈 ρ˜ eg〉 layer = ξ F˜ 2 as a function of the dimensionless energy x = ω/ξ at εD = 0, γ = 1, and α = 0.1, and 0.2 for the valenceband region (Fig. 3a) and bandgap region (Figs. 3b, 3c) of the substrate. Along with a decrease in the function f2(x) in the entire energy region, some features of changes in the DOS in the bandgap region of the substrate (Figs. 3b, 3c) should be noted. First, the peak of the function f2(x) is blurred and shifted from the position x *1 ≈ –0.377 to smaller absolute values of energy (Fig. 3b). This is due to the fact that the energy dependence of the reduced DOS is now defined not only by a steady variation in |Ω|, but by the functions A 1 (ω) and A 2 (ω). Second, the func tion f2(x) at small energies tends to zero more smoothly compared to the function in the defectfree case (Fig. 3c). Figure 3d refers to the case of γ = 5 ( x *1 ≈ –0.133). Comparison of the results shown in Figs. 3c and 3d shows that an increase in γ yields a shift of the peaks of f2(x) to the right along the energy axis in the defectfree and imperfect cases. It can be easily shown α the position of the peaks x max is given by the approxi mate expression α + x max ≈ x *1 – αa 1 ( x *1 + 0 ).

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8 (a)

0.6

6 α=0 α = 0.1 α = 0.2

0.2

f2

α=0 α = 0.1 α = 0.2 εD = 0 γ=1

0 –1.0

–0.8

–0.6

Haldane–Anderson DOS

–0.2

0 x

–0.1

Haldane–Anderson DOS

(d)

α=0 α = 0.1 α = 0.2

6

εD = 0 γ=1

0.4

–0.3

8

(c)

α=0 α = 0.1 α = 0.2

0.6

f2

εD = 0 γ=5

0 –0.4

–0.4 x

0.8

εD = 0 γ=5

4

2

0.2

0 –0.06

4

2

f2

f2

Haldane–Anderson DOS 0.4

(b)

Haldane–Anderson DOS

–0.04

–0.02

0 x

0 –0.16

–0.12

–0.08

–0.04

0 x

Fig. 3. The Haldane–Anderson model: the dependence of the reduced DOS of the EG layer f2 = πξ 〈 ρ˜ eg〉 layer on the dimension less energy x = ω/ξ at εD = 0, Eg/ξ = 1, and γ = 1, and α = (open circles) 0, (solid circles) 0.1, and (solid squares) 0.2. Only the lefthand portions of the even functions are shown.

From Figs. 1 and 2 it follows that, in the vicinity of α x 1* , we have the coefficient a 1 < 0, so that x max lies to the right of x *1 . 4. SEMICONDUCTOR SUBSTRATE: THE PARABOLIC MODEL Now, for the DOS of the substrate, we choose a model corresponding to the parabolic electron spec trum [12, 14, 15]. We are further interested in the energy region close to the band gap of the substrate. SEMICONDUCTORS

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Therefore, similarly to [14], we take the DOS of the substrate as ⎧ – ω – E g /2 , ⎪ ρ s ( ω ) = A ⎨ ω – E /2 , g ⎪ ⎩ 0,

ω < – E g /2, ω > E g /2,

(14)

ω ≤ E g /2,

where A is the coefficient of dimensionality eV–3/2 and zero energy is assumed to correspond to the middle of the band gap of the substrate. Then, as before, we have

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DAVYDOV

and F±(ω) = π ± ω + E g /2 . Thus, in contrast to the Haldane–Anderson model, the parabolic model con tains no divergences.

4 (a) 2

To calculate the coefficients a 1, 2 , we introduce the 2

additional dimensionless parameter c = πAV CS 2/ξ . Thus, the case of weak graphene–substrate coupling corresponds to the inequality c Ⰶ 1, and the case of strong graphene–substrate coupling to the inequality c Ⰷ 1. Assuming γ(x) = Γ(x)/ξ, λ(x) = Λ(x)/ξ, and as above, Eg = ξ, we obtain

– ai

0

εD = 0

1

c=1

–4

⎧ – x – 0.5 , ⎪ γ ( x ) = c ⎨ x – 0.5 , ⎪ ⎩ 0,

Parabolic DOS –8 –1.0

–0.8

–0.6

–0.4

–0.2

0 x

4 (b)

Parabolic DOS 4

(16)

x > 0.5, – 0.5 < x < 0.5 ,

⎧ – x + 0.5 , ⎪ λ ( x ) = c ⎨ – x + 0.5 – x + 0.5 , ⎪ ⎩ – x + 0.5 ,

x < – 0.5, – 0.5 < x < 0.5 ,(17) x > 0.5.

Here, as before, x = ω/ξ. The results of calculations of the coefficients a 1 and a 2 are shown in Fig. 4. Comparison of the results shown in Figs. 4a and 4b shows that, as the parameter c is increased, the values of | a 1 | and a 2 increase. Let us now compare the coefficients a 1 and a 2 obtained in the Haldane–Anderson model (Figs. 1, 2) with those in the parabolic model (Fig. 4). First, in the Haldane– Anderson model, the coefficients a 1 and a 2 at the edge of the band gap are zero, whereas in the parabolic model, the coefficient a 1 is always nonzero and the coefficient a 2 is zero, only if there is an energy gap in

2 0

– ai

1 εD = 0 –4

c = 0.2 c=5 3

–8 –1.0

x < – 0.5,

–0.8

–0.6

–0.4

–0.2

0 x

Fig. 4. The parabolic model: dependences of the reduced coefficients a 1, 2 = A 1, 2 /ξ on the dimensionless energy x = ω/ξ at εD = 0, Eg/ξ = 1. (a) c = 1, (1) a 1 , and (2) a 2 ; (b) c = 0.2, (1, solid circles) a 1 and (2, open circles) a 2 ; c = 5 (3, solid squares) a 1 and (4, open squares) a 2 . Only the lefthand portions of the odd function a 1 and the even function a 2 are shown. 2

Γ(ω) = π V CS ρs(ω) and, instead of (10), obtain Λ(ω) ≡ 2

A V CS Λ (ω), where ω < – E g 2, ⎧ F – ( ω ), ⎪ Λ ( ω ) = ⎨ F – ( ω ) – F + ( ω ) , – E g /2 ≤ ω ≤ E g 2, (15) ⎪ ω > E g 2, ⎩ –F+ ( ω ) ,

the DOS of defectfree EG, for which F 2 (ω) = 0 (Fig. 4a, curve 2; Fig. 4b, curve 4). Such a difference is not surprising, since the coefficients a 1, 2 in the Haldane–Anderson model are zero due to function (10) being logarithmically divergent at ω ± E g /2. Sec ond, in the case of weak coupling (γ = 0.2, c = 0.2), the coefficient a 2 is nonzero (there is no energy gap in the DOS of EG). Third, the dependences a 1, 2 (x) in the bandgap region are qualitatively the same. Figure 5 shows the results of calculations of the reduced DOS f2 = πξ〈 ρ˜ eg〉 layer = ξ F˜ 2 as a function of the dimensionless energy x = ω/ξ at εD = 0, c = 1, and α = 0.1 and 0.2. From Fig. 5a it follows that, as the vacancy concentration α is increased, the function f2(x) in the region of allowed states decreases; i.e., we have the same effect as in the Haldane–Anderson model (Fig. 3a). However, there is some difference: in SEMICONDUCTORS

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0.35 (a) 0.30

γm = 0.2 γm = 1 γm = 5

0.8 0.25 f2

– ai

a1 a2 × 0.1

α=0 α = 0.1 α = 0.2 εD = 0

0.20 Parabolic DOS 0.15

0.4

ρs = const

c=1 0.10 –1.0

–0.9

–0.8

–0.7

–0.6

–0.5 x

0 –0.5

–0.4

–0.3

–0.2

–0.1

0 x

8 (b)

Parabolic DOS

f2

6

Fig. 6. The Anderson model: dependences of the reduced coefficients a 1, 2 = A 1, 2 /ξ on the dimensionless energy x = ω/ξ at εD = 0, and γm = (squares) 0.2, (circles) 1, and (rhombs) 5. Open and solid symbols show the functions a 1

α=0 α = 0.1 α = 0.2 εD = 0

4

and a 2 , respectively. For γm = 5, the values of the coeffi cient a 2 are reduced tenfold. Only the lefthand portions of the odd function a 1 and the even function a 2 are shown.

c=1

5. METAL SUBSTRATE: THE ANDERSON MODEL

2

0 x

In accordance with the Anderson model in the approximation of an infinitely wide gap [16], we assume ρs(ω) = const, which gives Λ(ω) = 0 and Γ(ω) = Γm = const. Thus, the quantities appearing in expression (7) are Γα(ω) = Γm + αA1(ω) and Ωα = Ω + αA1(ω), where Ω = ω – εD. In what follows, we assume εD = 0 without loss of generality.

the Haldane–Anderson model, the function f2(x)

The dependences of the coefficients a 1, 2 (x) are shown in Fig. 6, where γm = Γm/ξ. It can be easily seen that, in all cases under consideration (γm = 0.2, 1, 5), the coefficients a 1 (x) increase, as |x| is decreased or, in other words, as the energy tends to the Dirac point, whereas the coefficients a 2 (x), on the contrary, decrease. It should be noted that, in the energy region under consideration, the dependences a 1, 2 (x) are slightly nonlinear only at γm = 0.2.

0 –0.4

–0.3

–0.2

–0.1

Fig. 5. The parabolic model: the dependence of the reduced DOS of the EG layer f2 = πξ 〈 ρ˜ eg〉 layer on the dimensionless energy x = ω/ξ at εD = 0, Eg/ξ = 1, and c = 1. α = (open circles) 0, (solid circles) 0.1, and (solid squares) 0.2. Only the lefthand portions of the even functions as shown.

( F 2 (ω)) at x –0.5 (ω –Eg/2) tends to zero logarithmically, whereas in the parabolic model, we have f2(x) ∝ – x – 0.5 ( F 2 (ω) ∝ – ω – E g /2 ). Sim ilarly, the dependences f2(x) in the bandgap region in the Haldane–Anderson model (Figs. 3b, 3d) and par abolic model are qualitatively the same (Fig. 5b). SEMICONDUCTORS

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Let us examine the coefficients a 10, 20 = a 1, 2 (0) in more detail. It can be easily shown that, in this case,

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Figure 7 shows the results of calculations of the reduced DOS f 2 = πξ〈 ρ˜ eg〉 layer = ξ F˜ 2 as a function of the dimensionless energy x = ω/ξ at εD = 0, γm = 1, and α = 0.1 and 0.2. From Fig. 7, it can be seen that, as the vacancy concentration is increased, the function f2 decreases. Let us consider the value of the function f20 = f2(0) at α 0. In the approximation linear in α, we obtain

1.0 α=0 α = 0.1 α = 0.2

0.9

f2

γm = 1 0.8

f 20 = f 10 + 2αB, 2

1 + γm f 20 = γ m ln  , 2 γ2

0.7

2 ⎛− + γ m a 10 – a 20 a 20 1 + γ m 1⎞ + ln   ± a 10 arctan ⎟ .    B ≈ ⎜  2 2 2 γ m⎠ ⎝ 1 + γm γm

ρs = const 0.6 –0.5

–0.4

–0.3

(21)

–0.2

–0.1

0 x

Fig. 7. The Anderson model: the dependence of the reduced DOS of the EG layer f2 = πξ 〈 ρ˜ eg〉 layer on the dimensionless energy x = ω/ξ at εD = 0, γm = 1, and α = (open circles) 0, (solid circles) 0.1, and (solid squares) 0.2. Only the lefthand portions of the even functions are shown.

the functions f 10 = F 1 (0)ξ and f 20 = F 2 (0)ξ take the form

6. CHARGE TRANSFER

EF

∫ 〈 ρ˜

eg ( ω )〉 layer dω,

(22)

W

f 20 ≈ 1/γm. Hence,

(19)

4 a 20 ≈ 1 – 2 . 9γ m

where EF is the Fermi level and W is the energy of the lower edge of the continuous spectrum of the graphene–substrate system. We begin our consideration from EG formed on a semiconductor substrate. In this case, the occupation number 〈ng〉 is conveniently represented as the sum of band contributions – E F /2

〈 n˜ gb〉 =

Under the condition of weak coupling, i.e., at γm Ⰶ 1, − 2 (1 – γmπ/2) and f 20 ≈ from (18) we obtain f 10 ≈ + –2γmlnγm, so that

γ m ln γ m . a 2 ( 0 ) ≈ –  2

noted that the maximum value of the function f 20 (γm) is reached at γ *m ≈ 0.505 and corresponds to f *20 ≈ 0.805.

〈 n˜ g〉 =

Under the condition of strong graphene–substrate 2 coupling, i.e., at γm Ⰷ 1, we obtain f 10 ≈ − + 2/3γ m and

1 π a1 ( 0 ) ≈ − +  ⎛⎝ 1 – γ m⎞⎠ , 2 2

α is increased, the value of f 20 decreases. It should be

(18)

1+ . f 20 = γ m ln  2 γm

4 ⎞ , − 2 ⎛ 1 –  a 10 ≈ + 2 3⎝ 9γ m⎠

2

–1/2 γ m and, at γm Ⰶ 1, we obtain B ≈ –π/4. Thus, as

The average occupation number of an atom in EG, 〈ng〉, is defined by the expression

1 f 10 = − + 2 ⎛⎝ 1 – γ m arctan ⎞⎠ , γm 2 γm

In deriving expression (21), we assume α Ⰶ γm. Now it can be easily shown that, at γm Ⰷ 1, we have B ≈



〈 ρ˜ eg ( ω )〉 layer dω

W

(23)

EF

+ θ ( E F – ω *1 )

∫ 〈 ρ˜

eg ( ω )〉 layer dω

ω *1

(20)

and the local contribution dΛ α ( ω ) –1  . 〈 n˜ gl〉 = θ ( E F – ω l ) 1 –  dω ωl SEMICONDUCTORS

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VACANCIES IN EPITAXIAL GRAPHENE

Here, ω *1 is the energy corresponding to the lower edge of the continuous spectrum of EG and lying in the bandgap region of the substrate (this contribution is nonzero, only if the Fermi level is higher than ω *1 , as provided by the unit step function θ(EF – ω *1 ), ωl is the energy of the local level lying below the Fermi level, as guaranteed by the unit step function θ(EF – ωl), and Λα(ω) = Λ(ω) – α A 1 (ω). It should be emphasized that the local level ωl can appear only in the energy gap region (–Eg/2, ω *1 ) induced in the DOS of EG by the band gap of the substrate. Since the average DOS 〈 ρ˜ egl〉 layer decreases with increasing vacancy concentration α, it is clear that the quantity 〈 n˜ gl〉 decreases as well. In this case, it is implied that the position of the Fermi level is defined by the substrate and only slightly dependent on α. Let us now turn to the local contribution 〈 n˜ gl〉 . The energy of the local level ωl is determined from the solution of the equation Ω α = Ω + αA 1 = 0.

(25)

At the same time, the condition for the appearance of an energy gap in the DOS of EG is the inequality |Ω| > ξ. Hence, for negative values of ω, we obtain α A 1 > ξ or α a 1 > 1. An impression can be formed for such a situation that, at α Ⰶ 1, it can occur in the Haldane–Anderson model in a narrowgap region adjacent to the top of the valence band, so that ωl = –(Eg/2) + δ, where δ Ⰶ Eg/2. However, as shown − Ω, above, at ω + E g /2, we have the function A 1 which converts Eq. (25) into the equation (1 + α)Ω = 0, in contradiction to the criterion of the formation of an energy gap. Thus, the local contribution to the occu pation number is lacking. Since the pz orbital of a carbon atom initially involves one electron, the charge transferred to the substrate can be estimated with the average charge of atoms of EG, i.e., with the quantity 〈 Z˜ g〉 = 1 – 〈 n˜ g〉 , where 〈 n˜ g〉 = 〈 n˜ gb〉 . Thus, with increasing α, the transferred charge 〈 Z˜ g〉 increases. This result is con sistent with the general conclusion of the theory of adsorption: with increasing concentration of interact ing adatoms (i.e., with decreasing vacancy concentra tion), the system is depolarized; i.e., the charge at ada toms decreases [17]. Let us now turn to EG formed on a metal substrate. In the Anderson model in the approximation of an infinitely wide band, the average occupation number of an atom is determined by expression (22), where W = –∞. It can be easily seen that, as in the case of a SEMICONDUCTORS

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semiconductor substrate, the DOS of EG decreases proportionally to the vacancy concentration (Fig. 7). This yields a decrease in 〈 n˜ g〉 and an increase in the charge 〈 Z˜ g〉 . 7. DISCUSSION In this study, a procedure is proposed for determin ing the coefficients A1, 2(ω) defining the dependence of the electron DOS of EG on the vacancy concentra tion. The procedure makes it possible to avoid the need to solve a set of two selfconsistent equations (20) which can be solved only numerically. Therefore, the approximation used is such that the product αA1, 2(ω) is replaced by α A 1, 2 (ω), where the coefficients A 1, 2 (ω) are no longer dependent on the vacancy con centration and can be calculated for ideal EG. It should be noted that, previously [7], the product 0

0

α A 1, 2 (ω) was replaced with α A 1, 2 , where α A 1, 2 (ω) was calculated for ideal SSG. In that case, the DOS of EG at the Dirac point was equal to zero. In this study, it has been possible to avoid this effect. Vacancies do not modify the shape of the DOS in energy regions corresponding to the continuous spec trum of the substrate. As to the bandgap region, there are some radical changes: instead of an Mshaped DOS, a DOS curve more spread in terms of energy is obtained, with maxima shifted with respect to those for defectfree EG. In the case of a metal substrate, vacancies do not introduce any radical changes in the DOS of EG. The general feature of all cases consid ered in the study is that the DOS of EG decreases pro portionally to the vacancy concentration. Such a decrease yields an increase in the average charge of a carbon adatom, which is in complete agreement with the theory of adsorption [17] and corresponds to the enhancement of electron transfer from graphene to the substrate. ACKNOWLEDGMENTS The study was supported by the Russian Founda tion for Basic Research, project no. 120200165a, and the Government Program of the support of lead ing universities in the Russian Federation, grant no. 074U01. REFERENCES 1. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2008). 2. J. Haas, W. A. de Heer, and E. H. Conrad, J. Phys: Condens. Matter 20, 323202 (2008).

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3. Y. H. Wu, T. Yu, and Z. X. Shen, J. Appl. Phys. 108, 071301 (2010). 4. D. R. Cooper, B. D’Anjou, N. Ghattamaneni, B. Ha rack, M. Hilke, A. Horth, N. Majlis, M. Massicotte, L. Vandsburger, E. Whiteway, and V. Yu, arXiv: 1110.6557. 5. N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73, 125411 (2006). 6. Z. Z. Alisultanov, Phys. Solid State 55, 1304 (2013). 7. S. Yu. Davydov, Fiz. Tekh. Poluprovodn. 49 (2015, in press). 8. Theory and Properties of Disordered Materials, Collec tion of Articles, Ed. by V. L. BonchBruevich (Mir, Moscow, 1977). 9. J. Ziman, Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems (Cambridge Univ., Cambridge, New York, 1979; Mir, Moscow, 1982).

10. F. D. M. Haldane, and P. W. Anderson, Phys. Rev. B 13, 2553 (1976). 11. S. Yu. Davydov, Semiconductors 48, 46 (2014). 12. S. Yu. Davydov, Tech. Phys. 59, 624 (2014). 13. S. Yu. Davydov, Tech. Phys. Lett. 39, 101 (2013). 14. S. Yu. Davydov, Semiconductors 47, 95 (2013). 15. S. Yu. Davydov, Semiconductors 45, 1070 (2011). 16. P. W. Anderson, Phys. Rev. 124, 41 (1961). 17. S. Yu. Davydov, A. A. Lebedev, and O. V. Posrednik, Elementary Introduction to the Theory of Nanosystems (Lan’, St.Petersburg, 2014) [in Russian].

Translated by E. Smorgonskaya

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