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Volumn 80, Issue 12, 2009, Pages

Conformal mapping and shot noise in graphene

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EID: 70350712333     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.80.125417     Document Type: Article
Times cited : (50)

References (64)
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    • Zero-energy solution of the Dirac equation in two dimensions has the general form Ψ= [ΨA (x+iy), ΨB (x-iy)] T, where the spinor components ΨA and ΨB are holomorphic functions. For the strip geometry, infinite-mass boundary conditions (ΨA □y=0 = ΨB □y=0, ΨA □y=W =- ΨB □y=W) are satisfied after applying the transformation ΨA = ΨA [z (w)], ΨB = ΨB [z (w□)], with z (w) □x (u+iv) +iy (u+iv) an analytical function that turns the coordinate system (u,v) into (x,y). The case of antiperiodic bc is discussed in Sec. 3.
    • Zero-energy solution of the Dirac equation in two dimensions has the general form Ψ= [ΨA (x+iy), ΨB (x-iy)] T, where the spinor components ΨA and ΨB are holomorphic functions. For the strip geometry, infinite-mass boundary conditions (ΨA □y=0 = ΨB □y=0, ΨA □y=W =- ΨB □y=W) are satisfied after applying the transformation ΨA = ΨA [z (w)], ΨB = ΨB [z (w□)], with z (w) □x (u+iv) +iy (u+iv) an analytical function that turns the coordinate system (u,v) into (x,y). The case of antiperiodic bc is discussed in Sec. 3.
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    • In the special case of antiperiodic bc, s=8 for all the modes. For periodic bc considered in Ref., the lowest fully transmitted mode has only a fourfold (spin and valley) degeneracy.
    • In the special case of antiperiodic bc, s=8 for all the modes. For periodic bc considered in Ref., the lowest fully transmitted mode has only a fourfold (spin and valley) degeneracy.
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    • We took ΦlI (r) = Hl (2) (Kr) + rl Hl (1) (Kr) for r> R2, Φl II (r) =a Cl (1) (kr) +b Cl (2) (kr) with Cl (1,2) given by Eq. 32 for R2 >r> R1, Φl III (r) = tl Hl (2) (Kr) for R1 >r, and solved the matching conditions ΦlI (R2) = Φl II (R2), Φl II (R1) = Φl III (R1), d ΦlI dr □ R2 = d Φl II dr □ R2, d Φl II dr □ R1 = d Φl III dr □ R1.
    • We took ΦlI (r) = Hl (2) (Kr) + rl Hl (1) (Kr) for r> R2, Φl II (r) =a Cl (1) (kr) +b Cl (2) (kr) with Cl (1,2) given by Eq. 32 for R2 >r> R1, Φl III (r) = tl Hl (2) (Kr) for R1 >r, and solved the matching conditions ΦlI (R2) = Φl II (R2), Φl II (R1) = Φl III (R1), d ΦlI dr □ R2 = d Φl II dr □ R2, d Φl II dr □ R1 = d Φl III dr □ R1.
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    • For a 2DEG and the limit of K□k, we find from Eq. 29 that Tl ≈64 (π2 K2 R1 R2) -1 □ Cl (1) (k R2) Cl (2) (k R1) - Cl (2) (k R2) Cl (1) (k R1) □-2, which vanishes as (1/K) 2.
    • For a 2DEG and the limit of K□k, we find from Eq. 29 that Tl ≈64 (π2 K2 R1 R2) -1 □ Cl (1) (k R2) Cl (2) (k R1) - Cl (2) (k R2) Cl (1) (k R1) □-2, which vanishes as (1/K) 2.
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    • For the half-Corbino disk (Θ=π), we have fixed the width of the wide lead attached from the right [see Fig. 7] at W∞ =560a (corresponding to N∞ =213 propagating modes for μ∞ =τ/2). The inner radius is varied in the range R1 =30-150a (N1 =23-114 modes). The remaining parameters are the outer radius R2 =160a (the diameter 2 R2 is noticeably smaller than W∞ to model the bulk-lead regime) and the edge-atoms chemical potential μA =- μB =0.01τ. For the quantum dot with circular edges [see Fig. 7], we have fixed the lead radius at R1 =40a (N1 =30 for μ∞ =τ/2), the edge radius is varied in the range 243a< R2 <853a, and the staggered-potential amplitude outside the dot is varied in the range | μA,B | =0.1-0.8τ. We find that the results are insensitive to the confinement strength.
    • For the half-Corbino disk (Θ=π), we have fixed the width of the wide lead attached from the right [see Fig. 7] at W∞ =560a (corresponding to N∞ =213 propagating modes for μ∞ =τ/2). The inner radius is varied in the range R1 =30-150a (N1 =23-114 modes). The remaining parameters are the outer radius R2 =160a (the diameter 2 R2 is noticeably smaller than W∞ to model the bulk-lead regime) and the edge-atoms chemical potential μA =- μB =0.01τ. For the quantum dot with circular edges [see Fig. 7], we have fixed the lead radius at R1 =40a (N1 =30 for μ∞ =τ/2), the edge radius is varied in the range 243a< R2 <853a, and the staggered-potential amplitude outside the dot is varied in the range | μA,B | =0.1-0.8τ. We find that the results are insensitive to the confinement strength.
  • 52
    • 70350714358 scopus 로고    scopus 로고
    • The chemical potential μj =0 in the undoped region (white areas in Fig. 9), except for the outermost edge atoms, where we put μj = μA,B, with μA =- μB =0.01-0.1τ. In the leads (shadow areas) μj = μ∞ =0.4τ, what corresponds to 60 propagating modes (for the lead width fixed at W=200a). The sample area length L is varied but kept equal to the width of undoped leads.
    • The chemical potential μj =0 in the undoped region (white areas in Fig. 9), except for the outermost edge atoms, where we put μj = μA,B, with μA =- μB =0.01-0.1τ. In the leads (shadow areas) μj = μ∞ =0.4τ, what corresponds to 60 propagating modes (for the lead width fixed at W=200a). The sample area length L is varied but kept equal to the width of undoped leads.
  • 60
    • 70350736039 scopus 로고    scopus 로고
    • We suppose the current density j= σ0 E=- σ0 □φ, where φ (x,y) is an electrostatic potential. The conductance G=I/ (φx=L/2 - φx=-L/2) is found by the analytical integration of the current passing the vertical symmetry axis of the system (x=0).
    • We suppose the current density j= σ0 E=- σ0 □φ, where φ (x,y) is an electrostatic potential. The conductance G=I/ (φx=L/2 - φx=-L/2) is found by the analytical integration of the current passing the vertical symmetry axis of the system (x=0).
  • 61
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    • Reference reports an exponential decay for a similar geometry, but it considers a far different leads arrangement, for which the current is effectively flowing along a narrow insulating nanoribbon in graphene.
    • Reference reports an exponential decay for a similar geometry, but it considers a far different leads arrangement, for which the current is effectively flowing along a narrow insulating nanoribbon in graphene.
  • 62
    • 70350726275 scopus 로고    scopus 로고
    • Equation 8 with s=4 and Λ=Λ (W/L) given by Eq. 49.
    • Equation 8 with s=4 and Λ=Λ (W/L) given by Eq. 49.
  • 63
    • 70350743057 scopus 로고    scopus 로고
    • Parameters of the nanoribbonlike system studied numerically are L=963a, W∞ =560a, μ∞ =τ/2, and | μA,B | =0.01τ. The narrow-lead width is varied in the range W/a=40-300.
    • Parameters of the nanoribbonlike system studied numerically are L=963a, W∞ =560a, μ∞ =τ/2, and | μA,B | =0.01τ. The narrow-lead width is varied in the range W/a=40-300.
  • 64
    • 70350702714 scopus 로고    scopus 로고
    • We use Hν (1) (ρ) ≈ 2/ (πρ) exp [i (ρ-νπ/2- π/4)] for ρ□1, and Hν (2) (ρ) = [Hν (1) (ρ)] □.
    • We use Hν (1) (ρ) ≈ 2/ (πρ) exp [i (ρ-νπ/2- π/4)] for ρ□1, and Hν (2) (ρ) = [Hν (1) (ρ)] □.


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