Negativity of the excess noise in a quantum wire capacitively coupled to a gate

F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert

Phys. Rev. B 75, 045332 – Published 19 January 2007

Abstract

The electrical current noise of a quantum wire is expected to increase with increasing applied voltage. We show that this intuition can be wrong. Specifically, we consider a single-channel quantum wire with impurities and with a capacitive coupling to nearby metallic gates and find that its excess noise, defined as the change in the noise caused by the finite voltage, can be negative at zero temperature. This feature is present both for large $(c ⪢ c_{q})$ and small $(c ⪡ c_{q})$ capacitive coupling, where $c$ is the geometrical and $c_{q}$ the quantum capacitance of the wire. In particular, for $c ⪢ c_{q}$ , negativity of the excess noise can occur at finite frequency when the transmission coefficients are energy dependent—i.e., in the presence of Fabry-Pérot resonances or band curvature. In the opposite regime $c ≲ c_{q}$ , a nontrivial voltage dependence of the noise arises even for energy-independent transmission coefficients: at zero frequency the noise decreases with voltage as a power law when $c < c_{q} ∕ 3$ , while, at finite frequency, regions of negative excess noise are present due to Andreev-type resonances.

Received 9 August 2006

DOI:https://doi.org/10.1103/PhysRevB.75.045332

Authors & Affiliations

F. Dolcini¹, B. Trauzettel^2,3, I. Safi⁴, and H. Grabert⁵

¹Scuola Normale Superiore and NEST-INFM-CNR, 56126 Pisa, Italy
²Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The Netherlands
³Department of Physics and Astronomy, University of Basel, 4056 Basel, Switzerland
⁴Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France
⁵Physikalisches Institut, Albert-Ludwigs-Universität, 79104 Freiburg, Germany

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Vol. 75, Iss. 4 — 15 January 2007

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Figure 1
Schematic of a quantum wire capacitively coupled to a nearby gate (with a capacitance per unit length $c$ ). The wire of finite length $L$ is connected at $x = \pm L ∕ 2$ to reservoirs with electrochemical potentials $μ_{L}$ and $μ_{R}$ .Reuse & Permissions
Figure 2
The case of large capacitive coupling $c ∕ c_{q} \to \infty$ : Voltage derivative of the shot noise (22) in units of $e^{3} ∕ 2 π ℏ$ (solid curve) and conductance (23) in units of $e^{2} ∕ 2 π ℏ$ (dotted curve) as a function of bias $V$ for a quantum wire with linear dispersion relation and two impurities of equal strength $Λ ∕ ℏ v_{F} = 3$ , located at the contacts $x_{1, 2} = \pm L ∕ 2$ . The electrochemical potentials are varied asymmetrically $μ_{L ∕ R} = E_{F} + γ_{L ∕ R} e V$ , where $γ_{L} = 3 ∕ 2$ and $γ_{R} = 1 ∕ 2$ , and the Fermi level $E_{F}$ lies near the resonance at $E_{F} = 5.9 ℏ ω_{L}$ with $ω_{L}$ given by Eq. (24). At low bias $V \to 0$ the voltage derivative of the noise and the conductance are positive [due to the fact that $S (ω = 0; V = 0) = 0$ , $S ⩾ 0$ , and $I (V = 0) = 0$ with $I ⩾ 0$ for $V ⩾ 0$ ]. Negativity of both quantities is, however, present for a wide range of voltage values up to $e V ≃ ℏ v_{F} ∕ L$ , indicating that the current and the shot noise can decrease with bias. Interestingly, a decreasing noise does not necessarily correspond to a decreasing conductance. Inset: energy dependence of the transmission coefficient showing resonances.Reuse & Permissions
Figure 3
The case of large capacitive coupling $c ∕ c_{q} \to \infty$ : For a quantum wire with linear dispersion relation and two impurities of equal strength $Λ ∕ ℏ v_{F} = 10$ located at positions $x_{1, 2} = \pm L ∕ 2$ the excess noise [in units of the energy $ℏ ω_{L}$ associated with the ballistic frequency (24)] is shown as a function of the noise frequency $ω$ and the applied voltage $V$ for a measurement point at $x = - 0.6 L$ . The values of the electrochemical potentials in the leads are $μ_{L ∕ R} = E_{F} \pm e V ∕ 2$ with $E_{F} = 6 ℏ v_{F} ∕ L$ . Regions of negative excess noise are clearly visible. This negativity originates from Fabry-Perot resonance phenomena.Reuse & Permissions
Figure 4
The case of large capacitive coupling $c ∕ c_{q} \to \infty$ : negativity of the excess noise due to band curvature effects for a quantum wire with one impurity. (a) Parabolic band curvature (30). The solid curve shows the excess noise [in units of $e^{2} ω_{L} ∕ 2 π$ with $ω_{L}$ given in Eq. (24)] as a function of the bias $V$ for a frequency $ω = ω_{L} ∕ 2$ at the measurement point $x = - L$ . The impurity of strength $Λ ∕ ℏ v_{F} = 1$ is located in the middle; the Fermi level is $E_{F} = 6 ℏ ω_{L}$ , and the bias weight is $γ_{L} = 1$ . The dotted curve shows the excess noise for a wire with linear dispersion and with the same transmission coefficient at the Fermi level. (b) Sinusoidal band dispersion (31) for the tight-binding model (16). The solid curve shows the excess noise (in units of $e^{2} τ ∕ 2 π ℏ$ ) as a function of the bias $V$ for a frequency $ω = 0.2 τ ∕ ℏ$ at the measurement point $j = - 51 a_{0}$ for a wire of length $L = 100 a_{0}$ . The impurity of strength $Λ ∕ a τ = 1$ is located in the middle; the Fermi level is $E_{F} = τ$ , and the bias weight is $γ_{L} = 1$ . The dotted curve has the same meaning as in (a). The slope $d S ∕ d V$ is shown as an inset.Reuse & Permissions
Figure 5
Excess noise for a gated wire with capacitance $c = c_{q} ∕ 10$ , with an impurity shifted by $L ∕ 4$ off the center of the wire. The measurement point is at $x = - 0.6 L$ . Regions of negative excess noise are visible at frequencies $ω \sim π ω_{L} ∕ g$ , where $ω_{L}$ is the ballistic frequency (24) and $g$ is the parameter defined in Eq. (32).Reuse & Permissions

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