Absorption in atomic wires

The transfer matrix formalism is implemented in the form of the multiple collision technique to account for dissipative transmission processes by using complex potentials in several models of atomic chains. The absorption term is rigorously treated to recover unitarity for the non-Hermitian Hamiltonians. In contrast to other models of parametrized scatterers we assemble explicit potentials profiles in the form of delta arrays, Pöschl-Teller holes, and complex Scarf potentials. The techniques developed provide analytical expressions for the scattering and absorption probabilities of arbitrarily long wires. The approach presented is suitable for modeling molecular aggregate potentials and also supports new models of continuous disordered systems. The results obtained also suggest the possibility of using these complex potentials within disordered wires to study the loss of coherence in the electronic localization regime due to phase-breaking inelastic processes.

Figure 1

(Color online) Scattering and absorption probabilities for one-species delta chains with length $N=15$ and parameters $(a∕a_1)=1.0$ (dashed lines) and $(a∕a_1)=1.0-0.015i$ (solid lines).

Figure 2

(Color online) Scattering process for disordered arrays of 15 deltas with complex couplings. The sequences of the real parts of the characteristic parameters are (a) $\mathrm{Re}(a∕a_j):3$,1,2,0.5,3,2,1,3,0.5,4,5,1,2,2,3 and (b) $\mathrm{Re}(a∕a_j):-1,-4,-3,-1,-2,-3,-4,-1,-2,-3,-1,-4,-4,-2,-3$. The imaginary part of each coupling has been chosen as $\mathrm{Im}(a∕a_j)=-0.01\mid \mathrm{Re}(a∕a_j)\mid$. The arrows in the legends mark the direction of incidence.

Figure 3

Pöschl-Teller potential (arbitrary units).

Figure 4

(Color online) Transmission through a double Pöschl-Teller hole with parameters ${\alpha }_1={\alpha }_2=2\left(x^{-1}\text{units}\right),{\lambda }_1={\lambda }_2=2.4,d_1=d_2=5\left(x\text{units}\right)$. The dashed line corresponds to a single potential hole. The inset shows the composite potential profile (arbitrary units).

Figure 5

(Color online) Transmission patterns for two symmetric composite potentials of ten units each. Their profiles are shown in the insets (arbitrary units). Parameters for the first five potentials of the sequences: (solid line) $\alpha =1$,1,0.5,1,1($x^{-1}$ units), $\lambda =1.66$,2.19,5.01,2.33,2.16, and $d=4$, 4,7,4,4($x$ units) and (dashed line) $\alpha =2$,2,1,1,1($x^{-1}$ units), $\lambda =1.66$,1.66,2.03,2.03,2.03, and $d=2$,2,4,4,4($x$ units).

Figure 6

Real and imaginary parts of the complex Scarf potential (arbitrary units).

Figure 7

(Color online) Characteristic scattering patterns for (a) a complex Scarf well $\alpha =1\left(x^{-1}\text{units}\right),V_1=-0.5,V_2=-0.4$ and (b) a complex Scarf barrier $\alpha =1\left(x^{-1}\text{units}\right),V_1=2,V_2=0.1$.

Figure 8

Maximum value of the absorption probability $A_{\max }$ vs $\mid V_2\mid$ for different values of $V_1$. For each graph the upper (lower) curve corresponds to a barrier (well). A dotted line is used in the forbidden values of $\mid V_2\mid$, nevertheless the whole curve is shown for continuity. The allowed ranges of $\mid V_2\mid$ and the position of the IP can be compared with the values in Table .

Figure 9

Scattering diagram for the complex Scarf potential in terms of the potential amplitudes. The physically acceptable ranges for $V_1,V_2$ correspond to the shaded zones. The curves are the inversion lines given by Eq. (31). The black points mark the correlated values of the amplitudes [Eqs. (30)] generating a fully transparent behavior.

Figure 10

(Color online) Scattering probabilities for double Scarf potentials with parameters $\alpha =1\left(x^{-1}\text{units}\right),V_1=2,V_2=1$ for barriers and $\alpha =1\left(x^{-1}\text{units}\right),V_1=-4,V_2=3.1$ for wells. The solid lines were obtained from the analytical composition technique and the dashed lines correspond to the exact numerical integration of the Schrödinger equation. The insets show the potential profile (in arbitrary units) for each case: a solid line for the real part and a dashed for the imaginary part. The vertical lines limit the portion of the potential that the composition technique takes into account. For numerical integration the whole potential profile was considered. Notice that for low $d$ the exact integration may lead to unphysical scattering because the conditions obtained for physical scattering are not valid for such $d$ values. As $d$ is increased an acceptable scattering is recovered and both methods start converging. Convergence is reached faster in the case of potential barriers where the analytical composition technique works impressively well even for very low values of $d$.

Figure 11

(Color online) Composition of Scarfs with different pair amplitudes describing IPs. Sequence with $\alpha =1\left(x^{-1}\text{units}\right)$ and $(V_1,V_2):(-4,\sqrt{13}),(-4.5,\sqrt{15}),(-5,\sqrt{17}),(-6,\sqrt{21}),(-7,2\sqrt{13})$. All the cutoff distances have been taken equal to $d=6\left(x\text{units}\right)$. The inset shows the potential profile with a solid (dashed) line for the real (imaginary) part in arbitrary units. Notice that the composition remains nondissipative.

Figure 12

(Color online) Periodic chain of 20 complex Scarfs with amplitudes $\alpha =1\left(x^{-1}\text{units}\right),V_1=2,V_2=0.02$ (solid lines), and $V_2=0.05$ (dashed lines) and cutoff distance $d=6\left(x\text{units}\right)$. The transmittivity is not changed much by the small imaginary part of the potentials.

Figure 13

(Color online) Scattering probabilities for a 10-Scarf structure with parameters $\alpha =1\left(x^{-1}\text{units}\right)$ and $(V_1,V_2):(1,0),$$(-0.5,-0.005),$ $(-1.3,0.013),$ $(1.8,0.01),$ $(-1.3,0.013),$ $(4,0.04),$ $(-2.4,-0.024),$ $(3.5,0.04),$ $(-3.1,0.031),$ $(-1.5,0.04)$. The cutoff distance is $d=6\left(x\text{units}\right)$ equal for all of them. The inset shows the real part of the potential profile (arbitrary units).

Figure 14

(Color online) Scattering probabilities for a 10-Scarf structure with parameters $\alpha =1\left(x^{-1}\text{units}\right)$ and $(V_1,V_2):(1,0.04),(1,0.05),(-2,0),(-1.5,0),(-1,0),(-1,0),(-1.5,0),(-2,0),(1,0.1),(1,0.1)$. The cutoff distance is $d=6\left(x\text{units}\right)$ equal for all of them. The inset shows the real part of the potential profile (arbitrary units).

Figure 15

(Color online) Transmittivity for different strengths of the imaginary part of the potentials in (a) a chain of 6-Scarf barriers $V_1=4$,2,2.5,3,3.5,4 and $\alpha =1\left(x^{-1}\text{units}\right)$, (b) a double Scarf well with $V_1=-1,-1$ and $\alpha =2\left(x^{-1}\text{units}\right)$. The cutoff distances in all cases are equal to $d=6\left(x\text{units}\right)$. Each imaginary amplitude reads $V_2=\epsilon \mid V_1\mid$, except the first barrier in example (a) which is maintained real. The insets show the evolution of $T_{\mathrm{int}}$ vs the strength of the imaginary amplitudes.