Introduction

Study of plasmon is one of the most attractive topics in the field of optoelectronics. Now, the properties of plasmons have already been widely applied in the areas of Plasmon rulers1, single-molecule sensing2, solar energy3, chemical reaction4 and cancer therapy5. And plasmon will be applicated in more technical fields due to their superior properties, so the excitation properties of plasmon in nanostructures will continue to attract more attentions. In the last few decades, many researchers have investigated the excitation properties of plasmon in a lot of nanostructure systems6,7,8,9,10,11,12,13,14,15,16,17,18. They have showed that the excitation properties of plasmon are highly affected by various factors, such as the shape, size, and electron filling of nanostructures, the direction and the frequency of external field. Their results are very helpful and provide basis for the theoretical study of plasmon in nanostructure systems. In these results, plasmons are mostly observed by the dynamical electronic response of nanostructure systems, which are mostly calculated based on the electromagnetic theory6,7,8,9, the random-phase approximation10,11,12,15,16 and the time-dependent density functional theory13,14,17,18. However, the modes of plasmon calculated by these methods is not complete under the applied external electric field, for instance, the longitudinal-mode plasmon can not be excited by transverse electric field13,14,17,18. Thus, in our previous work, we had used eigen-equation method to investigate plasmon in atomic cluster and found a new qudarupole mode of plasmon which is not excited by uniform electric field19,20. Then, we have realized that the plasmon modes will be affected by external electric fields. Such property has potential application value in the field of new logic optoelectronic device because it shows opportunity for controlling the modes of plasmon by a new way, ie, changing the distribution of external electric field. Motived by this property of plasmons, this paper will further investigate the influences of external field on the excitation properties of plasmon in linear atomic chain.

Through theoretical calculations of the energy absorption spectra and the eigen function, the fundamental physical properties of plasmon in linear atomic chain are given in this paper. The results show that the eigen frequency of the plasmon is independent on the external electric potential. However, the distribution of external electric potential strongly affects the excitation of plasmon modes, since the energy that sustains the collective electronic excitation are absorbed from the external electric field. The resonance properties of plasmon are also found to be influenced by the external field, especially, the external-field has effect on the evolution of plasmon-energy absorption with the atom number. Abnormal evolution of energy absorption with atom number is discussed.

Model and Theory

The model of linear atomic chain we consider here is shown in Fig. 1, where Lx = (Na + 1)a is the length of the atomic chain, a is the distance between the nearest two atoms, Na is the atom number, Vexe−iωt is the time-dependent external electric potential, and Vex is the space-distribution part of the potential. U is the on-site Coulomb interaction, and only nearest-neighbor Coulomb interactions V is considered21,22.

Figure 1
figure 1

The model diagram of linear atomic chain.

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Based on the tight-binding model and the mean field approximation, in spin-independent case, the one- band Hamiltonian of linear atomic chain system in the frequency space is

$$\begin{array}{ccc}{\rm{H}} & = & -{\rm{\gamma }}\sum _{l < {l}^{\text{'}}}({d}_{l}^{+}{d}_{{l}^{\text{'}}}+h.c.)+\sum _{l}{V}^{ex}(l,\omega ){d}_{l}^{+}{d}_{l}\\ & & +\,\sum _{l}(U\delta \langle n(l,\omega )\rangle +V\sum _{{l}^{\text{'}}}\delta \langle n({l}^{\text{'}},\omega )\rangle ){n}_{l}\,\end{array}$$
(1)

where, γ is the nearest-neighbor transfer energy, l and l′ are the positions of atoms. \({d}_{l}({d}_{l}^{+})\) is the annihilation(creation) operator on position l, and \({n}_{l}={d}_{l}^{+}({d}_{l})\). δ〈n(l, ω)〉 is the charge number induced by the external potential, and the total charge number is n(l, ω) = 〈n(l)0〉 + δ〈n(l, ω)〉, where 〈n(l)0 is the average occupied number in static case. Here, only the frequency-dependent part δ〈n(l, ω)〉 is considered because the frequency-independent part 〈n(l)〉0 doesn’t have effect on the dynamical response of the system. Moreover, the retardation effects arising from the distance between atoms is ignored here, since the interatomic distance is very small23,24.

For finite linear atomic chain, the eigen energy En and the corresponding wave function ψn(l) can be written as

$${E}_{n}=\,-\,2\gamma \,\cos (\frac{n\pi }{{N}_{a}+1})\,,\,n=1,\,2,\,3\,\ldots \,{N}_{a}$$
(2)

and

$${\psi }_{n}(l)=\sqrt{\frac{2}{{N}_{a}+1}}\,\sin (\frac{n\pi }{{N}_{a}+1}l)$$
(3)

Based on the linear response theory, the induced charge number under perturbation Vex(l, ω)e−iωt is

$$\delta \langle n(l,\omega )\rangle =\sum _{l^{\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega )[e{V}^{ex}(l^{\prime} ,\omega )+U\delta \langle n(l^{\prime} ,\,\omega )\rangle +V\sum _{l^{\prime\prime} }\delta \langle n(l^{\prime\prime} ,\,\omega )\rangle ]$$
(4)

where Lindhard function Π(l, l′, ω) is

$${\rm{\Pi }}(l,l^{\prime} ,\,\omega )=2\sum _{mn}\frac{f({E}_{m})-f({E}_{n})}{{E}_{m}-{E}_{n}-\omega -i\eta }{\psi }_{m}^{\ast }(l){\psi }_{n}(l){\psi }_{n}^{\ast }(l\text{'}){\psi }_{m}(l^{\prime} )$$
(5)

the 2 appears in the front of the equation above corresponds to two spins, f(Em) is the fermi function, parameter η is the scattering rate which makes the resonance of plasmon a broadening. For simplicity, Eq. (4) can be written as

$$\delta \langle n(l,\omega )\rangle -\sum _{l^{\prime} l^{\prime\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega ){v}_{l^{\prime} l^{\prime\prime} }\delta \langle n(l^{\prime\prime} ,\omega )\rangle =\sum _{l^{\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega )[e{V}^{ex}(l^{\prime} ,\omega )]$$
(6)

where

$${v}_{l^{\prime} l^{\prime\prime} }=\{\begin{array}{c}U,\,|l^{\prime} -l^{\prime\prime} |=0\\ V,\,|l^{\prime} -l^{\prime\prime} |=1\\ 0,\,\,else\end{array}$$
(7)

Take the induced charge Q(l, ω) = δn(l, ω) into the Eq. (6), have

$$Q(l,\omega )-\sum _{l^{\prime} l^{\prime\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega ){v}_{l^{\prime} l^{\prime\prime} }Q(l^{\prime\prime} ,\omega )={e}^{2}\sum _{l^{\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega ){V}^{ex}(l^{\prime} ,\omega )$$
(8)

Following Eq. (8), the energy absorption can be calculated by function

$$L({\rm{\omega }})=\frac{1}{2}\omega {\rm{Im}}\{\sum _{l}Q(l,\omega ){V}^{ex}(l,\omega )\}$$
(9)

and L(ω) shows a peak at plasmon frequency ω, and it should be noted that the energy absorption function depends on both the induced charge and the external potential.

Furthermore, we set Vex(l′, ω) = 0, and get the eigen-equation

$$\sum _{l^{\prime\prime} }[{\delta }_{ll^{\prime\prime} }-\sum _{l^{\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega ){v}_{l^{\prime} l^{\prime\prime} }]Q(l^{\prime\prime} ,\omega )=0$$
(10)

According to the Eq. (10), the plasmon frequency ω can be also calculated by

$$F({\rm{\omega }})={\rm{\det }}[{\delta }_{ll^{\prime\prime} }-\sum _{l^{\prime} }{\rm{\Pi }}(l,l^{\prime} ,\omega ){v}_{l^{\prime} l^{\prime\prime} }]=0\,$$
(11)

Here, the eigen-frequency of plasmon should be obtained by both Re(F(ω)) = 0 and Im(F(ω)) ≈ 0. If Im(F(ω)) ≫ 0, the collective excitation will decay rapidly into single particle excitation. So the plasmon frequency can be also found at the peak of the eigen function A(ω) = Im(1/F(ω)).

Results and Discussions

In the following, most of the results are calculated in the wannier representation except the charge distribution. The distribution of charge in the real space is calculated by \(Q(r,{\rm{\omega }})=\sum _{l}Q(l,\omega ){|\varphi (l,r)|}^{2}\), where ϕ(l, r) is a s-wave function which is identical to the wannier function in the tight-binding model, here, we set ϕ(l, r) = R40(r − l)Y00(θ, ϕ). In all calculations, we set parameters U = 3, V = 1, η = 0.01. The unit of frequency is the parameter γ, the unit of the space coordinate is the distance a. Arbitrary units are applied for the values of the eigen function, energy absorption and the charge response.

First, we focus on the influence of external-field distribution on the excitation of plasmon modes, which are given by the peaks of the eigen-function and the energy absorption spectra. In Fig. 2 (right y label), the eigen-function A(ω) shows three peaks at frequencies ω1 = 1.764, ω2 = 2.982 and ω3 = 4.176, respectively, which implies three eigen-modes of plasmon in the system of Na = 4, Ne = 2. Here, all modes of plasmon can be found by the peaks of the eigen-function, since all the plasmon frequencies are the solutions of the eigen equation19,20,25,26,27. However, the spectra of eigen function cannot be observed in experiment, so we give the spectra of energy-absorptions under potentials \({V}_{1}^{ex}-{V}_{5}^{ex}\). The external perturbations should be understood as Viexe−iωt, i = 1, 2, 3, …, and all these potentials are applied in the direction along the chain. Here, the forma of all the potentials are given in the space of wannier representation. In the real space, potentials are calculated by \({V}_{i}^{ex}({\bf{r}},{\rm{\omega }})=\sum _{l}{V}_{i}^{ex}(l,\omega ){|\varphi (l,{\bf{r}})|}^{2}\), i = 1,2,3, …, and the distributions of potentials \({V}_{1}^{ex}-{V}_{5}^{ex}\) in the x-y plane are showed in Fig. 2(b). Since s-wave function ϕ(l, r) is an odd function, V(r, ω) has the same symmetry as V(l, ω). Different forms of potentials are chosen to excite different modes of plasmon. In the space of wannier representation, \({V}_{1}^{ex}\) is similar to a potential of uniform electric field, \({V}_{2}^{ex}-{V}_{5}^{ex}\) are potentials of non-uniform electric field, especially, \({V}_{2}^{ex}\) is a symmetrical potential, \({V}_{4}^{ex}\,\,\)is an anti-symmetrical potential, \({V}_{3}^{ex}\) and \({V}_{5}^{ex}\) are asymmetrical potentials, and \({V}_{5}^{ex}\) is similar to the Yukawa potential. To display the energy-absorption peaks under different electric potentials at the same time, the amplitudes of various potentials are given with appropriate values. In Fig. 2 (left y label), the energy absorptions are changed with the frequency of external potentials, if applying potential \({V}_{1}^{ex}\) to the system, there is only one energy-absorption peak at frequency ω1, which implies that only the mode of plasmon at frequency ω1 is excited by \({V}_{1}^{ex}\). And even if the frequency of \({V}_{1}^{ex}\) equals to ω2 or ω3, the value of energy-absorptions is zero, which obviously implies the two modes of plasmon at frequencies ω2 and ω3 can not be excited by potential \(\,{V}_{1}^{ex}\). Similarly, this figure also show that only the mode of plasmon at frequency ω2 is excited by \({V}_{2}^{ex}\), only two modes of plasmon at frequencies ω1 and ω2 are excited by \({V}_{3}^{ex}\), only two modes of plasmon at frequencies ω1 and ω2 are excited by \({V}_{4}^{ex}\), and all three modes of plasmon at frequencies ω1, ω2 and ω3 are excited by \({V}_{5}^{ex}\). These results can be also understood as: the mode of plasmon at frequency ω1 can not be excited by potential \({V}_{2}^{ex}\); the mode of plasmon at frequency ω2 can not be excited by potentials \({V}_{1}^{ex}\) and \({V}_{{\rm{4}}}^{ex}\); the mode of plasmon at frequency ω3 can not be excited by potentials \({V}_{1}^{ex}\), \({V}_{2}^{ex}\,\)and \({V}_{3}^{ex}\). Such results show opportunity to control the plasmon modes by external electric potential. To control the plasmon mode, we must know whether the plasmon mode can be excited by an external potential or not? What kind of potential is needed to excite a plasmon mode. To answer these questions, this paper will further investigate the response of charge to the external field and show the answer by analyzing the charge distribution and the energy absorption at plasmon.

Figure 2
figure 2

(a) Spectra of energy absorptions L(ω) (left y label) and the eigen function A(ω) (right y label). Where, atom number Na = 4, electron number Ne = 2, external potentials \({V}_{1}^{ex}=\,-\,2l\), \({V}_{2}^{ex}=\,-\,\frac{2}{|l-{L}_{x}/2|}\), \({V}_{3}^{ex}=\frac{6}{l}\), \({V}_{4}^{ex}=\,-\,\frac{1}{l-{L}_{x}/2}\), \({V}_{5}^{ex}=\,-\,\frac{16}{l}{e}^{-l}\). (b) Distributions of potentials \({V}_{1}^{ex}-{V}_{5}^{ex}\) (top to bottom) in the real space.

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In Fig. 3(a), the eigen-charge distribution is showed at plasmon frequency ω1. The eigen charge \({Q}^{eg}(r,\omega )=\sum _{l^{\prime\prime} }{Q}^{eg}(l^{\prime\prime} ,\omega ){|\varphi (l^{\prime\prime} ,r)|}^{2}\), where Qeg(l′, ω) are calculated by Eq. (10) by setting Q(l″ = 1, ω) = 1. Qeg(l″, ω) is the eigenvector solution of Eq. (10), so each plasmon mode corresponding to one kind of eigen charge which is independent on the external potential. Here, only the real part of the charge is given in Fig. 3(a) (in the following figures, only the real parts of charge are plotted too, because the distribution of the imaginary parts of the charge is like the one of the real parts). In Fig. 3(a), the positive and negative charges asymmetrically gather in two sides of the atomic chain, respectively. Such eigen charge has dipole moment along the chain direction, which implies that the plasmon mode at frequency ω1 is a dipole mode. Figure 3(b–f) show the distributions of charge at frequency ω1 under potentials \({V}_{1}^{ex}\) − \({V}_{5}^{ex}\), respectively. In Fig. 3(b,d–f), the amplitudes of the charges are very large. Combine with the energy absorption peaks at ω1 in Fig. 1, we can know that the charges under potentials \({V}_{1}^{ex}\), \({V}_{3}^{ex}\), \({V}_{4}^{ex}\) and \({V}_{5}^{ex}\) are resonated at frequency ω1. And the distributions of these resonated charges are all characterized by dipole moment, which are like that of eigen charge at ω1. However, in Fig. 3(c), the charge distribution under potential \({V}_{2}^{ex}\) is symmetric among the atomic chain, which is different from the eigen-charge distribution at ω1. And the amplitude of the charge is very small, since the charge under potential \(\,{V}_{2}^{ex}\) is not resonated at frequency ω1. The non-resonant charge shows symmetric distribution because it is polarized by symmetric potential \(\,{V}_{2}^{ex}\).

Figure 3
figure 3

The charge distribution in the system of Fig. 2 at plasmon frequency ω1 = 1.764. (a) The eigen charge. (b–f) The charge induced respectively by external potentials \({V}_{1}^{ex}-{V}_{5}^{ex}\).

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In Fig. 4(a), the eigen charge at frequency ω2 is symmetric among the atomic chain, such eigen charge has quadrupole moment along the chain direction, which implies that the plasmon at frequency ω2 is a quadrupole mode. In Fig. 4(c,d,f), the charges under potentials \({V}_{2}^{ex}\), \({V}_{3}^{ex}\) and \({V}_{5}^{ex}\) are resonated at frequency ω2, and have the same distribution characteristic as the one of eigen charge at frequency ω2. However, in Fig. 4(b,e), due to the charges under potentials \({V}_{1}^{ex}\) and \({V}_{4}^{ex}\) are not resonated at frequency ω2, the non-resonant charges are polarized by potentials \({V}_{1}^{ex}\) and \({V}_{4}^{ex}\) respectively. And the distributions of these charges are different from eigen-charge distribution at frequency ω2.

Figure 4
figure 4

The charge distribution in the system of Fig. 2 at plasmon frequency ω2 = 2.982. (a) The eigen charge. (b–f) The charge induced respectively by external potentials \({V}_{1}^{ex}-{V}_{5}^{ex}\).

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In Fig. 5(a), the eigen charge is asymmetric among the atomic chain, such distribution of charge has both dipole and quadrupole moment along the chain direction, which implies that the plasmon at frequency ω3 is a normal high order mode. In Fig. 5(e,f), the charges under potentials \({V}_{4}^{ex}\) and \({V}_{5}^{ex}\) are resonated at frequency ω3 and have the same distribution characteristic as the one of eigen charge at frequency ω3. However, in Fig. 5(b,d), the charges under potentials \({V}_{1}^{ex}\), \({V}_{2}^{ex}\) and \({V}_{3}^{ex}\) are not resonated at frequency ω3, and the distributions of these charges are different from the distribution of eigen charge at frequency ω3. The non-resonant charges under potentials \({V}_{1}^{ex}\), \({V}_{2}^{ex}\) and \({V}_{3}^{ex}\) are polarized by potentials \({V}_{1}^{ex}\), \({V}_{2}^{ex}\) and \({V}_{3}^{ex}\), respectively.

Figure 5
figure 5

The charge distribution in the system of Fig. 2 at plasmon frequency ω3 = 4.176. (a) The eigen charge. (b–f) The charge induced respectively by external potentials \({V}_{1}^{ex}-{V}_{5}^{ex}\).

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The above results showed in Figs 3–5 imply that the distribution characteristics of the resonant charge will be the same as the one of eigen charge. Here, we assume the plasmon frequency in the atomic chain is ωP, then when the frequency of external potential is equal to ωP, according to the Eq. (8), we obtain that the energy absorption \(\,L({\omega }_{P})\propto \frac{1}{2}{\omega }_{P}{\rm{Im}}\{\sum _{l}{Q}^{eg}(l,{\omega }_{P}){V}^{ex}(l,{\omega }_{P})\}\), where, Qeg(l, ωP) is the intrinsic property of plasmon which only depends on the plasmon frequency ωP, and Qeg(l, ωP) has the same symmetry as Qeg(r, ωP). For a fixed plasmon mode, Qeg(l, ωP) is a fixed function, and the energy absorption only depends on the external potentials Vex(l, ωP). Thus, L(ωP) will be zero under some specific potentials Vex(l, ωP), and the collective oscillations will not be maintained with no energy absorption due to scattering. This is the reason why the mode of plasmon at frequency ω1 cannot be excited by \({V}_{2}^{ex}\) (see in Fig. 2), because the energy absorption L(ωP) between the antisymmetric eigen charge (see in Fig. 3(a)) and the symmetric potential \({V}_{2}^{ex}\) is zero. For the same reason, the mode of plasmon at frequency ω2 cannot be excited by \({V}_{1}^{ex}\) and \(\,{V}_{4}^{ex}\), the mode of plasmon at frequency ω3 cannot be excited by \({V}_{1}^{ex}\), \({V}_{2}^{ex}\) and \({V}_{3}^{ex}\). Hence, if the eigen-charge distribution at each mode of plasmon is calculated or observed, one can control the excitation of arbitrary plasmon mode by external field.

Next, we further investigate the influence of external-field distribution on the evolution of energy absorption with the atom number. To be simple, Fig. 6 only show the evolution of the dipole mode of plasmon, where Fig. 6(a,b) are obtained under potentials \({V}_{1}^{ex}\) and \({V}_{6}^{ex}\), respectively. Potential function |\({V}_{1}^{ex}\)| is a monotone increasing function of space coordinates of atoms, potential function |\({V}_{6}^{ex}\)| is a monotone decreasing function of space coordinates of atoms. In Fig. 6(a,b), the plasmon frequencies shift to red with the increase of the atom number which are the same as in refs13,14,15,16,19,20. However, in Fig. 6(b), the intensity of energy-absorption peak (ie, plasmon-energy absorption) monotonically decreases with the increase of the atom number, where this result is abnormal and is not consistent with the normal result13,14,15,16,19,20. The normal result in refs13,14,15,16,19,20 is consistent with the result in Fig. 6(a) that the plasmon-energy absorption increases with the atom number increasing. And we notice that the normal result commonly occurs under monotone increasing potential. Then, it’s easy to find out that the difference between Fig. 6(a,b) results from the effect of the potential on the intensity of collective oscillations: under monotone decreasing potential, the collective oscillations of charge will be weaker in more-atoms system, so less energy will be absorbed to sustain the weaker collective oscillations. Moreover, the influences of external field on the evolution of energy absorption at quadrupole and other modes of plasmon are also calculated, however, these parts of results are not shown in this paper since the calculated results are similar to the one of dipole mode.

Figure 6
figure 6

The evolution of energy absorption with atom number, where (a) under potential \({V}_{1}^{ex}\), (b) under potential \({V}_{6}^{ex}=\,-\,2(\frac{1}{l}+\frac{1}{l-{L}_{x}})\), (c) under potential \({V}_{7}^{ex}=\frac{1}{4}{V}_{1}^{ex}+{V}_{6}^{ex}\). The electron number is fixed to be Ne = 2.

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Inspired by the results of Fig. 6(b), under potential \({V}_{7}^{ex}=\frac{1}{4}{V}_{1}^{ex}+{V}_{6}^{ex}\), we have shown that the energy absorption (see Fig. 6(c)) at plasmon will not change monotonically with the atom number. So that, with the increase of atom number, there are three types of evolutions of energy absorption: the first one, the energy absorption increases monotonically; the second one, the energy absorption decreases monotonically; the third one, the energy absorption changes non-monotonically. And it’s easy to find that, the first type of evolutions occurs under monotone increasing potential, the second type of evolutions occurs under monotone decreasing potential. For quadrupole plasmon that excited by symmetric potential, we should point out that the first type of evolutions occurs as the absolute value of the external potential increases in the left half part of space. This result may provide a new idea for observing quantum mode of plasmon in tiny cluster system.

In Fig. 7, we further investigate the influence of external-field distribution on the evolution of energy absorption by setting different electron number and external potentials, where Fig. 7(a,b) correspond to \({V}_{1}^{ex}\) and \({V}_{6}^{ex}\), respectively. Here, besides differences in the intensity of energy absorption, the evolutions of energy absorptions are similar under potentials \({V}_{1}^{ex}\) and \({V}_{6}^{ex}\). So, regardless of any type of potential is applied to the atomic chain, the intensity of the energy absorptions at plasmon will increase with the increase of the electron number, because more electrons will participate into the collective oscillations. And result from the increase of the fermi energy, the plasmon frequencies will shift to blue with the electron number increasing.

Figure 7
figure 7

The evolution of energy absorption with electron number, where (a) under potential \({V}_{1}^{ex}\), (b) under potential \({V}_{6}^{ex}\). The atom number is fixed to be Na = 10.

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Conclusion

We have studied the plasmon excitation in linear atomic chains based on the tight binding model and self-consistent linear response theory. Results show that the plasmon modes are different under various types of external potentials. Each mode of plasmon corresponds to one kind of eigen-charge distribution. And, if the eigen-charge distribution at each plasmon mode is given, the excitation of arbitrary plasmon mode can be controlled by external field. Furthermore, the relationship between the plasmon-energy absorption and the atom number is influenced by external electric potential. Especially, under monotone decreasing potential, the plasmon-energy absorption may decrease with the increase of atom number. However, regardless of any type of potential is applied to the same atomic chain, the plasmon-energy absorptions will increase with the increase of the electron number. Controlling various modes of plasmon by external-field distribution has application value in new logic optoelectronic device. And the influence of external-field distribution on the evolution of energy absorption with electron number may provide a new idea for observing quantum mode of plasmon in tiny cluster system.