Abstract
We have calculated and analyzed the surface-state energies and wave functions in quasi-two dimensional (Q2D) organic conductors in a magnetic field parallel to the surface. Two different forms for the electron energy spectrum are used in order to obtain more information on the elementary properties of surface states in these conductors. In addition, two mathematical approaches are implemented that include the eigenvalue and eigenstate problem as well as the quantization rule. We find significant differences in calculations of the surface-state energies arising from the specific form of the energy dispersion law. This is correlated with the different conditions needed to calculate the surface-state energies, magnetic field resonant values and the surface wave functions. The calculations reveal that the value of the coordinate of the electron orbit must be different for each state in order to numerically calculate the surface energies for one energy dispersion law, but it has the same value for each state for the other energy dispersion law. This allows to determine more accurately the geometric characteristics of the electron skipping trajectories in Q2D organic conductors. The possible reasons for differences associated with implementation of two distinct energy spectra are discussed. By comparing and analyzing the results we find that, when the energy dispersion law obtained within the tight-binding approximation is used the results are more relevant and reflect the Q2D nature of the organic conductors. This might be very important for studying the unique properties of these conductors and their wider application in organic electronics.
1 Introduction
Surface states are important class of quantum states localized near the surface of a solid. These states are not present in the bulk, and their presence is correlated to the change in periodicity of the crystal lattice near the surface. The presence of a boundary significantly changes the character of the motion of the conduction electrons and leads to the appearance of a discrete spectrum of surface states if the electrons are almost specularly reflected from the surface of the sample [1], [2], [3], [4], [5], [6]. These states have properties which are quite different from the bulk electronic structure of the material. The quantum mechanical surface states are formed by the surface electrons that move along the surface on skipping trajectories by periodic specular reflection. In the presence of a magnetic field, due to action of the Lorentz force, the electrons are impelled toward the surface and hence are bound to the surface region. The oscillations of the surface impedance of metals in weak fields, discovered by M. Khaikin [7], occur due to resonant transitions between discrete magnetic surface levels of skipping electrons. Nee and Prange [1], [2], [3] have made use of the surface levels, by quantizing the periodic motion of the skipping electrons specularly reflected from the surface of the metal, to explain the oscillations of the impedance and shown that they can be treated as a cyclotron resonance due to the transitions between different magnetic surface levels. In contrast to the Landau levels that represent the electrons in the cyclotron electron orbits, the surface energy levels that represent the surface skipping electrons depend on the coordinate of the rotation center since the period of the skipping electrons depend on the coordinate of the center of rotation. The experimental studies of the surface impedance oscillations in metals [8], [9], [10], [11] have shown that there is a group of surface electrons that makes a significant contribution to the oscillations of the metal as a function of the magnetic field. Moreover, all the metals in which this effect is observed have almost cylindrical Fermi surface parts. Therefore, knowing the local geometry of the Fermi surface is important for studying the surface states and phenomena related to them.
Layered organic conductors are very attractive for researchers due to their unique properties including a sensitive response to weak structural changes and a large number of phase transformations under an external magnetic field [12]. A characteristic feature of the electronic properties of organic metals is a pronounced quasi-one (Q1D) or quasi-two (Q2D) anisotropy arising due to their crystal structure. The Fermi surface of most of the layered organic conductors is known as obtained from the transport experiments and has a form of slightly corrugated sheets for the Q1D and slightly corrugated cylinders for the Q2D organic conductors. On the other hand, high-frequency phenomena strongly suggest that in some families of organic conductors the Fermi surface is a union of Q2D cylinders and quasi-one dimensional sheets. All these properties make layered organic conductors excellent candidates for studding the surface states properties and the effects that occur as a result of the electronic transitions between the discrete surface levels.
In a previous work on the elementary properties of magnetic surface levels in layered organic conductors, the studies were conducted in the simplest case with respect to the geometry of the problem and the form of the energy spectrum [13]. However, due to the simplifications made for the electron energy spectrum, some of the characteristics of the Fermi surface concerning the number of electron orbits involved in the formation of the surface states are disregarded. Therefore, it is recommended that the surface states are also studied using the most general assumptions concerning the form of Q2D electron energy spectrum derived within the tight-binding approximation which is usually used for theoretical studies of the electron properties of organic conductors [12].
As a continuation of the previous work, in the present paper, we consider the surface states and their properties in organic conductors under the most general assumptions concerning the form of Q2D electron energy spectrum obtained in the frame of the tight-binding theory which assumes a weak coupling within the plane of the layers and a strong coupling approximation for the electrons belonging to adjacent layers. In addition, we are also considering the surface-state energies and wave functions obtained by using another form for the Q2D electron energy spectrum, previously used to investigate the nonmagneto–oscillatory surface effects in these conductors in order to reveal the differences that arise by implementation of different forms of the electron energy spectrum and how it reflects on the elementary properties of the surface states in organic conductors.
2 Formulation of the problem: numerical calculation of the magnetic surface levels
The usual approach in theoretical studies of the electronic properties of layered organic conductors is by using the energy spectrum of the form
or if the anisotropy of the energy spectrum in the plane of the layers is neglected, then it is given by the equation
A simplified form of the above equation obtained within the tight-binding approximation when only the first term in the sum over n is taken into account is widely used to describe the transport properties of layered organic conductors.
Here px and py are the in-plane momentum of the electron,
Previously, the non magneto–oscillatory surface resistance in Q2D organic conductors was studied by applying a different approach which involves a simpler form for the electron energy spectrum which corresponds to a corrugated Fermi surface with different profiles [14]. Here, we shall also consider this form for the electron energy spectrum in order to calculate the magnetic surface levels and compare them with those obtained by using the more general form of the energy spectrum derived within the tight-binding approximation. For the purpose of the investigation of the magnetic surface-state energies, we represent it in the following form
The above equation corresponds to a Fermi surface with a slightly corrugated cylinder open along the pz axis similar to that represented by Eq. (3).
In order to compare the energies of the surface levels obtained using the two different forms for the electron energy spectrum as well as to emphasize the differences due to applying a different approach in the calculations of the magnetic surface levels in organic conductors, we shall consider the simplest geometry for the problem that includes a magnetic field applied strictly parallel to the surface of the conductor along the y-axis,
2.1 The eigenvalue and eigenstate problem
To determine the surface-state energies and the corresponding wave functions, the Schrödinger equation for the problem must be solved. This involves the following Hamiltonian
where the potential V(z) is zero in the region z > 0 and an infinite surface potential barrier for z < 0. Using the Landau gauge, the magnetic field is described by the vector potential
and vanishes for
We consider the most important group among the surface electrons, the skipping electrons, in which the center of the classical orbit is located outside the conductor at a distance approximately equal to the radius of the electron trajectory
To better understand what orbits are involved in formation of the magnetic surface levels, we present in Figure 1 the Fermi surface of a Q2D organic conductor and the possible electron orbits when the magnetic field is along the y direction. The Fermi surface is an open and slightly corrugated along the pz-axis cylinder (Figure 1c) and the closed orbits which are cross-sections of the Fermi surface by the plane
In the following, we shall make some justified assumption in an attempt to find the eigenvalues and eigenstates, i.e., the surface-state levels and wave functions of the problem. First, since the motion of skipping electrons is in the proximity of the surface, in the skin layer with a depth δ, their z coordinate is small (
where
Second, an expansion in power series of
The solution of Eq. (9) is given by the Airy function [16]
where
The surface-state energy levels are obtained from the boundary condition
This yields the following relation for the quantity q that allows to calculate the surface-state energies εn of the Q2D organic conductors in a magnetic field
The above identity does not allow to analytically derive a relation for the surface-state energies, and therefore, they are calculated numerically by evaluating q and then using the expression
In a similar way, by using the energy dispersion law defined by Eq. (4) and keeping only the linear in z term in the expansion in power series of
which yields the surface wave functions of the form
with a normalization constant C
The corresponding magnetic surface levels in terms of the Fermi momentum are determined numerically from the following relation
2.2 The quantization condition
For the purpose of this study as well as for comparison, we shall explore all the possible ways available for determination of the surface-state energies in Q2D organic conductors. The motion of the skipping electrons along the z direction is periodic and therefore quantized. This allows us to determine the surface energy levels from the Bohr–Sommerfeld quantization condition by taking into account that in a magnetic field the quantities pz and Z are canonically conjugate variables. The quantization rule reads as
and can be applied within the quasi-classical approximation, which for the Q2D organic conductors is applicable when the condition
The value of γ is determined by the boundary conditions. For a complete cyclotron orbit,
Here, we make use of the boundary condition
where
Evidently, Eq. (22) is similar to the one that we derived from solving the eigenvalue and eigenstate problem (Eq. (14)). Hence, it is reasonable to expect similar values for the magnetic surface-state energies.
Similarly, applying the quantization rule in the case of energy spectrum defined by Eq. (4), the surface-state energies are obtained from the relation
The equations derived in this section for the surface-state energies and wave functions are valid in the limiting case
3 Discussion
In the following, we shall present the surface-state energies evaluated by implementing different energy dispersion law in a magnetic field parallel to the surface. In addition, we shall also calculate the resonance values of the magnetic field at which the surface electron excitation occurs. Due to the resonance transitions between the surface states, there appear peaks in the quantum oscillations of surface resistance at certain magnetic field values
3.1 The surface-state energies
In Table 1, we present the surface-state energies for n = 1 to n = 6 calculated for B = 3 T from the corresponding equations derived in the previous section. We shall analyze and compare the results in order to obtain information about the elementary properties of the magnetic surface states that can be of great significance for accurate experimental estimation of the values of the relevant quantities for the magnetic surface oscillations.
n | Eq. (14) | Eq. (18) | Eq. (22) | Eq. (23) |
---|---|---|---|---|
1 | 0.53679 | 0.53691 | 0.53679 | 0.53695 |
2 | 0.53683 | 0.53695 | 0.53684 | 0.53700 |
3 | 0.53686 | 0.53699 | 0.53688 | 0.53703 |
4 | 0.53688 | 0.53702 | 0.53690 | 0.53706 |
5 | 0.53690 | 0.53704 | 0.53693 | 0.53709 |
6 | 0.53692 | 0.53707 | 0.53695 | 0.53711 |
We find that there is a slight difference in the values of the surface-state energies obtained by applying different energy dispersion law and different approach in the calculations. Comparing the surface-state energies from the eigenvalue and eigenstate problem, we find that Eq. (18) yields slightly larger values than Eq. (14). The difference between them is of order of 10−4 eV and is slightly increasing while going from lower to higher surface states from 1.2 × 10−4 eV for n = 1 to 1.5 × 10−4 eV for n = 6. On the other hand, the quantization rule yields slightly higher values for the surface-state energies for both energy dispersion laws than those obtained by solving the Schrödinger equation. Apart from that, in this case, the difference in energy of a surface state with a quantum number n, obtained from Eqs. (22) and (23), is constant of order of 1.6 × 10−4 eV. The increasing difference in the surface-state energies obtained from the eigenvalue problem is correlated with the change of the value of the coordinate of the electron orbit in case when calculations are performed by using the energy dispersion law (3). Indeed, the energies are calculated numerically with each mathematical approach used but under different conditions. These conditions involve different allowed values for the coordinate of the electron orbit from the surface when using Eq. (14), i.e., in order to obtain the energies from Eq. (14) one has to apply different values for zc. The values for zc are the highest for the lowest state and decrease for higher states. The allowed skipping trajectories are determined from the expression
On the other hand, this is not the case when the other energy dispersion law (Eq. (4)) is used. On contrary, here the surface-state energies might be calculated by using one value for zc = 1.5 for both the eigenvalue problem and quantization rule. However, the expression
Using the values for the surface-state energies given in the second and third column in Table 1, we present the magnetic surface levels in Figure 2 for B = 3 T with the possible transitions between them indicated by arrows. It is evident by comparison of Figures 2a and 2b that there are apparent differences in the ordering of the magnetic surface levels in the Q2D organic conductors obtained with different models for the energy dispersion law. We find that the surface quantum levels of the skipping electrons are not equidistant as expected. Consequently, the distance between the neighboring surface levels
n | m | ||||
---|---|---|---|---|---|
1 | 2 | 0.3 | 0.3 | 1.4 | 1.6 |
1 | 3 | 0.15 | 0.15 | 0.7 | 0.8 |
1 | 4 | 0.1 | 0.1 | 0.5 | 0.5 |
1 | 5 | 0.07 | 0.07 | 0.4 | 0.4 |
1 | 6 | 0.06 | 0.06 | 0.3 | 0.3 |
2 | 3 | 0.6 | 0.6 | 1.4 | 1.6 |
2 | 4 | 0.3 | 0.3 | 0.7 | 0.8 |
2 | 5 | 0.2 | 0.2 | 0.5 | 0.6 |
2 | 6 | 0.16 | 0.15 | 0.4 | 0.4 |
3 | 4 | 0.9 | 0.85 | 1.4 | 1.6 |
3 | 5 | 0.4 | 0.4 | 0.7 | 0.8 |
3 | 6 | 0.3 | 0.3 | 0.5 | 0.6 |
4 | 5 | 1.2 | 1.1 | 1.4 | 1.6 |
4 | 6 | 0.6 | 0.6 | 0.7 | 0.8 |
5 | 6 | 1.4 | 1.3 | 1.4 | 1.6 |
The resonant magnetic fields are obtained for a fixed frequency ω = 60 GHz that corresponds to the transition between the second and first magnetic surface level, 2 → 1, for which the difference between the corresponding levels is the same and of order of
It is evident that the resonant magnetic fields obtained in the frame of a given energy dispersion law are similarly calculated by both the eigenvalue problem and quantization rule (the first and second column are obtained by using Eqs. (14) and (22) (
3.2 The surface-state wave functions
The corresponding wave functions for the surface states,
Thus, in Q2D organic conductors in general, the wave functions oscillate at relatively small distance from the conductor’s surface of order of 10−8 m and at low magnetic fields up to 2 T. The period of oscillations depends on the organic conductor characteristics, on the magnetic field magnitude and the number of the state. The distance z at which the surface wave functions are attenuated into the conductor is less than the allowed values for the maximal depth of penetration
In addition, as the electron motion is restricted in the proximity to the surface, they are less exposed to the influence of the bulk electrons that are situated on the so called skimming cyclotron orbits (orbit 2 in Figure 1b) and therefore are less scattered from them. In that regards, the electronic transitions take place only between the closed electron orbits in the immediate vicinity of the Fermi surface
A similar trend is also apparent in the corresponding distribution of the probability density for electrons situated in the n-th state with the distance z and the magnetic field B,
4 Conclusions
The energies and wave functions of the surface states in Q2D organic conductors are obtained by using two different forms for the Q2D electron energy spectrum. The first one derived within the tight-binding theory is usually used in studying the properties of the Q2D organic conductors, and the second one is the form previously used to investigate the surface effects in the Q2D organic conductors. The two different forms are used in order to investigate in detail the surface states in organic conductors and extract more information on their elementary properties. We also make use of both the Schrödinger equation and the quantization rule to calculate and compare the surface-state energies. We find that the energies differ in order of 10−4 and more importantly are obtained under different conditions when a different form of the energy dispersion law is used. When the energy dispersion law obtained in the frame of the tight-binding theory is applied, the surface-state energies (Eqs. (14)) and Eqs. (22)) are determined for different values for the coordinate of the electron orbit zc. On the other hand, with the second form for the energy dispersion law the surface-state energies (Eqs. (18)) and Eqs. (23)) and wave functions are obtained by using only one value for zc. This, in turn, allows to determine the geometric characteristics of the electron skipping trajectories in Q2D organic conductors. We ascribe the differences in the values for the surface energies to the different maximum electron velocity along the direction of periodic motion of the electrons. The lower ratio of the maximum velocity along the z direction and the Fermi velocity is obtained in the case of the first energy dispersion law, indicating that in this case the conditions for formation of the surface states in Q2D organic conductors are more favorable and therefore the values for the surface-state energies are more reliable than those obtained with the other form for the energy dispersion law. In this case, the resonant magnetic field values show that the electron transitions in Q2D organic conductors occur at weak magnetic fields. In addition, the analyzes of the corresponding surface wave functions further confirm that the more reliable results should be obtained if one makes use of the energy dispersion law derived in the frame of the tight-binding theory. We expect here presented results and observations to be of great significance for further studies of the surface states properties in Q2D organic conductors. Although by far there is no experimental evidence of the surface quantum oscillations in Q2D organic conductors, yet we believe that our results would provide necessary basics for future experimental studies. This would be very useful as the organic conductors are interesting for applications in the organic electronics.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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