The Thomas-Fermi and the Thomas-Fermi-Dirac Models in Two-Dimension- the Effect of Strong Quantizing Magnetic Field
Using Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) models, we have investigated the properties of electron gas inside two-dimensional (2D) Wigner-Seitz (WS) cells in presence of a strong orthogonal quantizing magnetic field. The electron-electron Coulomb exchange interaction in quasi-2D case is obtained. The exact form of exchange term in 2D is derived making the width of the system tending to zero. Further, using the exchange term, the Thomas-Fermi-Dirac equation in 2D is established. It has been observed that only the ionized WS cell can have finite radius in the Thomas-Fermi model, even in presence of a strong quantizing magnetic field. On the other hand, in the Thomas-Fermi-Dirac model a neutral WS cell can have finite radius.
pacs:31.15.pTwo-dimensional electron gas and 31.20.LrThomas-Fermi Model and 05.30.ChThomas-Fermi-Dirac Model and 71.70Landau levels
Using Fermi statistics, a formalism was developed by L.H. Thomas and E. Fermi to obtain the charge distribution and also the distribution of electric field in the extra-nuclear space inside heavy atoms TF1 (); TF2 (). This formalism is known as the Thomas-Fermi model. The electrons inside the atoms are assumed to be a degenerate Fermi gas. In this model the electron density is found to be nonuniform inside the atom, i.e., , where is the radial distance vector of the point from the nucleus situated at the center of the atom. The nucleus at the centre has been assumed to be a point particle. The electric potential and the corresponding electric field vector within the atom also vary with the radial vector. The electron density has been observed to be a smoothly varying function of radial coordinate , instead of having peaks according to shell model. Where the atoms are assumed to be of spherically symmetric in shape. The model was successful in predicting the binding energies of the atoms LIEB (). With some suitable modification the model has been successfully applied to molecules, solids and also to nuclei MARCH () to explain some of the experimentally observed phenomena. The electronic shell effect was also incorporated in the model. The model could also satisfactorily explain the thermodynamic properties of dense degenerate electron gas. For very high density matter, the electron gas surrounding the nucleus is assumed to be enclosed in a region, called WS cell LY (). Therefore in such situation, instead of atoms in a solid, there are regularly spaced WS cells, which are assumed to be charge neutral and spherically symmetric. There are also relativistic generalization of TF model for very high density electron gas and the model was found to be successful to investigate the thermodynamic properties of such high density degenerate electron gas REMO ().
Generalized versions of non-relativistic as well as relativistic form of Thomas-Fermi equations in presence of strong quantizing magnetic field, when the Landau levels for the electrons are populated have also been obtained NAG (); NAG1 (); NAG2 (). The thermodynamic properties of matter inside the magnetically deformed WS cells have also been investigated. In presence of strong quantizing magnetic field, the electron pressure becomes anisotropic inside the WS cells. As a result they will be deformed to ellipsoidal shape from their usual spherical structure GHOSH ().
However, all these investigations are associated with three dimensional degenerate electron gas, enclosed inside WS cells. There are only a few reported results on the study of two dimensional electron gas using Thomas-Fermi model R1 (); R2 (); R3 (). Further, to the best of our knowledge, no studies have been reported on the two dimensional Thomas-Fermi model for electron gas in presence of strong quantizing orthogonal magnetic field, in which the Landau levels are populated for the electrons. Again, in the three dimensional case, incorporating electron-electron exchange interaction, a modified form of Thomas-Fermi equation, called the Thomas-Fermi-Dirac equation has been developed both for non-relativistic as well as for relativistic electron gas with or without the presence of magnetic field LY (); ST (); NAG (); REMO (); NAG1 (). Unfortunately, no such formalism has been developed in the case of two dimension. To the best of our knowledge, the first reported result on two dimensional Thomas-Fermi model for degenerate electron gas in absence of magnetic field is by Bhaduri et. el. R1 () (see also R2 (); R3 ()).
Two dimensional electron gas has a lot of important applications in modern days condensed matter physics. The electrons in 2D are constrained to move in two dimensional sheet embedded in a three dimensional space. Such two dimensional electron gas may be realized in many semi-conductor devices R4 (). There are also possibility of having two dimensional electron gas on the surface of materials, e.g., liquid He R5 (). In such system electrons are free to move on the surface of liquid He but rigidly attached with the He atoms. There are also a kind of solid insulators, e.g., topological insulators R6 (), the surface of which supports conducting states of free electrons.
During the present days the most interesting two dimensional system in condensed matter physics is graphene. It is an almost ideal two dimensional material developed in the laboratory using graphite R7 (); R8 (); R9 (). It has been observed that graphene can support 2D electron gas. It has become a topic of current interest due to a large number of application of graphene. There are also a lot of academic interest on the theoretical investigation of graphene. In particular, application of quantum electrodynamics in graphene and study of the physics of mass-less electrons or chiral electrons using two component Dirac equation.
In this article we have investigated the properties of electron gas enclosed in two-dimensional WS cells, which are embedded in a three-dimensional space. The WS cells are assumed to be on plane and the strong magnetic field is along -axis. The electrons are constrained to move on plane. The presence of strong magnetic field along -direction makes the electron energy eigen value discrete. The motion of the electrons on plane are in quantized form. This is the well known Landau quantization in 2D. The electron energy therefore does not depend on , the component of momentum along -direction. However, the momentum component on plane changes in a discrete manner.
We have organized the various sections of this article in the following manner: In the next section we have developed the basic formalism for two dimensional Thomas-Fermi model for degenerate electron gas in presence of strong quantizing orthogonal magnetic field. In section 3 we have studied the thermodynamic properties of 2D degenerate electron gas inside a WS cell in presence of a strong quantizing magnetic field. In section 4 we have considered the electron-electron exchange interaction in absence of magnetic field and incorporate this result in Thomas-Fermi condition to obtain Thomas-Fermi-Dirac equation satisfied by degenerate electron gas in 2D. In section 5 we have shown explicitly that the same technique can not be followed to obtained Thomas-Fermi-Dirac equation for degenerate electron gas in presence of strong quatizing magnetic field. However, for the conventional three dimensional case, one can obtain exchange energy for electrons NAG1 (). Finally we present the conclusion of this work.
2 Basic Formalism
We assume that the constant external magnetic field is along z-direction. In presence of this strong external quantizing magnetic field, the Landau levels of the electrons are populated on plane. Since the electrons are constrained to move on plane, the momentum component along -direction, . The quantized form of electron energy eigen value is then given by , where , the cyclotron frequency and , the Landau quantum numbers, is the upper limit of Landau quantum number which is finite if the temperature of the system is zero or less than the corresponding Fermi temperature, whereas, if the temperature of the electron gas is greater than the Fermi temperature. Therefore is finite for strongly degenerate electron gas, whereas in the non-degenerate scenario, is infinitely large. In the strongly degenerate condition the number of electrons per unit surface area is given by
Assuming as the Fermi energy of the system, we have in natural units ,
Then it is a little algebra to check after putting the value of cyclotron frequency that the maximum value of Landau quantum number decreases with the strength of magnetic field. It becomes zero for , where G. In the present model calculation, there is an upper limit for the strength of magnetic, beyond which the Landau quantum number becomes unphysical (negative in nature). Then
Now for such degenerate electron gas confined inside a 2D WS cell, at the centre of which there is a nucleus of charge , the total negative charge carried by the electrons inside the cell is . Therefore the WS cells are charge neutral. Since both protons inside the nucleus and electrons within the WS cell carry electric charges, the Poisson’s equation satisfied by the electrostatic potential , produced jointly by the electrons and protons is given by
Here the nucleus is assumed to be a point object. The Thomas-Fermi condition in this particular situation is given by
In the present situation for , we have after substituting for , , gives the upper limit of the external magnetic field, beyond which the Landau quantum number becomes negative, which is unphysical. In terms of the critical values of the magnetic field for electrons at which the Landau levels for the electrons are populated, we have
where Gauss. Therefore the upper limit of external magnetic field depends on the radial distance from the centre of the Wigner-Seitz cell through the electrostatic potential . The number of electrons per unit surface area is then given by
Since is a function of radial coordinate , both and should also depend on the radial coordinate. Here for the sake of simplicity we assume circular symmetry for the WS cells. The above mentioned variables therefore do not depend on the angular coordinate . Then from the above equation we have in the extra-nuclear space of WS cell
and from the Poisson’s equation (eqn.(3)) we have for
This is a second order differential equation for the electron density . On substituting the value of cyclotron frequency, eqn.(8) can be re-expressed as
Hence for the 2D case, with circular symmetry, the Poisson’s equation in radial coordinate is given by
Defining the scaled radius parameter , given by
Which is the Thomas-Fermi equation in 2D in presence of a strong orthogonal quantizing magnetic field. The form of this equation is exactly identical with that of field free case R1 (), except the scaling parameter . Surprisingly the radius of the circular type WS cell does not change with the strength of magnetic field. However, in the usual three dimensional case, the radius of the spherical type WS cell decreases with the strength of magnetic field, e.g., in NAG1 (); NAG2 (), the scaling parameter . Hence the actual radius varies with magnetic field in the functional form . Further in the case of magnetically deformed cylindrical type WS cell, both the radial as well as the axial parameters decrease with the strength of magnetic field GHOSH (). Therefore in the present 2D scenario we may conclude that there will be no magnetic contraction of WS cells and even in presence of ultra strong quantizing magnetic field, there will be no magnetic distortion to elliptical shape. Now it is trivial to show that the general solution of eqn.(12) is given by R10 ()
where and are two unknown constants to be obtained from the initial and the boundary conditions and can be expressed in terms of , the surface value of , , the total number of electrons in the system and , the total positive charge within the nucleus situated at the centre of the WS cell. From the above solution it is quite obvious that as from the diverging nature of the modified Bessel function . (The modified Bessel function of second kind of order zero, ). This is also true in the case of three dimensional Thomas-Fermi equation, having a singular nature at the origin. The special technique developed by Feynman, Metropolis and Teller FMT () is used to solve the Thomas-Fermi differential equation in the three dimensional case.
Since the TF differential equation is exactly identical with eqn.(10) of R1 () then instead of repeating the calculations of R1 (), we shall use eqn.(17) to eqn.(20) from this article with our changed notations and are given by
The right hand side of this equation vanishes for , i.e., for a charge neutral two dimensional WS cell.
It should be noted that although the physical meaning of is same but its expression is quite differnt from R1 (). Finally
It is quite obvious from eqn.(17) that for , the radius of the WS cell is infinitely large. It is also to be noted further that for the finite value of the radius parameter , should be less than , i.e., the cell must be in ionized state with net positive charge. In fig.(1) we have plotted the variation of , the scaled radius parameter with . From the figure it is quite obvious that is finite for . Whereas for the ratio tending to unity, the scaled radial parameter increases sharply finally it diverges at . Therefore in 2D, in the TF case, the radius of a positively charged WS cell is finite, whereas for the charge neutral case the radius of the WS cell diverges. This is true even if there is stong quantizing orthogonal magnetic field in 2D.
3 Thermodynamics of 2D Electron Gas
The expressions for internal energy density and the corresponding kinetic pressure can be obtained from the first principle following the standard books on statistical mechanics. We define the -potential of Kramers in the form LSTAT ()
where is the kinetic pressure, is the surface area, is the temperature of the system, is the Boltzmann constant, , , with , the chemical potential of the electrons and . In presence of a strong orthogonal quantizing magnetic field, which populates the electron Landau levels, the energy eigen value corresponding to the th. Landau level is given by for . The total kinetic energy of the electron gas is then given by
where . For degenerate case, since the Fermi distribution function reduces to unity, we have
On summing over we can express the surface density of electron kinetic energy in the form
The kinetic pressure can also be obtained from -potential. Using the Euler summation formula (discussed in the Appendix) the degeneracy pressure for electron gas in 2D is given by
4 Electron Exchange Energy for in 2D
We shall now obtain the electron exchange energy (see LY () for three-dimensional case). For the sake of simplicity we start with an ideal two-dimensional system of degenerate electron gas. In the case of a purely two-dimensional electron gas, the electron-electron two body potential is logarithmic in nature and may be expressed in the form
where as stated before, has the dimension of energy. In 2D, the normalized form of free electron wave functions are given by
Then the exchange part of interaction may be written as
which after some straight forward algebra may be expressed in the form
Hence the expectation value for the electron exchange energy is given by
where , and . Then the angular part of the integral is given by
Now to evaluate this angular integral we use the following trivial algebraic relation
and use the standard relations R10 ()
where is the ordinary Bessel function of order . Then after some straight forward algebra we have the exchange interaction part
where we have used the relation . The integral given by eqn.(32) is obviously a diverging one for logarithmic form of coulomb potential as given in eqn.(23). The coulomb exchange energy therefore has infinite contribution in the case of ideal 2D electron gas, which is totally unphysical in nature. Hence, to obtain a finite contribution for the exchange part of interaction energy, we therefore follow an alternative approach. Instead of an ideal two-dimensional system, we consider a quasi two dimensional degenerate electron gas with an width along the third dimension and we shall make it tending to zero at the end QDOT (). The wave functions in the plane is as usual are given by eqn.(24), with replaced by and . In the -direction the wave function is assumed to be given by
where . The modified form of Thomas-Fermi condition is then given by
Which is the Thomas-Fermi-Dirac condition in 2D scenario. Hence
The other solution for Fermi momentum has been discarded for obvious reason. Let us now substitute
Then we have from eqn.(36)
Hence from the Poisson’s equation we can write down the Thomas-Fermi-Dirac equation in the following form
Writing , with , the dimensionless radius parameter and is an unknown constant, the above equation can be expressed in the following form:
The complementary function can be obtained from the solution of the homogeneous equation
To get an analytical solution, we put , with , where corresponds to surface value. Hence we have
where . The solution of this equation, which is the complementary function is then given by
where and are two unknown constants. The factor is put by hand to get the initial condition as or equivalently . In terms of this new variable z, the full form of Thomas-Fermi-Dirac equation is then given by
To obtain the particular integral, we follow the iterative technique. On the right hand side of the above equation we put the value of complementary function for which we call as the zeroth iteration term and obtain the equation
Defining , then
Hence using the standard technique of integrating factor, which in this case is , we have
Here to obtain the complete solution, we demand that as . Using the same integrating factor technique, one can solve the equation
Hence the complete solution is
Obviously as . It can further be shown that the boundary condition in -coordinate is given by
Hence we get . This particular value of has been used in the complete solution as given by eqn.(51). The constant can be obtained numerically, provided the surface value is known. Further, the surface value of scaled radius parameter and the constant can be solved numerically using the expression for total number of electrons, given by
and eqn.(52). In -coordinate eqn.(53) can be expressed as
In this case we have assumed that the number of electrons within the WS cell is equal to the number of protons within the nucleus situated at the centre of the cell. The WS cells are therefore charge neutral. As a consequence, the electric field at the surface vanishes exactly. We continue the iterative calculation for the particular integral, until the result converges. In fig.(2) we have plotted the variation of scaled surface radius parameter with the atomic number , which is also the number of electrons within the two-dimensional WS cell. It is obvious that initially for the low values of , the scaled parameter increases monotonically with and finally for large it saturates to some constant value. This type of variation has not been observed in the case of Thomas-Fermi model in 2D. Hence we may conclude that the radius of he charge neutral WS cell is finite in TFD model.
5 Electron Exchange Energy for in 2D
In presence of a strong quantizing magnetic field the semi-analytical expression for exchange energy can be obtained for electrons assuming that all of them occupy only their zeroth Landau level. Here again we consider a quasi-2D electron gas. In this case the numerically fitted form of exchange energy is given by NAG1 (); NAG2 ()
with the average values for and for the strength of magnetic field ranging from G to G. This range has special importance in the physics of strongly magnetized stellar electron gas. In the present situation, considering the exchange part inside a strongly magnetized neutron stars or magnetars, the Thomas-Fermi-Dirac condition is then given by
Hence one can obtain the numerically fitted form of Fermi momentum, given by
where the average values for and are for the same range of magnetic field. Here . Unfortunately, the Fermi momentum here is along the third dimension, i.e, along axis, which does not exist in the present scenario. Hence we can conclude that exchange interaction term does not contribute when a 2D electron gas is placed in a strong magnetic field which populates only the zeroth Landau level for the electrons.
To evaluate the kinetic pressure for degenerate electron gas in 2D in presence of strong quantizing magnetic field, we use the Euler summation formula as given bellow R10 ()
Then from eqn.(18) the q-potential can be re-written in presence of an electrostatic potential in the following form
where . Hence
After evaluating the integral by parts, we have
For , or , where is the Fermi temperature, given by , for , the second term exactly vanishes, whereas in the first part, putting the Fermi distribution , it gives , with . Then substituting the value of the integral in eqn.(61), we have
It is quite surprising that the form of Thomas-Fermi differential equation in 2D in presence of strong quantizing magnetic field is exactly identical with that of zero field case. The exchange part of electron energy does not exist in 2D in presence of strong quantizing magnetic field. The Fermi momentum obtained in this case is along -direction, which is suppressed in ideal 2D case. The Thomas-Fermi-Dirac equation therefore can not be obtained for a 2D electron gas in presence of a strong quantizing magnetic field. However, in absence of magnetic field, exchange energy can be obtained assuming a quasi 2D structure and finally making the width along the third dimension tending to zero. Hence one can formulate Thomas-Fermi-Dirac model in 2D scenario for an electron gas. We have noticed that the radius of a two dimensional charge neutral WS cell is infinitely large. On the other hand it is finite if , i.e., the cell is ionized and carrying some effective positive charge. In usual three dimensional case also the radius of an atom can not be finite in Thomas-Fermi model. However, in Thomas-Fermi-Dirac model the radius of a two dimensional charge neutral WS cell is found to be finite. We have also noticed that the size of circular shape WS cell does not depend on the strength of magnetic field, even if it is of astrophysical order. Because of two dimensional structure, there is no distortion of circular type WS cells to elliptical form. On the other hand, in the usual three dimensional case, because of pressure anisotropy within the WS cells, a distortion to ellipsoidal shape may occur in presence of strong quantizing magnetic field. In the usual three dimension, the anisotropy increases with the increase of the strength of magnetic field. Then in the extreme case, it can be shown that a WS cell acquires a needle like shape with its length along the direction of magnetic field.
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