There are a lot of literature dealing with the phenomenon of Quantum Hall Effect [1] - [8], and most of them use the Landau’s solution of the eigenvalue problem associated to the charged particle motion in a flat surface with static transversal magnetic field to the surface. This brings about the known Landau’s levels for the energies and a separable variable solution for the eigenfunctions [9]. However, it has been shown that a non separable of variables solution exists for this problem with the same Landau’s levels [10] [11], and these levels are numerable degenerated [12], determining the operators which causes this degeneration. In addition, the quantization of the magnetic flux appears naturally [10],

m ω c ℏ A = 2 π l ,   l ∈ Z ,   ω c = q B m c , (1)

where m is the mass of the charge q, c is the speed of light, ω c is the so called cyclotron frequency, B is the magnitude of the static magnetic field, A = L x L y is the area of the sample, and 2 π ℏ = h is the Planck’s constant. As we mentioned before, Landau’s separable solution is normally used to try to explain the so called Integer Quantum and Fractional Quantum Hall Effects (IQHE and FQHE) [4] [5] [6] [7], which were first discovered experimentally [1] [2] [3]. The IQHE is normally explained as a single particle phenomenon; meanwhile, the FQHE is explained as a many particle event [4] [5] [6]. Experimentally, both of them occur in highly impure samples, where these impurities have the effect of extending the range of magnetic field intensity where the resistivity is quantized [2] [3] [7]. The main characteristic of the IQHE or FQHE is the resistivity (or voltage) which appears on the transverse motion of the charges, so called Hall’s resistivity ρ H . This Hall’s resistivity acquires a constant value on certain regions of the magnetic field, and within these regions, the longitudinal resistivity is zero. The values of these constant ρ H turn out to be inverse to an integer number (IQHE) or proportional to an integer number (FQHE) multiplied by the constant h / q 2 , called von Klitzing constant [2] [3] ( h / q 2 ≈ 25812.80745   Ω ). In this paper, we calculate the quantum current and the expected value of the transverse and longitudinal resistivities for a single charged particle motion on a flat surface using the non separable solution in the lowest Landau level ( n = 0 ) and using the first wave function ( j = 0 ).

The Hamiltonian associated to the motion of a charge particle q with mass m on a flat surface of lengths L x and L y with transverse magnetic field B = ( 0,0, B ) and longitudinal electric field E = ( 0, E ,0 ) is given by

H ^ = 1 2 m ( p − q c A ) 2 + q V , (2)

where A is the vector potential, B = ∇ × A , and V is the scalar potential, E = − ∇ V . The Schrödinger’s equation,

i ℏ ∂ Ψ ∂ t = H ^ Ψ , (3)

can be written, using the operator p = − i ℏ ∇ , as

i ℏ ∂ Ψ ∂ t = 1 2 m [ − ℏ 2 ∇ 2 + i ℏ q c ( ∇ ⋅ A + A ⋅ ∇ ) + q 2 A 2 c 2 ] Ψ + q V Ψ . (4)

Taking the usual complex conjugated to this expression, a similar equation is gotten for the function Ψ * . Multiplying this one by Ψ , (4) by Ψ * and subtracting both, the following continuity equation is obtained

∂ ρ ∂ t + ∇ ⋅ J = 0 , (5)

where ρ and J are defined as

ρ = Ψ ⋅ Ψ * (6)

and

J = i ℏ 2 m ( Ψ ∇ Ψ * − Ψ * ∇ Ψ ) − q m c ρ A . (7)

Since Ψ is a scalar complex function, it can be written as Ψ = | Ψ | e i θ , where | Ψ | and θ are real functions, and θ is the argument of the function. Then, the current is given by

J = ( ℏ m ∇ θ − q m c A ) | Ψ | 2 . (8)

For the general solution of (3), the function θ can be very complicated expression of all variables. However, for a particular state solution of the system, say

ψ n ( x , t ) = e i ϕ n ( x , t ) f n ( x ) , (9)

the argument is just θ = ϕ n ( x , t ) , and the current associated to this state of the system is given by

J n = ( ℏ m ∇ ϕ n − q m c A ) | f n | 2 . (10)

The non separable solution of (3) using the Landau’s gauge A = B ( − y ,0,0 ) and the longitudinal constant electric field E = ( 0, E ,0 ) was given as

f n 0 = 1 2 n n ! L y ( m ω c π ℏ ) 1 / 4 e i ϕ n e − m ω c 2 ℏ ( x − c E t / B ) 2 H n ( m ω c ℏ ( x − c E t / B ) ) , (11a)

where E = q E , ω c is the cyclotron frequency (1), and ϕ n is given by

ϕ n = − [ ℏ ω c ( n + 1 2 ) − m c 2 E 2 B 2 ] t ℏ − m ω c ℏ ( x − c E t B ) ( y − m c 2 E q B 2 ) . (11b)

These functions are degenerated in the sense that for each Landau’s level ( ℏ ω c ( n + 1 / 2 ) ), one has a numerable solutions f n j = ( p ^ x ) j f n 0 , j ∈ Z . Thus, the expressions (11a) define the state of the system. Using this function ϕ n in (10) and for the index of degeneration j = 0 , we have

J n = [ c E B i ^ − ω c ( x − c E t B ) j ^ ] | f n 0 | 2 . (12)

In particular, for the ground state of Landau’s energy, it follows that the components of the current are

J 0 x = c E B | f 0 0 | 2 , (13)

and

J 0 y = − ω c ( x − c E t B ) | f 0 0 | 2 . (14)

The electric conductivity along the x-axis is called Hall’s conductivity and is given by

σ H = q E J 0 x = q c B | f 0 0 | 2 . (15)

Thus, the Hall’s resistivity is ρ H = 1 / σ H , and the expected value of the resistivity in the state f 0 0 is

〈 f 0 0 | ρ H | f 0 0 〉 = ∫ 0 L x ∫ 0 L y | f 0 0 | 2 σ H d x d y = B A q c . (16)

Now, multiplying and dividing this quantity by m ω c / ℏ and making some rearrangements, one gets

〈 f 0 0 | ρ H | f 0 0 〉 = ℏ q 2 ( m ω c ℏ A ) , (17)

and taking into consideration the magnetic field flux quantization (1), it follows that

〈 f 0 0 | ρ H | f 0 0 〉 = h q 2 l ,   l ∈ Z . (18)

The expected value in the state f 0 0 of the longitudinal resistivity ρ y is

〈 f 0 0 | ρ y | f 0 0 〉 = ∫ 0 L x ∫ 0 L y | f 0 0 | 2 d x d y σ y = E q ∫ 0 L x ∫ 0 L y | f 0 0 | 2 d x d y J 0 y (19)

= − E q ω c ∫ 0 L x ∫ 0 L y d x d y x − c E t B = − E L y q ω c ln ( 1 − L x B c E t ) ≈ 0 (20)

since one has normally in the experiments that L x B / c E t ≪ 1 , that is, the time in the experiments are such that

t ≫ L x B c E . (21)

For example, on the reference [2] and with respect the voltage gate V g , one has that B L x / c E = B A / c V g ~ 4.5 × 10 − 8 sec . So, the condition (21) is well satisfied in this experiment.

Note that the expression (18) implies a filling factor ν = 1 / l , which correspond to the IQHE phenomenon for l = 1 and to the FQHE phenomenon for l > 1 . However, this result is valid for an analysis of a single charged particle, and both QHE phenomena appear due to the quantization of the magnetic flux (1). In addition, one must note that this analysis is still valid for any n > 0 and j = 0 .

The quantization of the magnetic flux (1) arises from the periodicity of the solutions of the Hamiltonian [10], which can be expressed using (11a) for E = 0 as

f n 0 ( L x , y + L y , t ) = f n 0 ( L x , y , t ) . (22)

However (and also for E = 0 ), let us assume that L y = N l y where l y ≪ L y and N ∈ Z + , that is, the total area L x L y is covered with slices of area L x l y , with horizontal length L x and width l y . Let us impose the periodicity condition of the form

f n 0 ( L x , y + k l y , t ) = f n 0 ( L x , y , t ) ,   k ∈ Z , (23)

such that with the phase (11b), one gets

m ω c ℏ L x k l y = 2 π l ,   l ∈ Z (24)

which brings about the relation

m ω c ℏ a = 2 π l k ,     with       a = L x l y . (25)

Using (1) and making some rearrangements, the magnetic field can be given by

B = α l k ,     with     α = h c q a (26)

and using (25) in (17), the expected value of the Hall resistivity would be

〈 f 0 0 | ρ H | f 0 0 〉 = h q 2 l k ,   k , l ∈ Z , (27)

implying now a filling factor of ν = k / l , which represents the full IQHE (for l = 1 ) and FQHE (for l > 1 ). To determine the magnetic values B where these phenomena occur, one looks for the value B 0 where the first IQHE ( l = k = 1 ) appears, which intersect the normal linear dependence behavior straight line, and this defines α = B 0 . Then, one uses the resulting expression

B = B 0 l k (28)

to find the other quantized magnetic fields which correspond to IQHE or FQHE. For example, on the experimental data shown on the reference [3], one sees that B 0 ≈ 5   T for l = k = 1 (corresponding to an area a ≈ 8.27 × 10 − 4     μ m 2 ), and the other FQHE are matched quite well for l = 3 and k = 1 , that is B ≈ 15   T . Another example is shown on the reference [8] page 886, one sees that B 0 ≈ 9.8   T for l = k = 1 (corresponding to an area a ≈ 4.22 × 10 − 4     μ m 2 ), and the other IQHE and FQHE magnetic fields are matched quite well for l > 1 and k > 1 . In addition, on reference [13] page 207, one sees that B 0 ≈ 4.2   T for l = k = 1 (corresponding to an area a = 9.85 × 10 − 4     μ m 2 ), and the other IQHE and FQHE magnetic fields are matched quite well for l > 1 and k > 1 . Finally, on reference [14] page 156801-2, one sees that B 0 ≈ 5.3   T for l = k = 1 (corresponding to an area a = 7.8 × 10 − 4     μ m 2 ), and for the filling factor ν = 3 / 4 one gets B = 4 B 0 / 3 = 7.06   T , which is approximately the experimental value reported.

Using the known non-separable solution for the quantum motion of a charged particle in a flat surface with static fields, in the state n = 0 and j = 0 , the Hall and the longitudinal resistivities were calculated. For the quantization of the magnetic flux, which can appear from the simple periodicity on the y-direction, the results bring about the IQHE and FQHE phenomena since from the expression (18) it appears a filling factor of 1 / l for a single charged particle due to the quantization of the magnetic flux. If l = 1 , one gets the IQHE phenomenon, and if l > 1 , one gets the FQHE phenomenon. However, it is not possible to say anything about filling factors of the form ν = k / l . For a more extended quantization of the magnetic flux (25), which appears of the extended periodicity (23), one gets also IQHE and FQHE but with a filling factor of ν = k / l .

The authors declare no conflicts of interest regarding the publication of this paper.

López, G.V. and Lizarraga, J.A. (2022) Single Charged Particle Motion in a Flat Surface with Static Electromagnetic Field and Quantum Hall Effect. Journal of Modern Physics, 13, 1324-1330. https://doi.org/10.4236/jmp.2022.1311081