An analytically solvable model of confined electrons in a magnetic field and its relation to Landau diamagnetism

Physics Letters A 376 (2012) 1477–1480

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Physics Letters A www.elsevier.com/locate/pla

An analytically solvable model of confined electrons in a magnetic field and its relation to Landau diamagnetism M.L. Glasser a,b,∗ , N.H. March c,d , L.M. Nieto a a

Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain Department of Physics, Clarkson University, Potsdam, NY 13699, USA c Department of Physics, University of Antwerp, Antwerp, Belgium d Oxford University, Oxford, England, United Kingdom b

a r t i c l e

i n f o

Article history: Received 14 December 2011 Accepted 20 December 2011 Available online 13 March 2012 Communicated by V.M. Agranovich

a b s t r a c t Here we present analytic results for the Slater sum and the magnetic moment for arbitrary magnetic field strengths for an assembly of harmonically confined, but initially free, electrons. The relevance of the results to the generalized Landau diamagnetism of such confined electrons is emphasized. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In this brief report harmonically confined but initially free electrons will be treated analytically. In particular, results will be presented for (a) the Slater sum for such electrons in two dimensions in a transverse magnetic field of arbitrary strength and (b) the corresponding free energy and derived magnetic moment. The relevance to a generalization of Landau diamagnetism for such an inhomogeneous electron assembly is finally stressed. As a starting point, we refer to the work of March and Tosi [1] (MT) in which a single Wigner oscillator was studied in a magnetic field of arbitrary strength. The Hamiltonian adopted by MT takes the form

ˆ = H

1 2m

 − p

 eA

2

c

 1  + k x2 + y 2 , 2

(1)

(2)

MT then solved the Bloch equation for the canonical density matrix C , namely

ˆ C ( r , r0 , β) = − H

∂C ∂β

(3)

with β = (k B T )−1 and subject to the completeness boundary condition C ( r , r0 , β = 0) = δ( r −r0 ), generating thereby the pioneering

*

Corresponding author at: Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain. E-mail addresses: [email protected] (M.L. Glasser), [email protected] (L.M. Nieto). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.01.016

pf =



C ( r , r , β) dr =

S ( r , β) dr

(4)

where S ( r , β) denotes the Slater sum – the first focus of this Letter. Using the canonical density matrix of MT, it will be convenient to write the Slater sum, first of all, in the form following from Eq. (A.1):









(5)

Putting the confining potential in (1) as

V ( r ) =



 = − 1 H y , 1 H x, 0 . A 2 2



S ( r , β) = f (β) exp −4 x2 + y 2 h(β) .

 was chosen specifically to be where the vector potential A



formula of Sondheimer and Wilson [2], who examined, however, only the case k = 0 of (1). We merely record in Appendix A the shape of C ( r , r0 , β) as derived by MT and turn next to the partition function

1 2

x2 + y 2



(6)

Eq. (5) has the form



8  S ( r , β) = f (β) exp − h(β) V (r ) k

(7)

where, from the work of MT the function h(β) has the explicit form

f (β) =

B sinh α b

(8)

where B = mωb/2π h¯ , ω = e H /2mc, b = (1 + k/mω2 )1/2 , α = h¯ ωβ and β = (k B T )−1 . The other function, h(β), entering (7) is given in Eq. (A.3).  This special case of the MT model for the partition function S ( r , β) dr can, in fact, be traced back at least to Darwin [3]. The

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3. Discussion of the Slater sum and generalized Landau diamagnetism To gain insight into the Slater sum S ( r , β) [5] given above let us consider first the limit of Eq. (7) as the magnetic field H is allowed in Eq. (A.5) of Appendix A to tend to zero. From Eq. (A.3) one readily finds that

8h(β) k

→β

(15)

and the Slater sum (5) reduces to the semi-classical Thomas–Fermi (TF) form





S TF ( r , β) = S (0, β) exp −β V ( r )

(16)

where V ( r ) is given explicitly in (6). S (0, β) is retained as in the Fig. 1. (Color online.) The dimensionless magnetization | M |/μ0 N plotted against ω/Ω , where Ω = k/m, for several values of the temperature T in units of k B /¯hΩ (from upper curve to lower curve), T = 10, T = 7, T = 5, T = 3.5, T = 2, T = 1, and T = 0.1.

1 2(cosh α b − cosh α )



S 0 ( r , β, H = 0) =

result is given in Eq. (17) of [1] as

pf =

Sondheimer and Wilson work [2]. Here, we note that the Slater sum for H = 0 is that for two-dimensional harmonic confinement and is known to be [5]

(9)

.

The divergence of p f at b = 1 is merely a reflection that the partition function is proportional to the total area, in this case, which is infinite. 2. Magnetization including harmonic confinement We next calculate the classical free energy from

F = − Nk B T ln p f

(10)



m



ω

2π h¯

sinh β h¯ ω

m ωr 2 β h¯ ω × exp − tan h¯ 2

(17)

where ω2 = k/m. Turning finally to the generalization of Landau diamagnetism due to confinement, one has to use the inverse Laplace transform technique to go from M in Eq. (13) into a result for M in terms of the Fermi energy. When one switches off the confinement by letting k → 0, one regains Landau’s result that the orbital diamagnetism of such free electrons is −1/3 of the Pauli paramagnetism. To complete this discussion we show in the second part of Appendix A the insensitivity of the Landau diamagnetism to ‘soft’ harmonic confinement.

by inserting Eq. (9). Then the magnetization M is given by the thermodynamic relation

Acknowledgements

∂ F

M =− . ∂ H T

M.L.G. thanks Professor L.M. Nieto for arranging his visit to Valladolid. N.H.M. wishes to acknowledge a Visiting Fellowship at the University of Valladolid very kindly arranged by Professor J.A. Alonso. He also thanks Professors D. Lamoen and C. Van Alsenoy of the University of Antwerp (UA) for making his continuing affiliation with UA possible through BOF/NOI. This work has been supported by the Spanish Ministerio de Educación y Ciencia (Project MTM2005-09183).

(11)

After straightforward calculations we find the free energy F from (10) and (11) as





F = Nk B T ln 2(cosh α b − cosh α ) .

(12)

In Eq. (12) it is clear that while the quantized energy levels corˆ in Eq. (2) are reflected responding to the Hamiltonian operator H directly by the presence of Planck’s constant, this free energy F is a quite explicit function of the magnetic field strength H entering in (2) via (3), the force constant k and the thermal energy k B T = β −1 . Inserting the exact result (12) for the free energy into Eq. (11) gives the magnetization M as

M =−

Neh¯ sinh α b − b sinh α 2mcb cosh α b − cosh α

.

M ≈−

6mc k B T

March and Tosi [1] generalize the solution of Sondheimer and Wilson [2] for C in the limit k = 0 to yield the general structure of the density matrix as

 

   × exp − (x + x0 )2 + ( y + y 0 )2 h(β)  − i (x0 y − y 0 x)ϕ (β) .



m ωb 2h¯

(14)

so there is no magnetism then in accord with the Bohr–van Leeuwen Theorem [4]. (See Fig. 1.)



(A.1)

By using the Bloch equation MT then determine the functions g and h to be

g (β) =

→0



C ( r0 , r , β) = f (β) exp − (x − x0 )2 + ( y − y 0 )2 g (β)

(13)

In deriving (13) we have used the fact that b − ω∂ b/∂ ω = 1/b. It is an easy consequence from (13) that in the classical limit h¯ → 0 the magnetization is

Neh¯ h¯ ω

Appendix A. Shape of the canonical density matrix for Hˆ in Eq. (2) and Landau diamagnetism with soft confinement

and

h(β) =

πB 2

−

π 2



H coth α b +

coth α b −

cosh α sinh α b

π 2

H

cosh α sinh α b

(A.2)

.

(A.3)

M.L. Glasser et al. / Physics Letters A 376 (2012) 1477–1480

The phase

ϕ (β) =

ϕ (β) in Eq. (A.1) is given by

2π B sinh α sinh α b

For Fermi statistics the free energy is given by [7]

(A.4)

.

Finally, MT relate the function f (β) entering (A.1) to the partition function p f in (6) [Eq. (17) in MT] and the known h(β) in (A.3) by

f (β) =

4

π

h(β) p f

(A.5)

which completes the specification of C ( r0 , r , β) in (A.1). A.1. Insensitivity of Landau diamagnetism to harmonic confinement The aim of this section of the appendix is to generalize the theory of free-electron Landau diamagnetism so as to include harmonic confinement. In this way we move from classical statistics to the degenerate Fermi limit. We shall present below an exact treatment of this aspect of the Darwin–March–Tosi two-dimensional model. When the force constant k is set to zero the result has been treated exactly by Sondheimer and Wilson [2]. This work is summarized by Jones and March [6]. The result for what is related to the integrated density of states, written by Sondheimer and Wilson as Z ( E ) is

 Z SW ( E ) =

2π m

3/2

h2

−

8

3π

Z ( E , r , H , k) =

1 2π i

c −i ∞

dβ

β2

e E β S ( r , β, H , k)





3/2

(A.7)

(A.8)

3π

(A.9)

To compare this with the Sondheimer–Wilson result (A.6) we average (A.9) over the effective area of the system, which we take as the cross-sectional area where 12 k(x2 + y 2 ) = E. We obtain

Z¯ TF ( E , H , k) =

2π m h2

−

3/2

16 E 7/2

9π

4π i

c −i ∞



2

(b2

− 1) s 4 1 

720

c +i ∞

1 2π i

c −i ∞

when

ds

e (ζ /¯hω)s

s2

cosh bs − cosh s

2 3/2 . ( μ H ) E 0 1/2

.

(A.12)

1−

 1  2 b + 1 s2 12 



3b4 + 8b2 + 3 s4 + · · · .

(A.13)

ds

t ν −1

s

Γ (ν )

ets = ν

(A.10)

The fact that the right-hand side of (A.10) is independent of k indicates that there is little or no effect of confinement on the magnetic susceptibility. We show below more rigorously that this is true in two dimensions.

(A.14)

ν is positive, we find

2  ζ3 ζ h¯ ω F − Nζ = − + . 12 6 h¯ Ω 6(h¯ ω)2

(A.15)

Here b2 = 1 + (Ω/ω)2 has been used with Ω 2 = k/m. Apart from the fact that the contribution from the non-real singularities of the integrand in (A.12), which will be examined below, has been neglected, (A.15) is exact since the remaining integrals are all derivatives of δ(ζ /¯hω) which vanish since the argument is non-zero. The total magnetic moment of the system is

∂F μ0 h¯ ω =− . ∂ h¯ ω 3 (h¯ Ω)2

(A.16)

One cannot properly determine the magnetization or susceptibility without dividing by an effective area A. This we shall take as the area of the cross-section of the potential well where V ( r ) = ζ and can be written A = π mζ /Ω 2 . Thus the magnetization is independent of the confinement parameter k and the magnetic susceptibility

χ =−

105π 1/2

2

(A.11)

1

M = −μ0



5/2 Z TF ( E , r , H , k) ≈ E − V ( r ) 2 1 / 2 h 15π

1/2 1  2 − E − V ( r ) (μ0 H ) . 1/2 8

(0 < c < 1),

Inserting (A.13) into (A.12) and using the formula

where V ( r ) = 12 k(x2 + y 2 ). Using the lower order terms given explicitly in (A.6) we find



c +i ∞

h¯ ω

ζ

Z TF ( E , r , H , k) = Z SW E − V ( r )

2π m

Z (β s)e βζ s

s2 [cosh bs − cosh s]

(A.6)

and is what we term the integrated local density of states. Here we shall evaluate (A.7) in the semi-classical approximation in which the exact Slater sum is replaced by its Thomas–Fermi (TF) counterpart



s sin π s

For low fields, ζ /¯hω  1, only the small s behavior of the non-exponential portion of the integrand in (A.12) contributes significantly:

+

The terms indicated by the order of magnitude estimate include the de Haas–van Alphen oscillations. The desired generalization of the present study of harmonic confinement reads

∞

c −i ∞

F − Nζ = −



  (μ0 H )2 E 1/2 + O H 5/2 . 1/2

β

ds

where the chemical potential ζ is determined by ∂ F /∂ζ |β = N, presumed to be macroscopic, and Z denotes the partition function, which in our case is given in (9). As Sommerfeld has shown, for non-interacting Fermions, finite temperature effects are represented by an expansion in powers of the small quantity k B T /ζ . Therefore, we shall simply examine the zero temperature limit of (A.11)

E 5/2

1

c +i ∞

i

F − Nζ =

=

15π 1/2

1479

μ20 3π mζ h¯ 2

(A.17)

is as well. Hence, it appears that “soft” confinement has no effect on Landau diamagnetism. Note that the very weak field dependence of the chemical potential, usually of order H 2 , has been neglected. We next turn to de Haas–van Alphen oscillatory behavior in the magnetization. The integrand in (A.12) has simple poles and the points 2lπ i /(b ± 1), l = ±1, ±2, . . . along the imaginary axis

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M.L. Glasser et al. / Physics Letters A 376 (2012) 1477–1480

having residues (b ± 1) exp[2lπ i (ζ /¯hω)/(b ± 1)] csc[lπ (b ∓ 1)/(b ± 1)]/4l2 π 2 . Therefore, evaluating (A.12) by closing the integration path by a large semicircle to the left, which is allowed since the integrand decays exponentially into the left-hand s-plane, and summing the residues we have

F − Nζ =

∞ 1 1



(b + 1) sin[2π l(ζ /h¯ ω)/(b + 1)] sin[π l(b − 1)/(b + 1)] l =1  (b − 1) sin[2π l(ζ /h¯ ω)/(b − 1)] . + sin[π l(b + 1)/(b − 1)]

4π 2

approaches 1, the amplitude of the first term in (A.18) will grow without bound as expected, since the free energy has not been normalized to unit area. Finally, we refer to [6] for the stronger effect of surface confinement on Landau diamagnetism. References

l2

(A.18)

The structure of the resulting oscillations is quite complex since b is strongly field dependent, and includes spikes where (b ± 1)/(b ∓ 1) possesses integer values. At very high fields, so b

[1] [2] [3] [4]

N.H. March, M.P. Tosi, J. Phys. A 18 (1985) L643. E.H. Sondheimer, A.H. Wilson, Proc. Roy. Soc. A 210 (1951) 173. C.G. Darwin, Proc. Camb. Phil. Soc. 27 (1931) 85. J.H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford Univ. Press, 1932. [5] N.H. March, I.A. Howard, Phys. Stat. Solidi (b) 237 (2003) 265. [6] W. Jones, N.H. March, Theoretical Solid State Physics, vol. 1, Dover, 1985. [7] M.L. Glasser, J. Math. Phys. 5 (1964) 1150.