Size effects of a nanoobject in magnetic field
B. A. Lukiyanets D. V. Matulka


Biermann battery effects on the turbulent dynamo in a colliding plasma
jets produced by high-power lasers
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    December 30, 2023
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A theoretical analysis of physical properties of the effect of size of a nanoobject in the form of a rectangular parallelepiped whose sides a, b, c, are oriented along the OX, OY, OZ, respectively, is carried out. In the framework of the perturbation theory, changes in the electronic spectrum of the nanoobject caused by an external magnetic field B⃗, depending on its size, are analyzed. We consider two cases of the fields which are described 1) by the Landau gauge, A⃗(r⃗)=(0,Bx,0) (B⃗ is oriented along the side c) and 2) by  A⃗(r⃗)=(Bz,0,α By) (α is a parameter; at  α  = 0, B⃗ is directed along OX axis, and  at α = 1, B⃗ is directed along the diagonal in XOY plane). Firstly, it is shown that the first correction to the spectrum is zero, regardless of B⃗ orientation. Secondly,  it is established that, in contrast to the case of the field orientation 1), where the correction does not depend on the length of c, in the case 2) such  correction  depends both on c and on its ratios to the lengths of a and b. There  was found the existence of such nanoobject sizes in XOY plane at which the corrections to the spectrum are the same for different lengths of c of the nanoobject.

anoobject, magnetic field, electronic spectrum, size effect

73.22.-f







§ INTRODUCTION


The influence of the shape and size of nanoobjects on their physical properties (electronic, optical, optoelectric, and others) is an important problem in modern nanotechnology.

The electronic spectrum of crystals is the basis of microelectronics. It is known that the   spectrum  of an infinite crystal is quasi-continuous, while the   spectrum  of a nanosized crystal  is noticeably discrete. Such a quantized character can be practically used in microelectronics. Moreover, if the size of a nanoobject is equal to the length of the coherence of electronic excitations then, by changing its size and/or shape, we can manipulate with its basic characteristics. Such changes in combination with optoelectric characteristics create the basis of such a promising scientific and technological direction as femtosecond optics <cit.>.

There are two methods of fabrication of nanoobjects in modern nanotechnology <cit.>: 
1) top-down nanotechnology — the transition from bulk crystal to nanosized one by removing its layers;
2) bottom-up nanotechnology — fabrication of a nanoobject at an atomic level by addition of atom to atom, molecule to molecule or cluster to cluster. Until the 90s of the last century, semiconductor quantum dots (zero-dimensional objects) were fabricated as dopants in glasses and in solvates or as epitaxially grown objects. While in the former case the quantum dots were of a nearly spherical shape with a diameter of 1…10 nm, epitaxial quantum dots were of an elongated shape of the order of nm and a cross-section of about 10 nm. However, such nanoobjects had a wide-size spread, which limited the accuracy of their experimental studies. After 90s,  high-quality, practically monodisperse quantum dots were achieved by injection of highly reactive organometallic precursors into solvent.

Modern technology allows us to change their size and shape, and thus, purposefully change their optical and photoelectric characteristics, cathodoluminescence, Raman scattering, thermodynamics. An analysis of such effects occupies a significant place in the monographs <cit.>. Cognition of the physics of sizes and shapes in nanoobjects remains urgent.  In <cit.>, the theoretical investigation of changes in the band gap of the semiconductor nanocrystals XY (X — Cd, Zn, Y — S, Se, Te)  related to their shape and sizes is presented. Similar changes are recorded in <cit.> in the study of GaP nanocrystals. In <cit.>, electron and hole states of InAs nanocrystals of the same size, but of different ellipsoidal shape, were analyzed. It is shown that the changes in the energy states lead to significant differences in the spectra of radiation with different polarizations. Results <cit.> showed differences in the optical and electrical characteristics of the colloidal semiconductor nanocrystals fabricated in the form of spherical quantum dots, nanowires, two-dimensional nanoplates, and nanosheets. In <cit.>, it was found that silver nanoparticles of different shapes (sphere, cube, rod), but of the same size, have different biological characteristics.  An analysis of the differences between the electronic spectra of quantum dots and quantum wells is given in <cit.>. Recent works <cit.> present the results of investigations of the influence of a magnetic field on the electronic spectrum and on optical properties of nanoobjects. Even such an incomplete list of publications on the effects of shapes and sizes in nano-sized materials is a convincing proof of its relevance.

Herein below we will analyze, within the perturbation theory, the change of the electronic spectrum of a nanoobject due to an applied magnetic field, depending on the size of the nanoobject.




§ MODEL. CALCULATIONS

Let us consider a nanoobject of the shape of a rectangular parallelepiped with the sides a, b, c located along  OX, OY, OZ axes, respectively. Let the potential energy of an electron in it be the sum of three terms

    U(r⃗)=U(x)+U(y)+U(z),

where each of them is an infinite deep square well, i.e.,



    U(s)=
    0,   if s=x ∈ [0,a] or y ∈[0,b] or z ∈ [0,c],
    ∞,   otherwise.




In this case, the electronic spectrum  is a solution of the time-independent Schrödinger equation with the Hamiltonian


    H_0 =p⃗^ 2/2m+U(r⃗)

and equals <cit.>

    E(n_a n_b n_c)=∑ _s=a,b,cħ ^2 ^2/2mn_s^2/s^2⇒ħ ^2 ^2/2m(n_a^2/a^2 +n_b^2/b^2 +n_c^2/c^2),

where n_s =1, 2, 3,… are principal quantum numbers. It is seen that the spectrum (<ref>) is a set of discrete energy levels separated by a distance, dependent on the widths of the wells. Moreover, at a=b=c, each state is triply degenerate, at a=b c, a=c b or b=c a each state is doublet plus singlet level, whereas at a b c, each state is a set of three singlet levels.

The wave functions of an electron in such states are as follows:

    ψ(xyz)=√(2/a)sinn_a/a x·√(2/b)sinn_b/b y·√(2/c)sinn_c/c z.


 Let us place the nanoobject in a uniform magnetic field  whose vector potential is A⃗(r⃗).
 It is known <cit.> that, in the general case, the Hamiltonian of the electronic subsystem in the magnetic field B⃗ takes the following form:

    Ĥ=[p⃗+eA⃗(r⃗)]^2/2m +U(r⃗)+[eħ/2mB⃗+α⃗(r⃗)]σ⃗.

Here, σ⃗=(σ _xσ _yσ _z) where σ _x =([ 0 1; 1 0 ]), σ _y =([  0 -i;  i  0 ]), σ _z =([  1  0;  0 -1 ])  are Pauli matrix; and  the expression α⃗(r⃗)=ħ/4m^2{∇⃗U(r⃗)· [p⃗+eA⃗(r⃗)]}σ⃗ describes spin-orbital interaction.


After transformations, the Hamiltonian (<ref>) can be rewritten as follows:

    Ĥ=Ĥ_0 +Ĥ_p ,

where

    Ĥ_p =e/2m[p⃗A⃗(r⃗)+A⃗(r⃗)p⃗]+e^2/2m A^2(r⃗)+eħ/2mB⃗σ⃗+α⃗σ⃗

is the perturbation Hamiltonian which describes the interaction of an electron with the magnetic field.

The second term in (<ref>) can be neglected in comparison with the first one. The last term in (<ref>), taking into account the potential (<ref>), is absent, because ∇⃗U(r⃗)=0. Therefore,

    Ĥ_p =e/2m[p⃗A⃗(r⃗)+A⃗(r⃗)p⃗]+eħ/2mB⃗σ⃗.

The perturbation theory criterion, i.e., ⟨ f |Ĥ_1|i⟩≪|E_f-E_i |, in nanoobjects with their sharp discrete spectrum, permits consideration of a problem with magnetic fields in a wide range of their values. Then, according to <cit.>,

    ⟨ n_a n_b n_c|Ĥ| n_a n_b n_c⟩   =⟨ n_a n_b n_c| Ĥ_0+eħ/2mB⃗σ⃗| n_a n_b n_c⟩ +⟨ n_a n_b n_c|Ĥ_1| n_a n_b n_c⟩
       +∑_n_x n_y n_z⟨ n_a n_b n_c| Ĥ_1 | n_x n_y n_z⟩⟨ n_x n_y n_z| Ĥ_1|n_a n_b n_c⟩/E_n_an_bn_c-E_n_x n_yn_z ,

where Ĥ_1=Ĥ_p-eħ/2mB⃗σ⃗. The 1st term is the sum of the matrix elements ⟨ n |Ĥ_0| n⟩ and  ⟨ n |eħ/2mB⃗σ⃗| n⟩  (here, |n ⟩ = |n_an_bn_c⟩|s ⟩; |s ⟩ = |↑⟩ or  |↓⟩ are spin indices), and the last two ones describe the perturbation up to the second order correction. In the 2nd correction, summation is carried out in all states, with the exception of  n_xn_yn_z=n_an_bn_c.
The above-mentioned matrix element ⟨ n |eħ/2mB⃗σ⃗| n⟩  in Landau  gauge case, due to the absence of spin-orbital interaction after summation over spin indices, takes the form: 
⟨ n |eħ/2mB⃗σ⃗| n⟩⇒⟨ n |eħ/2m B_zσ _z | n⟩⇒±eħ/2m B_z⟨ n | n ⟩⇒±eħ/2m B_z. The obtained result does not depend on the size of the nanoobject and  it is a value of the displacement of any state E( n_1n_2n_3)   (<ref>) up and down with the spin  |↑⟩  and |↓⟩,  respectively. Thus, only the corrections with Ĥ_1  contain the size of the nanoobject.



§ SHIFT OF LEVELS BY A MAGNETIC FIELD

Using (<ref>), we consider the shift of stationary states caused by the magnetic field:

    Δ E (n_a n_b n_c)   ≡⟨ n_a n_b n_c|Ĥ| n_a n_b n_c⟩-⟨ n_a n_b n_c|Ĥ_0+eħ/2mB⃗σ⃗| n_a n_b n_c⟩ =⟨ n_a n_b n_c|Ĥ_1| n_a n_b n_c⟩
       +∑_n_x n_y n_z⟨ n_a n_b n_c|Ĥ_1| n_x n_y n_z⟩⟨ n_x n_y n_z|Ĥ_1| n_a n_b n_c⟩/E_n_a n_b n_c-E_n_x n_y n_z.


Let us analyze two cases of the magnetic field B⃗ orientation:
a) the Landau gauge case, A⃗(r⃗)=(0,Bx,0). Taking into account the relationship B⃗=A⃗(r⃗)⇒(0,0,B), the field is oriented along OZ axis b) the gauge A⃗(r⃗)=(Bz,0,α By) — magnetic field B⃗ is in XOY plane with different orientation, depending on parameter α, in particular, at α = 1, B⃗ is oriented along the diagonal in  XOY plane, and at α=0 along OY axis.




 §.§ Case a) A⃗(r⃗)=(0,Bx,0).


The 1st correction, taking into account that p_y A_y=A_y p_y, is:

    ⟨ n |Ĥ_1| n ⟩ =e/2m⟨ n |Ĥ_1| n ⟩⇒e/2m⟨ n | (A_yp̂_y +p̂_y A_y )| n ⟩⇒eB/mħ/⟨ n | x∂/∂ y| n ⟩⇒eB/mħ/ i_1 i_2 i_3.


In equation (<ref>),  we used the denotation ⟨ n | =⟨ n_a n_b n_c|; it is taken into account that the term ⟨ n |eħ/2mB⃗σ⃗| n ⟩ equals zero due to the absence of spin-orbital interaction; with the denotation q=n_a/a x, using the integrals tables <cit.>,

    i_1 =2/a·(a/n_a)^2·∫ _0^n_aq·sin ^2  q· q⇒2/a·(a/n_a)^2. (q^2/4-q/4sin 2q-1/8cos 2q )|_0^n_a =a/2 ,


    i_2 =2/b·∫ _0^bsinn_b/d  y∂/∂ ysinn_b/d y· y⇒2/b·1/2.(sin ^2n_b/b y)|_0^b =0,


    i_3 =2/c·∫ _0^csin^2n_c/c z · z=1.

Thus, the 1st correction (<ref>) is zero.

Let us consider the 2nd correction.

We start with the calculation of the matrix elements in equation (<ref>)

    ⟨ n |Ĥ_1| m ⟩ =e/2m⟨ n |Ĥ_1| n ⟩⇒eB/mħ/⟨ n | x∂/∂ y| m ⟩⇒eB/mħ/ i_4 i_5 i_6 ,

where

    i_4   =2/a·∫ _0^ax·sinn_1/a  x·sinm_1/a x· x⇒2/a·  x1/2. [cos(n_1-m_1)/a x-cos(n_1 +m_1)/a x]|_0^a
       ⇒a/ ^28n_1 m_1/(m_1^2 -n_1^2)^2[(-1)^n_1 +m_1-1],


    i_5 =2/b·∫ _0^bsinn_2/b  y∂/∂ ysinm_2/b y· y⇒(-1)4/bn_2 m_2/m_2^2 -n_2^2[(-1)^n_2 +m_2 -1],


    i_6=2/c·∫ _0^csinn_3/c  z·sinm_3/c z· z=Δ _n_3m_3

(Δ _n_3 m_3 is the Kronecker symbol).

Thus, the 2nd correction is non-zero only at even n_1+m_1, n_2+m_2, i.e., n_i+m_i =2k_i+1 (k_i =1, 2, 3,…).

In particular, the correction Δ _2 E(111) to the ground state E(111) in terms of the units  [-(eB/m)^2·(32/ ^2)^2·2· 10^-18· m/ ^2] is of the following form:

    Δ _2 E(111)=(a/b)·∑_k_1 k_2{2k_1/[(2k_1)^2 -1]^2·2k_2/[(2k_2)^2 -1]}^2[(2k_1)^2-1/a^2 +(2k_2)^2 -1/b^2]^-1.


It is seen that such a correction does not depend on the length of c of the nanoobject, along which the magnetic field B⃗ is directed. Figure <ref> shows the dependence of the correction on the size of the nanoobject in XOY plane, which is perpendicular to the direction of B⃗. The correction monotonously increases with an increase of a for any value of b from the interval 1–5 nm, and vice versa, monotonously decreases with an increase of b for any value  of a from the interval 1-5 nm.



We will evaluate the correction to the energy state E(111) of the nanoobject for B= 10 T and  for different values of a=b. Table <ref> shows the relative magnitudes of the correction (Δ _2 E(111)/E(111))_a.



The obtained results indicate a sharp dependence of the spectrum in a magnetic field on the nanoobject size.



 §.§  Case b) A⃗(r⃗)=(Bz,0,α By).



In this case, the perturbation Hamiltonian is as follows:

    H_1 =e/2m(A⃗p⃗+p⃗A⃗)⇒e/m(A_x p_x +A_z p_z)⇒eB/mħ/(z∂/∂ x +α y∂/∂ z).


In the case  of α = 0, i.e., with B⃗ directed along OX, the results and conclusions qualitatively coincide with the similar ones in the case A⃗(r⃗)=(0,Bx,0). Herein below we will consider the case of α = 1.

Here, the 1st correction, similarly to the case a), equals zero.

The calculations, similar to the above ones, in terms of the same units, give the following 2nd correction to the ground state:

    Δ'_2 E(111)=(c/a)^2·∑_k_1 k_3{2k_1/[(2k_1)^2 -1]·2k_3/[(2k_3)^2 -1]^2}^2[(2k_1)^2 -1/a^2 +(2k_3)^2 -1/c^2]^-1


    +α·(b/c)^2·∑_k_2 k_3{2k_3/[(2k_3)^2 -1]·2k_2/[(2k_2)^2 -1]^2}^2[(2k_2)^2 -1/b^2 +(2k_3)^2 -1/c^2]^-1 .


This correction, in contrast to the analogous one in the case a), contains the dependence of the length of c, as well as of its ratios to a and b, namely (c/a), (b/c). Figure <ref> shows the analogous to figure <ref> dependences of the 2nd correction for the size family c. It is shown that the greater c, the greater Δ'_2 E(111). For all surfaces . Δ'_2 E(111)|_c, for any fixed value of a, the correction monotonously increases with an increase in b and weakly decreases with an increase in a at fixed b.





Figure <ref> shows the dependencies Δ'_2 E(111)=f(a,b) for the family c_1=1 and c_2=3 (a) and c_1 =1 and c_2 =4 (b). There is clearly seen an intersection of two pairs of planes. The points a_i, b_i on the line of intersection are those in which the corrections corresponding to c_1, c_2 are of the same values. From the difference in the intersections in the cases a) and b), we can make a conclusion regarding the possible intersections outside the intervals a∈[1,5] and b∈[1,5].





§ CONCLUSIONS


The analysis of the electron spectrum of a nanoobject and of the shape of rectangular  parallelepiped placed in an external magnetic field, depending on the size of the object,  indicates that



  * in the framework of perturbation theory,  the correction to the spectrum appears only from the second term onward;

  * the magnitudes of the corrections depend both on the magnitude of the magnetic field B⃗ (it is greater for greater fields) and on its orientation relative to the nanoobject;

  * in the case of Landau gauge  (B⃗ is directed along OZ axis), the correction does not depend on the length of the nanoobject in this direction;

  * in the case A⃗=B(z,0,α  y), for any parameter α from the semi-interval (0,1], unlike in the former case, the correction  depends on the three lengths of the nanoobject;

  * it is established that in the case of   A⃗=B(z,0,y), there exists such a set of dimensions  of the  nanoobject  in XOY plane for which the corrections are the same as those for their certain values of length along OZ axis.


Thus, in order to purposefully change the electronic spectrum of the  nanoobject by the magnetic field B⃗, one should take into account not only  the orientation of  B⃗ in the nanoobject, but also its size and relationship between its geometrical  characteristics.





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