5. The differential form of transcendental equations

We shall indicate now an approach for solving the eigenvalue Eqs. (18)–(20) and (22)–(24) providing an exact solution, written as a series expansion. We shall first illustrate this method with the function ζ1ð Þp :

Taking the derivative with respect to p in both sides of the equation

$$p\zeta\_1(p) = \sin\zeta\_1(p)\tag{40}$$

we get

Solving this equation for y

92 Heterojunctions and Nanostructures

parabolic approximation."

in Eq. (34), namely,

proposed in [30].

x ¼ 2n þ

<sup>2</sup><sup>n</sup> ð Þ¼ x 2n þ

and making the change x \$ y, we get for the root ζ2<sup>n</sup> [27]:

ζ ð Þ par

1 2 � �

π �

1 2 � �

0 < x <

ζ ð Þ par

obtained in [27] are simple, but cumbersome, and will not be given here.

It is possible to improve the parabolic approximation in two ways:

tary, but cumbersome, and will not be given here [29].

π �

Following exactly the same steps, similar expressions can be obtained for ζ2nð Þ x < 0 and for all the functions ζq, ξ<sup>q</sup> their parabolic approximations can be obtained. A special case is ζ<sup>1</sup> :

<sup>2</sup><sup>n</sup> ð Þ¼ <sup>x</sup> <sup>π</sup> ffiffiffiffiffiffiffiffiffiffiffi

The method cannot be applied, evidently, for ξ1, as the function to be inverted has no bump.

The explicit expressions of the parabolic approximation for the functions ξ<sup>n</sup> ð Þ n > 1 and ζn,

(1) To express the numerical coefficients in formulas similar to Eq. (36) using analytic approximations for the roots of the equations tan x ¼ x and tan x ¼ �1=x: Actually, these transcendental equations can be transformed in approximate, tractable, algebraic equations, using the algebraic approximations of the tan function, proposed by de Alcantara Bonfim and Griffiths [26] and generalized by other authors [28]. This approach is sometimes called "improved

(2) To approximate the bumps of the functions sin x=x and cos x=x with a cubic curve (polynomial); this approach is sometimes called cubic approximation. The calculations are elemen-

For an algebraic approximation of ζ1, we can use a formula similar to the cos approximation

<sup>x</sup> <sup>≃</sup> <sup>1</sup> � <sup>x</sup>

The finite square well is a good starting point for similar quantum mechanical problems, i.e., the asymmetric well (when the walls of the well, see Figure 1, have different heights), the

π � �<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin x

π2 <sup>4</sup> � <sup>y</sup>

> π2 <sup>4</sup> � <sup>x</sup>

1 <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> 2 � �π

s

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>4</sup> <sup>2</sup><sup>n</sup> <sup>þ</sup>

π3

1 2 � �

π3

<sup>1</sup> � <sup>x</sup> <sup>p</sup> (38)

<sup>1</sup> <sup>þ</sup> <sup>0</sup>:2x<sup>2</sup> <sup>p</sup> (39)

(36)

, (37)

<sup>4</sup> <sup>2</sup><sup>n</sup> <sup>þ</sup>

$$\frac{d\zeta\_1(p)}{dp} = \frac{\zeta\_1(p)}{\cos \zeta\_1(p) - p} \tag{41}$$

Using Eq. (40) and taking into account that we are in the second quadrant

$$\cos \zeta\_1(p) = -\sqrt{1 - p^2 \zeta\_1(p)^2} \tag{42}$$

we obtain the differential form of the equation for ζ1ð Þp :

$$\frac{d\zeta\_1(p)}{dp} = -\frac{\zeta\_1(p)}{\sqrt{1 - p^2 \zeta\_1(p)^2} + p}, \quad p \in [0, 1], \quad \zeta\_1(p) \in \left(\frac{\pi}{2}, \pi\right) \tag{43}$$

with the initial condition:

$$
\zeta\_1(0) = \pi \tag{44}
$$

Putting

$$X\_{2n}(p) = \zeta\_n(p), X\_{2n-1}(p) = \xi\_n(p), \ n = 1, 2, \ldots,\tag{45}$$

replacing p by x and relaxing the restriction p > 0, the equations for the eigenvalues of the wave vectors can be written in a unitary form:

$$\frac{dX\_n(\mathbf{x})}{d\mathbf{x}} = -\frac{X\_n(\mathbf{x})}{\sqrt{1 - \mathbf{x}^2 X\_n(\mathbf{x})^2} + \mathbf{x}}\tag{46}$$

with the initial condition:

$$X\_n(0) = \frac{n\pi}{2} \tag{47}$$

satisfactorily explains the quantum scale effects (QSEs) appearing in such systems and

If, for thin films, such theoretical models can be successfully applied, for ultrathin films, with only few (typically, less than 5) monolayers, obtained experimentally in the last two decades, the approximation of the infinite well is inadequate. This is why in such cases we have to use the exact solutions for the bound-state energy of the finite well or, at least, their analytic approximations. In order to make clear the differences between the predictions of the two models—the first one is based on the infinite well, and the second one is based on the finite well—we shall evaluate some QSE for an ultrathin metallic film for three potentials: infinite,

Let us consider a rectangular metallic films, with edges Lx, Ly, Lz, where Lx, Ly are macroscopic or mesoscopic and Lz is nanoscopic. If the metallic film is placed between two semi-infinite dielectrics, we can presume that the conduction electrons move freely in the plane of the film (defined by the axes Ox, Oy), and in transversal direction ð Þ Oz , the potential can be approxi-

> ℏ<sup>2</sup> k !2

> > Lx

The differences between the values taken by the integers nx, ny and nz are due to the fact that, along the directions Ox and Oy, the quantization is obtained imposing cyclic conditions, and along the direction Oz by "rigid wall" conditions, specific to the infinite well, with impenetra-

For ultrathin films, the discrete spectrum of kz can be easily observed experimentally, and the conduction electrons constitute a quasi-2D multiband electron gas, characterized by a

<sup>2</sup><sup>D</sup> ¼ kx; ky

nx; <sup>2</sup><sup>π</sup> Ly

ny; <sup>π</sup> Lz nz (55)

nx, ny ¼ �1, � 2, …; nz ¼ 1, 2, … (56)

V ¼ LxLyLz (53)

Quantum Wells and Ultrathin Metallic Films http://dx.doi.org/10.5772/intechopen.74150 95

<sup>2</sup><sup>m</sup> (54)

and by a quantized wave vector kz <sup>¼</sup> nz <sup>π</sup>

Lz :

predicted theoretically in the pioneering papers of Sandomirskii [4] and Schulte [5].

6.1. The infinite well model for the quantum well in an ultrathin metallic film

mated by an infinite rectangular well. The film has the volume:

k !

¼ kx; ky; kz

!

<sup>¼</sup> <sup>2</sup><sup>π</sup>

semi-infinite, and finite wells.

and the electron energy is

where we put

ble walls.

quasi-continuum, 2D wave vector k

With Eq. (46), we can obtain the derivatives of any order of Xnð Þx in an arbitrary point x<sup>0</sup> and, consequently, write down the Taylor series for this function, with arbitrary accuracy. Choosing x<sup>0</sup> ¼ 0, we get the following series expansion for Xnð Þx :

$$X\_n(\mathbf{x}) = \sum\_{m=0}^{\infty} q\_m \left(\frac{n\pi}{2}\right) \mathbf{x}^m \tag{48}$$

The parameters qm are polynomials in the variable

$$\frac{m\pi}{2} \equiv b:\tag{49}$$

$$q\_0(b) = b, \quad q\_1(b) = -b, \quad q\_2(b) = b \tag{50}$$

$$q\_3(b) = -b \left( 1 + \frac{b^2}{6} \right), \quad q\_4(b) = b \left( 1 + \frac{2b^2}{3} \right) \tag{51}$$

$$q\_5(b) = -b\left(1 + \frac{5}{3}b^2 + \frac{3}{2^3 \cdot 5}b^4\right), \quad q\_6(b) = b\left(1 + \frac{2 \cdot 5}{3}b^2 + \frac{2^3}{3 \cdot 5}b^4\right) \tag{52}$$

and so on. For the explicit expression of qmð Þb , m < 17, see [13]. The first three terms correspond to the Barker approximation.

Let us also remark that, in spite of the fact that the equivalence of Sturm-Liouville problems for electromagnetic fields and for wave functions was noticed many years ago, the results obtained for the finite rectangular well remain unused by the researchers studying wave propagation in wave guides or in other simple geometries. Reciprocally, the very detailed solutions of the equations for the normal modes of electromagnetic waves (see, for instance, the references [90, 92] in [31]) were apparently overlooked by researchers working in quantum mechanics.

### 6. Applications to the statistical physics of ultrathin metallic films

With few exceptions, the physics of ultrathin metallic films can be satisfactorily explained using different types of infinite well for the potential of electrons moving normally to the film plane. The model of the infinite well can be improved, for instance, by the phase accumulation theory [11, 12, 32], quite popular among the scientist working in surface physics. The theory satisfactorily explains the quantum scale effects (QSEs) appearing in such systems and predicted theoretically in the pioneering papers of Sandomirskii [4] and Schulte [5].

If, for thin films, such theoretical models can be successfully applied, for ultrathin films, with only few (typically, less than 5) monolayers, obtained experimentally in the last two decades, the approximation of the infinite well is inadequate. This is why in such cases we have to use the exact solutions for the bound-state energy of the finite well or, at least, their analytic approximations. In order to make clear the differences between the predictions of the two models—the first one is based on the infinite well, and the second one is based on the finite well—we shall evaluate some QSE for an ultrathin metallic film for three potentials: infinite, semi-infinite, and finite wells.

#### 6.1. The infinite well model for the quantum well in an ultrathin metallic film

Let us consider a rectangular metallic films, with edges Lx, Ly, Lz, where Lx, Ly are macroscopic or mesoscopic and Lz is nanoscopic. If the metallic film is placed between two semi-infinite dielectrics, we can presume that the conduction electrons move freely in the plane of the film (defined by the axes Ox, Oy), and in transversal direction ð Þ Oz , the potential can be approximated by an infinite rectangular well. The film has the volume:

$$V = L\_x L\_y L\_z \tag{53}$$

and the electron energy is

$$\frac{\hbar^2 \overline{k}^2}{2m} \tag{54}$$

where we put

with the initial condition:

94 Heterojunctions and Nanostructures

x<sup>0</sup> ¼ 0, we get the following series expansion for Xnð Þx :

The parameters qm are polynomials in the variable

q5ð Þ¼� b b 1 þ

spond to the Barker approximation.

mechanics.

q3ð Þ¼� b b 1 þ

� �

5 3 <sup>b</sup><sup>2</sup> <sup>þ</sup> Xnð Þ¼ <sup>0</sup> <sup>n</sup><sup>π</sup>

With Eq. (46), we can obtain the derivatives of any order of Xnð Þx in an arbitrary point x<sup>0</sup> and, consequently, write down the Taylor series for this function, with arbitrary accuracy. Choosing

> m¼0 qm nπ 2 � �

nπ

b2 6 � �

and so on. For the explicit expression of qmð Þb , m < 17, see [13]. The first three terms corre-

Let us also remark that, in spite of the fact that the equivalence of Sturm-Liouville problems for electromagnetic fields and for wave functions was noticed many years ago, the results obtained for the finite rectangular well remain unused by the researchers studying wave propagation in wave guides or in other simple geometries. Reciprocally, the very detailed solutions of the equations for the normal modes of electromagnetic waves (see, for instance, the references [90, 92] in [31]) were apparently overlooked by researchers working in quantum

6. Applications to the statistical physics of ultrathin metallic films

With few exceptions, the physics of ultrathin metallic films can be satisfactorily explained using different types of infinite well for the potential of electrons moving normally to the film plane. The model of the infinite well can be improved, for instance, by the phase accumulation theory [11, 12, 32], quite popular among the scientist working in surface physics. The theory

3 23 � <sup>5</sup> b4

Xnð Þ¼ <sup>x</sup> <sup>X</sup><sup>∞</sup>

<sup>2</sup> (47)

xm (48)

23 3 � 5 b4 (51)

(52)

<sup>2</sup> � <sup>b</sup> : (49)

� �

q0ð Þ¼ b b, q1ð Þ¼� b b, q2ð Þ¼ b b (50)

2b<sup>2</sup> 3 � �

> 2 � 5 <sup>3</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup>

, q4ð Þ¼ b b 1 þ

, q6ð Þ¼ b b 1 þ

$$\overrightarrow{k} = \begin{pmatrix} k\_x, k\_y, k\_z \end{pmatrix} = \begin{pmatrix} 2\pi \\ \overline{L\_x} n\_x, & \frac{2\pi}{L\_y} n\_y, & \frac{\pi}{L\_z} n\_z \end{pmatrix} \tag{55}$$

$$n\_{\mathbf{x}} \quad n\_{\mathbf{y}} = \pm 1, \quad \pm 2, \quad \ldots; \ n\_{\mathbf{z}} = 1, \ 2, \ \ldots \tag{56}$$

The differences between the values taken by the integers nx, ny and nz are due to the fact that, along the directions Ox and Oy, the quantization is obtained imposing cyclic conditions, and along the direction Oz by "rigid wall" conditions, specific to the infinite well, with impenetrable walls.

For ultrathin films, the discrete spectrum of kz can be easily observed experimentally, and the conduction electrons constitute a quasi-2D multiband electron gas, characterized by a quasi-continuum, 2D wave vector k ! <sup>2</sup><sup>D</sup> ¼ kx; ky and by a quantized wave vector kz <sup>¼</sup> nz <sup>π</sup> Lz : The number nz ¼ q plays the role of an subband index. So, in the 3D reciprocal space, the spectrum is formed by planes of allowed states (subbands), parallel to the xOy plane, and separated along the z direction, by segments of length Δkz ¼ π=Lz:

Let us consider a numeric example. For a metallic film with two atomic monolayers, the typical values are Lz � <sup>0</sup>:<sup>6</sup> nm, so <sup>Δ</sup>kz � <sup>5</sup> nm�<sup>1</sup> and kF <sup>¼</sup> <sup>16</sup> nm�<sup>1</sup>: Therefore, only three plans cut the Fermi (hemi-)sphere, or—in other words—only the first subbands are occupied, corresponding to p ¼ 1; 2; 3: Let us mention that there is no band corresponding to p ¼ 0, as, in this case, the amplitude of the wave function would be zero.

We shall compute the number of occupied electronic states and the Fermi wave vector of the ultrathin film. The total number of subbands, which cut the Fermi sphere is Q, defined by

$$Q = \text{int}\left[\frac{k\_{\text{f}}}{\Delta k\_{z}}\right] \tag{57}$$

and introducing the number density of electrons n ¼ N=V, we get

k2 F � �

subbands Q, and on the electron number density n.

<sup>12</sup> <sup>Q</sup>ð Þ <sup>4</sup><sup>Q</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>Q</sup> � <sup>1</sup> h i<sup>1</sup>=<sup>3</sup> 1

> k 2 F � �

<sup>3</sup> <sup>¼</sup> <sup>80</sup><sup>π</sup>

<sup>2</sup> ¼ 40πLz þ

<sup>3</sup> Lz <sup>þ</sup>

<sup>4</sup> <sup>¼</sup> <sup>20</sup>πLz <sup>þ</sup> <sup>7</sup>:<sup>5</sup> � <sup>π</sup>

k 2 F � �

k 2 F � �

k 2 F � �

with the same length. In other words

<sup>1</sup> ¼ 80πLz þ

5 2

14 3

π Lz � �<sup>2</sup>

π Lz � �<sup>2</sup>

Lz � �<sup>2</sup>

6.2. The semi-infinite well model for the quantum well in an ultrathin metallic film

where kF is measured in nm�<sup>1</sup>: The expressions (63)–(66) clearly illustrate the QSE on the Fermi

As already mentioned (see Eq. (45) and the remark just below Eq. (27)), the relation between the solutions of the eigenvalue Eq. (46), namely, the functions X, and the wave vector k is

<sup>k</sup> ! <sup>2</sup>

and the bound states of the semi-infinite well are described by the odd states of a finite well

Introducing Eq. (61) in Eq. (57), we get

π

electronic gas 1, 2, 3, or 4 subbands:

ultrathin film.

wave vector.

<sup>Q</sup> ¼ n

2πLz Q þ

giving the dependence of the Fermi wave vector on the thickness Lz, on the number of

<sup>n</sup><sup>1</sup>=<sup>3</sup> <sup>⩽</sup> Lz <sup>&</sup>lt;

The last two equations define the QSEs on the Fermi wave vector; they can be considered as the starting point of all other similar QSEs of various physical quantities characterizing the

Choosing <sup>n</sup> <sup>¼</sup> <sup>4</sup> � 1022cm�<sup>3</sup> <sup>¼</sup> <sup>40</sup> nm�<sup>3</sup>, we get the expression of the Fermi wave vector for an

π Lz � �<sup>2</sup>

π Lz

π

<sup>12</sup> <sup>Q</sup>ð Þ <sup>4</sup><sup>Q</sup> <sup>þ</sup> <sup>5</sup> ð Þ <sup>Q</sup> <sup>þ</sup> <sup>1</sup> h i<sup>1</sup>=<sup>3</sup> 1

� �<sup>2</sup> <sup>Σ</sup>1ð Þ <sup>Q</sup>

<sup>Q</sup> (61)

Quantum Wells and Ultrathin Metallic Films http://dx.doi.org/10.5772/intechopen.74150

, Lz < 0:5 nm (63)

, 0:5 nm ⩽Lz < 0:8 nm (64)

, 0:8 nm ⩽ Lz < 1:1024 nm (65)

, 1:1024 nm ⩽ Lz < 1:4006 nm (66)

<sup>L</sup> <sup>X</sup> (67)

<sup>n</sup><sup>1</sup>=<sup>3</sup> (62)

97

where int½ � x is the largest integer smaller than x: In our particular case, discussed in the previous example, Q ¼ 3, so there are only three distinct subbands, occupied at T ¼ 0: For films with few monolayers, the subbands are separated by energies of about 1 eV, so we can consider that T ¼ 0:

As the occupied states belonging to the subband of index q are situated inside circles cut by the Fermi sphere, of radius kF, <sup>q</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>F</sup> � ð Þ <sup>q</sup>Δkz <sup>2</sup> h i<sup>1</sup>=<sup>2</sup> (these circles are the intersection of the subband plane with the Fermi sphere), and the area corresponding to one electronic state k in each subband is 2ð Þ π 2 =LxLy ¼ ð Þ 2π 2 Lz=V, there are

$$\frac{\pi k\_{\text{F},q}^2}{\left(2\pi\right)^2 L\_z/V} = \frac{V}{\left(2\pi\right)^2 L\_z} \pi k\_{\text{F},q}^2 \tag{58}$$

occupied states in the subband q. The number of electrons N inside the Fermi sphere is obtained by summing up over the occupied subbands:

$$N = 2\frac{V}{(2\pi)^2 L\_z} \pi \sum\_{q=1}^{Q} k\_{F,q}^2 \tag{59}$$

$$= \frac{V}{2\pi L\_z} \left[ Qk\_F^2 - \left(\frac{\pi}{L\_z}\right)^2 \sum\_{q=1}^{Q} q^2 \right]$$

where the factor of 2 is due to the electron spin. Putting

$$\Sigma\_1(Q) = \sum\_{q=1}^{Q} q^2 = \frac{1}{6}Q(Q+1)(2Q+1) \tag{60}$$

and introducing the number density of electrons n ¼ N=V, we get

$$\left[k\_F^2\right]\_Q = n\frac{2\pi L\_z}{Q} + \left(\frac{\pi}{L\_z}\right)^2 \frac{\Sigma\_1(Q)}{Q} \tag{61}$$

giving the dependence of the Fermi wave vector on the thickness Lz, on the number of subbands Q, and on the electron number density n.

Introducing Eq. (61) in Eq. (57), we get

The number nz ¼ q plays the role of an subband index. So, in the 3D reciprocal space, the spectrum is formed by planes of allowed states (subbands), parallel to the xOy plane, and

Let us consider a numeric example. For a metallic film with two atomic monolayers, the typical values are Lz � <sup>0</sup>:<sup>6</sup> nm, so <sup>Δ</sup>kz � <sup>5</sup> nm�<sup>1</sup> and kF <sup>¼</sup> <sup>16</sup> nm�<sup>1</sup>: Therefore, only three plans cut the Fermi (hemi-)sphere, or—in other words—only the first subbands are occupied, corresponding to p ¼ 1; 2; 3: Let us mention that there is no band corresponding to p ¼ 0, as, in this case, the

We shall compute the number of occupied electronic states and the Fermi wave vector of the ultrathin film. The total number of subbands, which cut the Fermi sphere is Q, defined by

<sup>Q</sup> <sup>¼</sup> int kF

where int½ � x is the largest integer smaller than x: In our particular case, discussed in the previous example, Q ¼ 3, so there are only three distinct subbands, occupied at T ¼ 0: For films with few monolayers, the subbands are separated by energies of about 1 eV, so we can

As the occupied states belonging to the subband of index q are situated inside circles cut by

subband plane with the Fermi sphere), and the area corresponding to one electronic state k in

<sup>F</sup> � ð Þ <sup>q</sup>Δkz <sup>2</sup> h i<sup>1</sup>=<sup>2</sup>

Lz=V, there are

Lz=<sup>V</sup> <sup>¼</sup> <sup>V</sup> ð Þ 2π 2 Lz πk 2

occupied states in the subband q. The number of electrons N inside the Fermi sphere is

q¼1 k2

> X Q

3 5

q¼1 q2

Δkz � �

(57)

(these circles are the intersection of the

F, <sup>q</sup> (58)

F, <sup>q</sup> (59)

Q Qð Þ þ 1 ð Þ 2Q þ 1 (60)

separated along the z direction, by segments of length Δkz ¼ π=Lz:

amplitude of the wave function would be zero.

consider that T ¼ 0:

96 Heterojunctions and Nanostructures

each subband is 2ð Þ π

the Fermi sphere, of radius kF, <sup>q</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

2

=LxLy ¼ ð Þ 2π

obtained by summing up over the occupied subbands:

where the factor of 2 is due to the electron spin. Putting

2

πk 2 F, q

<sup>N</sup> <sup>¼</sup> <sup>2</sup> <sup>V</sup> ð Þ 2π 2 Lz π X Q

> 2 4

Qk<sup>2</sup> <sup>F</sup> � <sup>π</sup> Lz � �<sup>2</sup>

<sup>q</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> 6

ð Þ 2π 2

<sup>¼</sup> <sup>V</sup> 2πLz

<sup>Σ</sup>1ð Þ¼ <sup>Q</sup> <sup>X</sup>

Q

q¼1

$$\left[\frac{\pi}{12}Q(4Q+1)(Q-1)\right]^{1/3}\frac{1}{n^{1/3}}\lessdot L\_z < \left[\frac{\pi}{12}Q(4Q+5)(Q+1)\right]^{1/3}\frac{1}{n^{1/3}}\tag{62}$$

The last two equations define the QSEs on the Fermi wave vector; they can be considered as the starting point of all other similar QSEs of various physical quantities characterizing the ultrathin film.

Choosing <sup>n</sup> <sup>¼</sup> <sup>4</sup> � 1022cm�<sup>3</sup> <sup>¼</sup> <sup>40</sup> nm�<sup>3</sup>, we get the expression of the Fermi wave vector for an electronic gas 1, 2, 3, or 4 subbands:

$$\left[k\_F^2\right]\_1 = 80\pi L\_z + \left(\frac{\pi}{L\_z}\right)^2, \quad L\_z < 0.5 \text{ nm} \tag{63}$$

$$\left[\left[k\_{\rm F}^{2}\right]\_{2} = 40\pi L\_{z} + \frac{5}{2}\left(\frac{\pi}{L\_{z}}\right)^{2}, \quad 0.5 \text{ nm} \lessapprox L\_{z} < 0.8 \text{ nm} \tag{64}$$

$$\left[k\_F^2\right]\_3 = \frac{80\pi}{3}L\_z + \frac{14}{3}\left(\frac{\pi}{L\_z}\right)^2, \quad 0.8 \text{ nm} \lessapprox L\_z < 1.1024 \text{ nm} \tag{65}$$

$$\left[k\_F^2\right]\_4 = 20\pi L\_z + 7.5 \cdot \left(\frac{\pi}{L\_z}\right)^2, \quad 1.1024 \text{ nm} \lessapprox L\_z < 1.4006 \text{ nm} \tag{66}$$

where kF is measured in nm�<sup>1</sup>: The expressions (63)–(66) clearly illustrate the QSE on the Fermi wave vector.

#### 6.2. The semi-infinite well model for the quantum well in an ultrathin metallic film

As already mentioned (see Eq. (45) and the remark just below Eq. (27)), the relation between the solutions of the eigenvalue Eq. (46), namely, the functions X, and the wave vector k is

$$k \to \frac{2}{L}X \tag{67}$$

and the bound states of the semi-infinite well are described by the odd states of a finite well with the same length. In other words

$$k\_{2n} \to \frac{2}{L\_z} \zeta\_n(p) \tag{68}$$

k2 <sup>F</sup> <sup>¼</sup> <sup>2</sup>πLz

<sup>&</sup>lt; <sup>p</sup> <sup>¼</sup> <sup>1</sup>

ζ2 <sup>Q</sup>ð Þp ⩽

6.2.1. The finite well model for the quantum well in an ultrathin metallic film

there is at least a solution for each value of p. Eqs. (71)–(73) are replaced by

p > 1 or Lz <

6:5 � Lz

1

2 ð Þ Q � 1=2 π

Therefore, instead of Eq. (62), we have

2 Lz � �<sup>2</sup> ζ2 <sup>Q</sup>ð Þp ⩽k 2 <sup>F</sup> <sup>¼</sup> <sup>2</sup>πLz

must be considered together with Eqs. (71)–(73).

1

relation:

or

Eq. (77).

<sup>Q</sup> <sup>n</sup> <sup>þ</sup>

1 Q

2 Lz � �<sup>2</sup>

Let us presume that the electron gas contains exactly Q subbands, which is equivalent to the

<sup>6</sup>:<sup>5</sup> <sup>&</sup>lt; Lz <sup>&</sup>lt; ð Þ <sup>Q</sup> � <sup>1</sup>=<sup>2</sup> <sup>π</sup>

1 Q

The term corresponding to the r.h.s. of the inequality (62) is missing in this case, as the number of roots (solutions) is completely determined by the condition imposed to Lz, according to

Replacing the electron number density with a typical value <sup>n</sup> <sup>¼</sup> <sup>40</sup> nm�<sup>3</sup> and using Eq. (63), we get (we took advantage of the fact that, incidentally, the numeric factor is 0:25128 ≃1=4)

This restriction on p, which can be verified using, for instance, the cubic approximation for ζn,

The situation is quite similar to the previous one—the semi-infinite well. However, in this case,

1

one state, X<sup>1</sup> ¼ ξ1;

1 Q X Q

q¼1

1 <sup>4</sup>Qp<sup>3</sup> <sup>þ</sup>

2 Lz � �<sup>2</sup> X Q

q¼1

<sup>Q</sup> <sup>n</sup> <sup>þ</sup>

X Q

<sup>ζ</sup>qð Þ<sup>p</sup> � �<sup>2</sup> (75)

Quantum Wells and Ultrathin Metallic Films http://dx.doi.org/10.5772/intechopen.74150 99

< 1, or 0:16 nm < Lz < 1:2467 nm (76)

<sup>13</sup> (77)

<sup>ζ</sup>qð Þ<sup>p</sup> � �<sup>2</sup> ==<sup>78</sup> (78)

<sup>ζ</sup>qð Þ<sup>p</sup> � �<sup>2</sup> (79)

<sup>6</sup>:<sup>3</sup> <sup>¼</sup> <sup>0</sup>:<sup>16</sup> nm, (80)

<sup>π</sup> <sup>¼</sup> <sup>0</sup>:<sup>31831</sup> <sup>&</sup>lt; p, or Lz <sup>&</sup>lt; <sup>0</sup>:<sup>31831</sup> nm, (81)

q¼1

It is convenient to define

$$w = \left(\frac{m\mathcal{U}}{2\hbar^2}\right)^{1/2} \tag{69}$$

So, the inverse strength of the quantum well, similar to Eq. (10), can be defined as

$$p = \frac{1}{wL\_z} \tag{70}$$

According to Eq. (68), the wave vector depends on both Lz and U (or w). As U is a material dependent quantity, related, in principle, to the work function, we shall replace it, for this numerical example, with the typical value of U ¼ 5 eV; in this case, Eq. (70) gives

$$p = \frac{1}{6.5 \times \overline{L\_z}}\tag{71}$$

with Lz in nanometers.

An important difference which occurs at semi-infinite wells, compared to the infinite wells, is that it keeps a finite number of bound states. Consequently, the energy spectrum of the electron gas of the metallic film contains a finite number of subbands, in dependence of the value of p: The well keeps at least one state if

$$p < 1 \quad \Rightarrow \overline{L\_z} > \frac{1}{6.3} = 0.16 \quad nm \tag{72}$$

and exactly one state ζ<sup>1</sup> if

$$\frac{2}{3\pi} = 0.21221 < p < 1,\text{ or } 0.16\text{ mm} < \overline{L\_z} < 0.74794\text{ mm} \tag{73}$$

This corresponds, usually, to a film with one or two monolayers. We have two states in the well, ζ<sup>1</sup> and ζ2, if

$$\frac{2}{5\pi} = 0.12732 < p < 1,\text{ or } 0.16\text{ }nm < \overline{L\_z} < 1.2467\text{ }nm\tag{74}$$

This corresponds, usually, to a film with up to four monolayers, etc. These conditions are purely mathematical, i.e., consequences of the specific form of the eigenvalue equations.

Now, we shall impose physical conditions, due to the p� or Lz� dependence of the Fermi wave vector and of the number of subbands. Taking into account Eq. (67) and using an argument similar to Eq. (57), we find

Quantum Wells and Ultrathin Metallic Films http://dx.doi.org/10.5772/intechopen.74150 99

$$k\_F^2 = \frac{2\pi L\_z}{Q}n + \frac{1}{Q}\left(\frac{2}{L\_z}\right)^2 \sum\_{q=1}^{Q} \left(\zeta\_q(p)\right)^2\tag{75}$$

Let us presume that the electron gas contains exactly Q subbands, which is equivalent to the relation:

$$\frac{2}{(Q - 1/2)\pi} < p = \frac{1}{6.5 \times \overline{L\_z}} < 1,\text{ or } 0.16 \text{ mm} < \overline{L\_z} < 1.2467 \text{ mm} \tag{76}$$

or

<sup>k</sup>2<sup>n</sup> ! <sup>2</sup> Lz

<sup>w</sup> <sup>¼</sup> mU 2ℏ<sup>2</sup> <sup>1</sup>=<sup>2</sup>

> <sup>p</sup> <sup>¼</sup> <sup>1</sup> wLz

According to Eq. (68), the wave vector depends on both Lz and U (or w). As U is a material dependent quantity, related, in principle, to the work function, we shall replace it, for this

> <sup>p</sup> <sup>¼</sup> <sup>1</sup> 6:5 � Lz

An important difference which occurs at semi-infinite wells, compared to the infinite wells, is that it keeps a finite number of bound states. Consequently, the energy spectrum of the electron gas of the metallic film contains a finite number of subbands, in dependence of the

1

This corresponds, usually, to a film with one or two monolayers. We have two states in the

This corresponds, usually, to a film with up to four monolayers, etc. These conditions are purely mathematical, i.e., consequences of the specific form of the eigenvalue equations.

Now, we shall impose physical conditions, due to the p� or Lz� dependence of the Fermi wave vector and of the number of subbands. Taking into account Eq. (67) and using an argument

<sup>3</sup><sup>π</sup> <sup>¼</sup> <sup>0</sup>:<sup>21221</sup> <sup>&</sup>lt; <sup>p</sup> <sup>&</sup>lt; <sup>1</sup>, or <sup>0</sup>:<sup>16</sup> nm <sup>&</sup>lt; Lz <sup>&</sup>lt; <sup>0</sup>:<sup>74794</sup> nm (73)

<sup>5</sup><sup>π</sup> <sup>¼</sup> <sup>0</sup>:<sup>12732</sup> <sup>&</sup>lt; <sup>p</sup> <sup>&</sup>lt; <sup>1</sup>, or <sup>0</sup>:<sup>16</sup> nm <sup>&</sup>lt; Lz <sup>&</sup>lt; <sup>1</sup>:<sup>2467</sup> nm (74)

So, the inverse strength of the quantum well, similar to Eq. (10), can be defined as

numerical example, with the typical value of U ¼ 5 eV; in this case, Eq. (70) gives

p < 1 ) Lz >

It is convenient to define

98 Heterojunctions and Nanostructures

with Lz in nanometers.

and exactly one state ζ<sup>1</sup> if

similar to Eq. (57), we find

well, ζ<sup>1</sup> and ζ2, if

value of p: The well keeps at least one state if

2

2

ζnð Þp (68)

<sup>6</sup>:<sup>3</sup> <sup>¼</sup> <sup>0</sup>:<sup>16</sup> nm (72)

(69)

(70)

(71)

$$\frac{1}{6.5} < \overline{L\_z} < \frac{(Q - 1/2)\pi}{13} \tag{77}$$

Therefore, instead of Eq. (62), we have

$$\chi\left(\frac{2}{L\_z}\right)^2 \zeta\_Q^2(p) \ll k\_F^2 = \frac{2\pi L\_z}{Q}n + \frac{1}{Q}\left(\frac{2}{L\_z}\right)^2 \sum\_{q=1}^Q \left(\zeta\_q(p)\right)^2 \quad / / 78\tag{78}$$

The term corresponding to the r.h.s. of the inequality (62) is missing in this case, as the number of roots (solutions) is completely determined by the condition imposed to Lz, according to Eq. (77).

Replacing the electron number density with a typical value <sup>n</sup> <sup>¼</sup> <sup>40</sup> nm�<sup>3</sup> and using Eq. (63), we get (we took advantage of the fact that, incidentally, the numeric factor is 0:25128 ≃1=4)

$$
\zeta\_Q^2(p) \lessapprox \frac{1}{4Qp^3} + \frac{1}{Q} \sum\_{q=1}^Q \left(\zeta\_q(p)\right)^2 \tag{79}
$$

This restriction on p, which can be verified using, for instance, the cubic approximation for ζn, must be considered together with Eqs. (71)–(73).

#### 6.2.1. The finite well model for the quantum well in an ultrathin metallic film

The situation is quite similar to the previous one—the semi-infinite well. However, in this case, there is at least a solution for each value of p. Eqs. (71)–(73) are replaced by

$$\left| \begin{array}{c} p > 1 \text{ or } \overline{L\_z} < \frac{1}{6.3} = 0.16 \text{ } nm \text{.} \end{array} \right. \tag{80}$$

one state, X<sup>1</sup> ¼ ξ1;

$$\frac{1}{\pi} = 0.31831 < p\_{\prime} \quad \text{or} \quad \overline{L\_z} < 0.31831 \text{ mm},\tag{81}$$

$$\begin{aligned} \text{two states,} \quad &X\_1 = \xi\_1 \quad X\_2 = \zeta\_1\\ &\frac{2}{3\pi} = 0.21221 < p < 1, \text{ or } \overline{L\_z} < 0.74794 \quad \text{mm,} \\ &\text{three states,} \quad X\_1 = \xi\_1 \quad X\_2 = \zeta\_1 \quad X\_3 = \xi\_2 \end{aligned} \tag{82}$$

Acknowledgements

Author details

Victor Barsan

References

Lett.91.226801

101-106

\$30.00

DOI: 10.1119/1.2976792

The author acknowledges the financial support of the IFIN-HH–ANCSI project PN 16 42 0101/

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2016 and of the IFIN-HH–JINR grant 04-4-1121-2015/2017.

Address all correspondence to: vbarsan@theory.nipne.ro

IFIN and the UNESCO Chair at HHF, Bucharest-Magurele, Romania

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and so on. In Eqs. (76) and (77), the replacement ζ<sup>q</sup> ! Xq must be done. Eq. (74), with ζ<sup>q</sup> ! Xq, gives the QSE for the Fermi wave vector.

These solutions, or their analytic approximations (for instance, the cubic one), can be used directly in the models already proposed for the infinite well [33], in order to obtain the electron density, the surface free energy, the surface dipolar moment, or other similar quantities, in the more realistic case of a finite rectangular well.
