Mathematical Derivation

To determine the density of allowed quantum states as a function of energy, we need to consider an appropriate mathematical model. Electrons are allowed to move relatively freely in the conduction band of a semiconductor, but are confined to the crystal. As a first step, we will consider a free electron confined to a three-dimensional infinite potential well, where the potential well represents the crystal. The potential of the infinite potential well is defined as

(1)

where the crystal is assumed to be a cube with length a. Schrodinger's wave equation in three dimensions can be solved using the separation of variables technique. Extrapolating the results from the one-dimensional infinite potential well, we can show that

(2)

where n_x, n_y, and n_z are positive integers. (Negative values of n_x, n_y,  and n_z yield the same wave function, except for the sign, as the positive integer values, resulting in the same probability function and energy, so the negative integers do not represent a different quantum state.)

We can schematically plot the allowed quantum states in k space. Figure 1.1a shows a two-dimensional plot as a function of k_x and k_y. Each point represents an allowed quantum state corresponding to various integral values of n_x and n_y. Positive and negative values of k_x, k_y, or k_z have the same energy and represent the same

Figure 1.1 (a) A two-dimensional army of allowed quantum stales in k space. (b) The positive one-eighth of the spherical k space.

energy state. Since negative values of k_x, k_y, or k_z do not represent additional quantum states, the density of quantum states will be determined by considering only the positive one-eighth of the spherical k space as shown in Figure 1.1b.

The distance between two quantum states in the k_x direction, for example, is given by

(3)

Generalizing this result to three dimensions, the volume V_k of a single quantum state is

(4)

We can now determine the density of quantum states in k space. A differential volume in k space is shown in Figure 1.1b and is given by 4πk² dk, so the differential density of quantum states in k space can he written as

(5)

The first factor, 2, takes into account the two spin states allowed for each quantum stale; the next factor, 1/8, takes into account that we are considering only the quantum states for positive values of k_x, k_y, and k_z. The factor 4πk² dk, is again the differential volume and the factor (π/a)³ is the volume of one quantum state. Equation (5) may be simplified to

(6)

Equation (6) gives the density of quantum states as a function of momentum, through the parameter k. We can now determine the density of quantum states as a function of energy E. For a free electron, the parameters E and k are related by

(7a)

or

(7b)

The differential dk is

(8)

Then, substituting the expressions for k² and dk into Equation (6). the number of energy states between E and E + dE is given by

(9)

Since h = h/2π, Equation (9) becomes

(10)

Equation (10) gives the total number of quantum states between the energy E and E + dE in the crystal space volume of a³. If we divide by the volume a³, then we will obtain the density of quantum states per unit volume of the crystal. Equation (10) then becomes

(11)

The density of quantum states is a function of energy E. As the energy of this free electron becomes small, the number of available quantum states decreases. This density function is really a double density, in that the units are given in terms of states per unit energy per unit volume.

مواضيع ذات صلة

الاصرة الايونية ionic bond

تأثير تصغير مقاييس الحبيبات على خواص المادة

الحبيبات

غياب المثالية عن الترتيب الذري

الشبكات البلورية

الإشعاع السينكروتروني Synchrotron Radiation

مراكز الالوان Color Centers

الأشعة وحيدة الموجة في تجارب الحيود من البلورات

شاشات عرض البلورات السائلة ذات المصفوفات الفعالة: «AMLCD «Active Matrix Liquid Crystal Display

استخدام البلورات السائلة في أجهزة العرض

السلوك الكهروبصري لأغشية (PDLC)

الخواص الكهروبصرية لأغشية البلورات السائلة المنتشرة داخل بوليمر «PDLC»