о

♦

о

о

OS m

Ь

С

in m

* ч

о -

а л

о

>

£ О

to О

-о

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X-rays

I I I

Gamma rays

Ultraviolet

Infrared

Radio spectrum

I fm

\ pm -h-

1 nm -h-

1 fim

I mm

1 km -h-

1 Mm

-12 ,Q-10 10-9

1 THz -h-

1 GHz -h-

10 1 MHz

10 I kHz

Wavelength (m)

1 Hz Frequency (Hz)

10 10 10 10 iO

Figure 2.4 I The electromagnetic frequency spectrum.

EXAMPLE 2.2

ultraviolet and visible range. These wavelengths are very short compared to the usual radio spectrum range.

Objective

To calculate the de Broglie wavelength of a particle.

Consider an electron traveling at a velocity of 10 cm/sec = lO m/s.

The momentum is given by

p=.mv = (9.11 X 10- )(10) = 9,11 X 10 Then, the de Broglie wavelength is

h 6.625 X 10-- .

X = ~ = ----- = 7.27 X 10 m

p 9-11 X 10--*

л = 72.7 A

Comment

This calculation shows the order of magnitude of the de Broglie wavelength for a typical electron.

In some cases electromagnetic waves behave as if they are particles (photons) and sometimes particles behave as if they are waves. This wave-particle duality

TEST YOUR UNDERSTANDING

Е2Л Determine the energy of a photon having wavelengths of (й) л = 10,000 A and (h) A = lOA. [ЛЭ tOl > PZ\ JOf 01 X 664 (Ф -AVZ-} J<f 6[-0l X 66 I W suv]

E2.2 (a) Find the momentum and energy of a particle with mass of 5 x 10~- kg and a de Broglie wavelength of 180 A. An electron has a kinetic energy of 20 meV. Determine the de Bmghe wavelength. iv i*98 = Y уш-2ц ,. DI x P9l = (cf) t-Ol X 9t*8 JO f [- oi X g£\ = 3 s/m-g>i .qi x 89T = W uvl

2Л.З The Uncertainty Principle

The Heisenberg uncertainty principle, given in 1927, also applies primarily to very small particles, and states that we cannot describe with absolute accuracy the behavior of these subatomic particles. The uncertainty principle describes a fundamental relationship between conjugate variables, including position and momentum and also energy and time.

The first statement of the uncertainty principle is that it is iinpossible to simultaneously describe with absolute accuracy the position and momentum of a particle. If the uncertainty in the momentum is Ap and the uncertainty in the postion is Дх, then the uncertainty principle is stated as

Ap Ax > n (2.4)

where h is defined as Ti - h/2jt = 1.054 x lO - J-s and is called a modified Plancks constant. This statement inay be generalized to include angular posirion and angular momentum.

The second statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of fime the particle has this energy. Again, if the uncertainty in the energy is given by AE and the uncertainty in the time is given by At, then the uncertainty principle is stated as

A£ At>h (2.5)

One way to visualize the uncertainty principle is to consider the simultaneous measurement of position and momentum, and the simultaneous measurement of energy and time. The uncertainty principle iinplies that these simultaneous measurements

In some texts, the uncertainty principle is stated as Д/? A.v > hjl. We are interested here in the order of magnitude and will not be concerned with small differences.

principle of quantum mechanics applies primarily to small particles such as electrons, but it has also been shown to apply to protons and neutrons. For very large particles, we can show that the relevant equations reduce to those of classical mechanics. The wave-particle duality principle is the basis on which we will use wave theory to describe the motion and behavior of electrons in a crystal.

are in error to a certain extent. However, the modified PI ancles constant fi is very small; the uncertainty principle is only significant for subatomic panicles. We must keep in mind nevertheless that the uncertainty principle is a fundamental statement and does not deal only with measurements.

One consequence of the uncertainty principle is that we cannot, for example, determine the exact position of an electron. We will, instead, determine the probability of finding an electron at a particular position. In later chapters, we will develop a probability density function that will allow us to determine the probability that an electron has a particular energy. So in describing electron behavior, we will be dealing with probability functions.

TEST YOUR UNDERSTANDING

E2.3 The uncertainty in position of an electron is 12 A. Determine the minimum

uncertainty in momentum and also the corresponding uncertainty in kinetic energy. (ЛЭ 90*0 = .?V s/m-3Ti y 0l X 6/:8 = V *suv)

E2.4 An electrons energy is measured with an uncertainty of 1.2 e V. What is the minimum uncertainty in time over which the energy is measured? щ 01 x 61*5 = suy)

2.2 I SCHRODINGERS WAVE EQUATION

The various experimental results involving electromagnetic waves and particles, which could not be explained by classical laws of physics, showed that a revised formulation of mechanics was required. Schrodinger, in 1926. provided a formulation called wave mechanics, which incorporated the principles of quanta introduced by Planck, and the wave-particle duality principle introduced by de Broglie. Based on the wave-particle duality principle, we will describe the motion of electrons in a crystal by wave theory. This wave theory is described by Schrodingers wave equation.

2Л, 1 The Wave Equation

The one-dimensional, nonrelativistic Schrodingers wave equation is given by

---+ Пх)Фи, 0 - Jh (2.6)

2m dx at

where Ф(х /) is the wave function, V{x) is the potential function assumed to he independent of time, m is the mass of the particle, and / is the imaginary constant There are theoretical arguments that justify the form of Schrodingers wave equation, but the equation is a basic postulate of quantum mechanics. The wave function

/) will be used to describe the behavior of the system and, mathematically, Ф(х t) can be a complex quantity.

We may determine the time-dependent portion of the wave function and the position-dependent, or time-independent, portion of the wave function by using the

technique of separation of variables. Assume that the vave function can be vritten in the form

Пх)г(х)ф(П (2.7)

where -ф{х) is a function of the position x only and 0{t) is a function of time t only. Substituting this form of the solution into Schrodingers wave equation, we obtain

(OV + УММФ() = jnf(x) (2.8)

Zm ox at

Jf we divide by the total wave function, Eiuation (2.8) becomes

-----г- + V(.v) - /П - -- (2.9)

Since the left side of Equation (2.9) is a function of position x only and the right side of the equation is a function of time t only, each side of this equation must be equal to a constant. We will denote this separation of variables constant by /?. The time-dependent portion of Equation (2.9) is then written as

where again the parameter rj is called a separation constant. The solution of Equation (2Л0) can be written in the form

фи)е- (2.11)

The form of this solution is the classical exponential form of a sinusoidal wave where T}/h is the radian frequency ш. We have that E - hv or E = hoj/ln. Then o) - i)lh= E/hso that the separation constant is equal to the total energy E of the particle.

The time-independent portion of Schrodingers wave equation can now be written from Equation (2.9) as

2m tj/ix) dx

where the separation constant is the total energy E of the particle. Equation (2.12) may be written as

В^Ф(х) 2m

-{E-V(xMU) = 0

(2ЛЗ)

where again m is the mass of the particle, У{х) is the potential experienced by the particle, and £ is the total energy of the particle. This time-independent Schrodingers wave equation can also be justified on the basis of the classical wave equation as

shown in Appendix E. The pseudo-derivation in the appendix is a simple approach but shows the plausibility of the time-independent Schrodingers equation.

2.2.2 Physical Meaning of the Wave Function

We are ultimately trying to use the wave function Ф (jc, /) to describe the behavior of an electron in a crystal. The function Ф(д:, /) is a wave function, so it is reasonable lo ask what the relation is between the function and the electron. The total wave function is the product of the position-dependent, or time-independent, function and the time-dependent function. We have from Equation (2.7) that

-jit:/h]t

(2.14)

Since the total wave function Ф(х, /) is a complex function, it cannot by itself represent a real physical quantity.

Max Born postulated in 1926 that the function Cv, f )P d is the probability of finding the particle between x and x + Jjc at a given time, or that Ф(;с, 01 is a probability density function. We have that

(Ф(;с,Ор-Ф(х,г)-vi*(x,0 (2.15)

where Ф*(л, /) is the complex conjugate function. Therefore

Then the product of the total wave function and its complex conjugate is given by

-j(E/h)n

i/4x)e] = V(x)iA4x) (2.16)

Therefore, we have that

(x.r)\ = i/{x)rM\i/ix)

(2Л7)

is the probability density function and is independent of time. One major difference between classical and quantum mechanics is that in classical mechanics, the position of a particle or body can be determined precisely, whereas in quantum mechanics, the position of a particle is found in terms of a probability. We will determine the probability density function for several examples, and, since this property is independent of time, we will, in general, only be concerned with the time-independent wave function.

2.2.3 Boundary Conditions

Since the function (;r, /)p represents the probability density function, then for a . single particle, we must have that

/ОС -ОС

(2.18)

The probabiUty of finding the particle somewhere is certain. Equation (2.18) allows us to normalize the wave function and is one boundary condition that is used to determine some wave function coefficients.

The remaining boundary conditions imposed on the wave function and its derivative are postulates. However, we may state the boundary conditions and present arguments that justify why they must be imposed. The wave function and its first derivative must have the following properties if the total energy E and the potential V (x) are finite everywhere.

Condition 1. i/Cv) must be finite, single-valued, and continuous. Condition 2. д'ф(х)1дх must be finite, single-valued, and continuous.

Since I T/f(jc) is a probability density, then V(x) must be finite and single-valued. If the probability density were to become infinite at some point in space, then the probability of finding the particle at this position would be certain and the uncertainty principle would be violated. If the total energy E and the potential V{x) are finite everywhere, then from Equation (2.13), the second derivative must be finite, which implies that the first derivative must be continuous. The first derivative is related to the particle momentum, which must be finite and single-valued. Finally, a finite first derivative implies that the function itself must be continuous. In some of the specific examples that we will consider, the potential function will become infinite in particular regions of space. For these cases, the first derivative will not necessarily be continuous, but the remaining boundary conditions will still hold.

2.3 1 APPLICATIONS OF SCHRODINGERS WAVE EQUATION

We will now apply Schrodinger*s wave equation in several examples using various potential functions. These examples will demonstrate the techniques used in the solution of Schrodingers differential equation and the results of these examples will provide an indication of the electron behavior under these various potentials. We will utilize the resulting concepts later in the discussion of semiconductor properties.

2Л1 Electron in Free Space

As a first example of applying the Schrodingers wave equation, consider the motion of an electron in free space. If there is no force acting on the particle, then the potential function V(jc) will be constant and we must have E > V{x), Assume, for simplicity, that the potential function V{x) =0 for all x. Then, the time-independent wave equation can be written from Equation (2.13) as

г^ф(х) 2m E

(x)-0

3x fi

The solution to this differential equation can be written in the form

(2Л9)

ф(х) = А exp

jx/bnE

-h В exp

-]хл/2тЕ h

(2.20)

Recall that the time-dependent portion of the solution is Then the total solution for the wave function is given by

Ф(;с, t) = A exp

XxlmE - Et) -hfiexp --{x/lmE + Et)

(2.22)

This wave function solution is a traveling wave, which means that a particle moving in free space is represented by a traveling wave. The first term, with the coefficient A. is a wave traveling in the -\-x direction, while the second term, with the coefficient is a wave traveling in the -x direction. The value of these coefficients will be deter mined from boundary conditions. We will again see the traveling-wave solution for an electron in a crystal or semiconductor material.

Assume, for a moment, that we have a particle traveling in the -I-jc direction, which will be described by the -\-x traveling wave. The coefficient 6 = 0. We can write the traveling-wave solution in the form

Ф(х, i) = A exp {j{kx - ojt)\ (2.23)

where /: is a wave number and is

2л

(2.24)

The parameter Л is the wavelength and, comparing Equation (2.23) with Equation (2.22), the wavelength is given by

(2-25)

л/2ш£

From de Broglies wave-particle duality principle, the wavelength is also given by

A = - (2.26)

A free particle with a well-defined energy will also have a well-defined wavelength and momentum.

The probability density function is Ф(х, ОФ*(Л) - АЛ*, which is a constant independent of position. A free particle with a well-defined momentum can be found anywhere with equal probability. This result is in agreement with the Heisenberg uncertainty principle in that a precise momentum implies an undefined position.

A localized free particle is defined by a wave packet, formed by a superposition of wave functions with different momentum or к values. We will not consider the wave packet here.

2-3.2 The Inflnite Potential Well

The problem of a particle in the infinite potential well is a classic example of a bound particle. The potential V{x) as a function of position for this problem is shown in

2 - 3 Applications of Schrodingers Wave Equatbn

DO

Region !

Region П

Region 111

Figure 2,5 ! Potential function of the infinite potential well

Figure 2.5. The particle is assumed to exist in region II so the particle is contained within a finite region of space. The time-independent Schrodingers wave equation is again given by Equation (2.13) as

Ч-(£:-v(x))(x)-o

(2ЛЗ)

where E is the total energy of the particle. If E is finite, the wave function must be zero, or iA(Jc) 0, in both regions I and III. A particle cannot penetrate these infinite potential barriers, so the probability of finding the particle in regions I and III is zero.

The nme-independent Schrodingers wave equation in region П, where V - 0, becomes

dif(x) 2mE

(2.27)

A particular form of solution to this equation is given by

ф(х) - Ai cos Kx -h Л2sin Kx

(2.28)

where

2m E

(2.29)

One boundary condition is that the wave function \{/{х) must be continuous so

that

r{x = 0) = fixa)-=Q

(2.30)

Applying the boundary condition at .v := 0, we must have that Ai = 0. At л: = a, we have

(x =a) = 0 A2sinKa (2.31)

This equation is valid if Ka = птт, where the parameter n is a positive integer, or = 1, 2, 3.....The parameter n is referred to as a quantum number. We can write

К - (2.32)

Negative values of n simply introduce a negative sign in the wave funcdon and yield redundant solutions for the probability density function. Wc cannot physically distinguish any difference between -\-n and -n solutions. Because of this redundancy, negative values of n are not considered.

The coefficient A2 can be found from the normalization boundary condition that was given by Equation (2.18) as ф(х)ф*(х) dx = 1. If we assume that the wave function solution i/(x) is a real function, then -ix) - ф^{х). Substituting the wave function into Equation (2.18), we have

I Alm-Kxdx - 1 (2.33)

Evaluating this integral gives

(234)

Finally, the time-independent wave solution is given by

, 2 /nnx\

fix) = - sm f - 1 where /г = 1, 2, 3, ... (2.35)

This solution represents the electron in the inhnite potential well and is a standing wave solution. The free electron was represented by a traveling wave, and now the bound particle is represented by a standing wave.

The parameter in the wave solution was defined by Equations (2.29) and (2.32). Equating these two expressions for K, we obtain

2m E nn

= - (2.36)

nr a-

more thorough analysis shows that lA2l = 2/a, so solutions for the coefficient A2 include -\-y/2/a, -yfTfa, -j2/a, -jyjlja, or any complex number whose magnitude is Since the wave

function itself has no physical meaning, the choice of which coefficient to use is immaterial: They all produce the same probability density function.

The total energy can then be written as

E = E -

2ma

where = 1,2,3,... (2.37)

For the particle in the infinite potential well, the wave function is now given by

f{x) = s:mKx (2.38)

where the constant К must have discrete values, implying that the total energy of the particle can only have discrete values. This result means that the energy of the particle is quantized. That is, the energy of the particle can only have particular discrete values. The quantization of the particle energy is contrary to results from classical physics, which would allow the particle to have continuous energy values. The discrete energies lead to quantum states that will be considered in more detail in this and later chapters. The quantization of the energy of a bound particle is an extremely important result.

Objective example 2.3

To calculate the first three energy levels of an electron in an infinite potential well.

о

Consider an electron in an infinite potential well of width 5 A. Solution

From Equation (2,37) we have

- -2;;шГ- - 2(9.11 X 10---)(5 X IQ-i =

= ---- /Г(1.51) eV

Then,

£, 1.51 eV, £2 = 6.04 eV, £3 = 13.59 eV

Comment

This CELlcuiation shows the order of magnitiide of the energy levels of a bound electron.

Figure 2.6a shows the first four allowed energies for the particle in the infinite potential well, and Figures 2.6b and 2.6c show the corresponding wave functions and probabihty functions. We may note that as the energy increases, the probability of finding the particle at any given value of x becomes more uniform.