This book contains a comprehensive account of application of WKB method to pure Physics of nanostructures containing single or symmetric double barrier V(x) in their band model in presence of longitudinal magnetic field applied along x direction. It concentrates on effects on transmission coefficient of single and symmetric double barriers by three dimensional electron gas (3DEG). Analytical expressions for longitudinal magnetic field dependent transmission coefficient of single and symmetric double barrier of general shape are obtained first. These general expressions are then used to obtain analytical expressions of longitudinal magnetic field dependent transmission coefficient of single and symmetric double barriers of many different shapes we encounter in studying nanostructure Physics. This is followed by thorough numerical investigation to bring out effects of longitudinal magnetic field on transmission coefficient of all these barriers. Comparisons with standard results where available showed excellent agreements. Results of numerical investigation have been explained completely. The book makes well documented, with thorough calculation and discussion, pure Physics of semiconductor nanostructures.

**Obtaining analytical expressions**for tunneling regime of

**longitudinal magnetic field dependent transmission coefficient**

**Numerical investigation of**

**longitudinal magnetic field dependent transmission coefficient**

**1.1 WKB approximation**

**1.2 WKB solution of (1 dimensional) Schroedinger equation**

**1.3 Classical turning point**

**Figure 1.1**Diagram to help describe classical turning point if V is an increasing

**Figure 1.2**Diagram to help describe classical turning point if V is a decreasing

**Figure 1.3**Diagram showing classical turning points (x = a and b) for a particle of

**Figure 1.4**Diagram showing classical turning points (a and b) for a particle of total

**1.4 WKB connection formulae: case I**

**Figure 1.5**Diagram to help describe WKB connection formulae. We consider V(x)

**1.5 WKB connection formulae: case II**

**Figure 1.6**Diagram to help describe WKB connection formulae. We consider V(x)

**[3]**which contains an extensive account of WKB method and its

**2.1 Lagrange's equation, Lagrangian function and generalized potential**

**2.2 Construction of Lagrangian function for a charged particle in an electric**

**field and a magnetic field**

**2.3 Construction of Hamiltonian function for a charged particle in an electric**

**field and a magnetic field**

**3.1 Description of the problem**

**(a)**

**(b)**

**Figure 3.1**

**(a)**Single tunnel barrier of general shape V(x) in applied magnetic field

*longitudinal*magnetic field,

**(b)**symmetric double barrier of general shape

*longitudinal*magnetic field.

*longitudinal*magnetic field applied along x

**In Chapter IV, we repeat calculations of this chapter using another Landau**

*general*analytical

**3.2 The general eigenvalue equation of energy**

**3.3 Simplification of the general eigenvalue equation of energy**

**3.4 Adapting the general eigenvalue equation of energy to our problem**

*Landau gauges*: (

**3.5**

**commutes with Hamiltonian operator of our problem**

**3.6 Reducing 3D eigenvalue equation to 1D eigenvalue equation and obtaining**

**eigenvalue spectrum and identification of different parts of eigenvalue spectrum**

**does not**

*Landau levels*, and these are allowed values of energy of electron for motion in YZ

**3.7 Landau level index n is a constant of motion**