Nanoelectronics Chapter 5 Electrons Subjected to a Periodic Potential – Band Theory of Solids

2 STM image of a nickel surface

5.1 Crystalline Materials Solid materials may be classified as crystalline, polycrystalline or amorphous. Crystalline solid consists of a periodic array of atoms called the lattice. Polycrystalline solid has a well-defined structure in each of many small regions, each region differs from its neighboring regions. Amorphous solid does not exhibit any sort of regularity, such as glass and plastic.

5.1 Crystalline Materials Most interested electronic materials are crystalline solids, e.g., semiconductors, such as Si, GaAs, and conductors, such as Al, Cu and Au Here, we consider the effect of periodic lattice on electronic properties.

Crystal Types The fundamental property of a crystal is regularity in its atomic structure; the atoms in a crystal arranged in a regular (periodic / repeated) array. A lattice: is a set of points that form a periodic structure. we need to understand its basis. Here, we consider: simple cubic (sc), body- centered cubic (bcc) and face-centered cubic (fcc).

Crystal Types The 7 lattice systems (From least to most symmetric) 14 Bravais lattices

Simple cubic (sc) is the simplest lattice A sc lattice consists of points equally spaced at the corners of a 3D cube. Fundamental translation vectors identical Atomic arrangement at r ’ and r u 1, u 2 and u 3 are integers.

Simple cubic (sc) is the simplest lattice The set of three vectors: form a parallelepiped. The parallelepiped with the smallest volume is called the primitive cell. The primitive cell has only one lattice point, and a crystal can be constructed from repetitions of the primitive cell. However, usually other unit cells is more convenient to work with.

Simple cubic (sc) is the simplest lattice Not necessary that a single atom is located at each lattice point. a group of atoms, called a basis, is placed at each lattice point.

Body-Centered Cubic (bcc) Primitive translation vectors Examples: Na and W lattices

Face-Centered Cubic (fcc) Primitive translation vectors

Face-Centered Cubic (fcc) How many atoms in a conventional cell? Examples: metals such as Cu, Au, Ag, Ni and semiconductors such as Si, GaAs and Ge The volume of conventional cell is a 3. How about the primitive cell? The basis consists of two atoms The two atoms are identical  diamond structure: Si, Ge and C The two atoms are not identical  zinc blende structure GaAs and AlAs

Diamond crystal structure The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated 2-atom pattern. 2 fcc merge along ¼ diagonal direction.

5.2 Electrons in a Periodic Potential Consider a one-dimensional example Ionized atom / ion Coulomb potential of an electron:

5.2 Electrons in a Periodic Potential Obviously, the potential is periodic In three-dimensional system: Potential energy is periodic r is the position vector of the electron, T is crystal translation vector

5.2 Electrons in a Periodic Potential Bloch’s Theorem: If the potential is periodic, The solution of Schrodinger’s eq.    Electrons can propagate through a perfect periodic medium without scattering

5.3 Kronig-Penney Model of Band Structure If we set V 0 as infinitely large, it will become delta function.

5.3 Kronig-Penney Model of Band Structure where

5.3 Kronig-Penney Model of Band Structure where

In one dimension: The Bloch wavevector k to be determined: a is the period of the lattice: 5.3 Kronig-Penney Model of Band Structure

5.3 Kronig-Penney Model of Band Structure

If 0 < E < V 0 If E > V Kronig-Penney Model of Band Structure where

5.3 Kronig-Penney Model of Band Structure γ(E)

5.3 Kronig-Penney Model of Band Structure

Dispersion diagram Because:  5.3 Kronig-Penney Model of Band Structure Note: first Brillouin Zone second Brillouin Zone

Free electron continuous band  Electron energy band with gap 5.3 Kronig-Penney Model of Band Structure

5.3.1 Effective Mass We view an electron quantum mechanically as a wavepacket, with electron’s velocity being its group velocity for free-electron model consider the importance of electrons near bandedges:

5.3.1 Effective Mass Effective mass can be defined as or agree with classical mechanics

Band Structures of Ge, Si and GaAs

Silicon Band Structure

Germanium Band Structure

First Brillouin Zone

5.3.1 Effective Mass To simplify the Schrodinger’s equation, we can use m * to replace m Effective mass is different in different direction

5.3.1 Effective Mass For SiO 2 Effective mass is modeled as Depends on interface

5.4 Band Theory of Solid One electrons in partially filled energy band can contribute to conduction

5.4 Band Theory of Solid

5.4.1 Doping in Semiconductors donor

5.4.1 Doping in Semiconductors

5.4.1 Doping in Semiconductors Acceptor, B, typically E a – E v is about 45 meV

5.4.2 Interacting systems Model

5.4.2 Interacting systems Model

5.4.2 Interacting systems Model

Direct Bandgap Semiconductors

Indirect Bandgap Semiconductors