Introduction | Mathematical Analysis | Applet Tutorial | Applet Worksheet | Quiz | SPICE/CAD | References | Feedback | How To Print

(Energy Band, Fermi Level, and Doping Concentration Virtual Lab)

Who should study this applet and this Introduction – Students who have been introduced to the Energy Band concepts may skip this Introduction page and proceed directly to the Worksheet page (an example textbook that deals with Energy Band in detail: Streetman/Banerjee “Solid State Electronic Devices”).  However, those students who are learning from a textbook with emphasis of circuit models of electronic devices (an example textbook that does  not deal with Energy Band at all: Sedra/Smith “Microelectronic Circuits”) will need to read this Introduction page of this applet before proceeding to the Worksheet page.

Definition of Energy Band – Energy Bands of a solid state semiconductor crystal consists of Conduction Band, Valence Band, and Energy Gap.  The charge carrier electrons exist in the conduction band (CB); the charge carrier holes exist in the valence band (VB); and the two bands (CB & VB) are separated in energy by the energy gap.  The lowest energy of CB is called the conduction band edge, denoted as Ec, and is equal to the top of energy gap; and the highest energy of VB is called the valence band egde, denoted as Ev, and is equal to the bottom of energy gap.

In the energy bands (CB and VB), the quantum mechanical energy states exist in almost continuum, both in the ‘vertical’ energy scale and in the ‘horizontal’ spatial dimension.  Upon application of electric field or for thermal motion, carriers move to the nearby (both in energy and in spatial distance) available state.  Within the energy gap there are no states for the mobile carriers.   However, energy states that are associated with the donor and acceptor impurities do exist within the gap, as well as the energy states that act as carrier traps or as recombination centers.

Charge Carriers in Energy Bands of Intrinsic Semiconductor – An intrinsic semiconductor is a pure material without any chemical impurities.  At absolute zero, there are no charge carriers.  This situation, in the energy band terms, is that the CB is completely empty of electrons, and the VB is fully occupied by the valence electrons (and therefore no empty states or holes).  Note that the valence electrons are the same as the covalent bonds and are not mobile (they are localized).
At finite temperature, the thermal energy can break some of  the covalent bonds, which sets free the electrons that were originally tied to, or localised in, the covalent bonds.  This, in the energy band terms, means that a valence electron (in VB, localized and not mobile) is thermally excited into the CB (where the electron is mobile), leaving behind an unoccupied state in the valence band.  The unoccupied state in the valence band is called hole.  A hole carries an effective charge of +q.  A covalent bond , once thermally broken, is equivalent to one electron in the conduction band plus one unoccupied state (or hole) in the valence band. In intrinsic semiconductor, for every electron there is a corresponding hole (the empty state in VB).  Therefore, the electron concentration is equal to the hole concentration, and this concentration is called the intrinsic carrier concentration, denoted as ni.

Charge Carriers in Energy Bands of Extrinsic Semiconductor – The relative concentration of electrons and holes in a semiconductor is controlled by the kind and concentration of chemical impurity atoms that are intentionally introduced (or doped) into the material during or after the crystal growth. Chemical impurities that contribute conduction electrons are called donor impurities or donors; and the impurity atoms that take away an electron from the semiconductor is called acceptor impurities or acceptors.  Donors in Silicon are As, P, or SB.  Acceptor example in Silicon is B.  Semiconductor doped with donor is n-type.  In an n-type semiconductor, electrons are the majority carriers and the electron concentration n is equal to the donor impurity concentration less any acceptor impurity concentration.  In a p-type semiconductor, holes are the majority carriers and the hole concentration p is equal to the acceptor impurity concentration less any donor impurity concentration.  The minority carrier concentration is then determined by the so-called the Mass-Action Law: n * p = ni2.

Fermi Energy Level and Carrier Concentration – Under thermal equlibrium the electron and hole concentrations in a Semiconductor can be obtained from a single physical parameter called Fermi energy level, denoted as Ef.    For most semiconductors, Ef is in the band gap, that is, Ef is below Ec and above Ev: Ec > Ef > Ev.  The electron and hole concentrations are determined from Ef by

n = ni exp[(Ef-Ei)/kT], and
p = ni exp[-(Ef-Ei)/kT] where
ni is the intrinsic concentration, k = 8.62x10-5 [eV/K] is the Boltzman constant, T is the absolute temperature, and Ei is the Fermi level position in the band gap of an  intrinsic material.  In an intrinsic material the electron and hole concentrations are equal: n = p.  It can be shown that Ei is approximately mid way between Ec and Ev:  Ei = (Ec+Ev)/2.