EE 562 Intro to Solid State Physics Notes Introduction to Solid State Physics Solid State Physics Basics

Fall 00

Band Theory and Band Diagrams Recall from physics that electron energy is quantitized; electrons of a given element can only occupy specific discrete energies, and those closest to the nucleus have the least energy and are most tightly bound to the atom. In some elements (like diamond, carbon, silicon, etc.), when many atoms are brought together to form a solid, a crystal lattice structure is created which has many unique properties (note: this is the origin of the term solid-state). In a crystal lattice, the available electron energy levels get blurred a bit and we find the discrete energy levels becomes thick bands of energy which electrons can occupy. This phenomena is shown in Figure 1.1. Band diagrams are a tool used in solid-state physics to illustrate what is going on in the outer electron shells (energy levels) of a group of atoms held together in a tightly-packed crystal structure (like a silicon wafer). Band diagram help illustrate where the outer electrons are in a given situation. Look at Figure 1.2 which shows typical band diagrams for an insulator, a conductor, and a semiconductor. All have a band at the top which is the conduction band and a band at the bottom which is the valence band . These bands represent the outermost energy levels that electrons can occupy. Generally speaking (i.e. without getting into too many details) electrons the valence band are still tightly bound to the atom and cannot move around to other atoms in the crystal. However, electrons in the conduction band have enough energy that they can freely move throughout the crystal material thus acting as mobile charge carriers which can conduct current. (Recall that electrons try to occupy the lowest energy levels available, and an electron at a higher energy level will always move to a lower level if it is available. Just remember that electrons are happiest when they are as close as possible to the nucleus.

Another important factor in the band diagram is the energy gap, Eg, which is the energy difference between the valance and conduction bands. Because electrons can only occupy energies within the two bands, the gap between the two bands is a forbidden region called the band gap . The band gap is significant because it tells us how much energy electrons need before they will jump up to the conduction band and be available to conduct current. The band gap is a constant for each material (although it varies slightly with temperature, a fact that is exploited in a class of circuits appropriately called band gap circuits which we will discuss near the end of the semester). For Si at room temperature, Eg = 1.1eV. Prof. Andrew Mason Handout 1 1

EE 562 Intro to Solid State Physics Notes

Fall 00

What is a semiconductor? In an insulating material, all of the electrons are held within the valence band, and no electrons are in the conduction band and free to move around to create currents thus an insulator. In a conducting material, the outer electrons fill the valence band but they also occupy the conduction band, so some electrons are available to move around and create currents thus a conductor. Also, in an insulator, the band gap is large meaning that an electron must gain a lot of energy to move up to the conduction band. In a conductor, these bands are much closer (i.e. the band gap is smaller) and electrons are more likely to have enough energy to change bands. A semiconductor is much like an insulator in that its outer electrons fill the valence band but do not occupy the conduction band; however, the band gap is much smaller than in an insulator. As a result, electrons can obtain enough energy to jump to the conduction band through a variety of mechanisms including ambient thermal energy (e.g. heat at room temperature). When this happens, the electrons in the conduction band can move around and the semiconductor can conduct an electrical current. Thus, a semiconductor is a material that can act as both an insulator and a conductor depending on the conditions around the material. What is a hole? In an intrinsic (pure) semiconductor material at 0 K, all of the electrons are in the valence band. When an electron gets enough energy to move up to the conduction band, it leaves behind a space that an electron can occupy. This empty space is called a hole , and the process of an electron moving from the valence band into the conduction band is called electron-hole pair generation as illustrated in Figure 1.3. This is important in electronics because a hole acts as a positive charge carrier. For example, when an electron jumps out of the valence band leaving behind a hole, another electron can move into that hole (which commonly occurs in the presence of an electric field). Now there is a hole left where the second electron came from and a third electron will jump into its place leaving behind another hole. And on and on, until we see this one hole move across the material. Thus, it acts just like an electron, only it has a positive charge. It might help to imagine that electrons must move to fill this hole, and electrons will move in the opposite direction from the hole, so the net current is negative of the electron current. Its a strange concept, I know, but it is very important in how circuits are modeled.

the white spaces in the valence band are holes left by the electrons that have moved up to the conduction band

Prof. Andrew Mason

Handout 1

EE 562 Intro to Solid State Physics Notes

Fall 00

Fermi Level The information above gives a very simplified explanation of what is really going on in an solid material, and to understand it better we must introduce something called the Fermi level. The Fermi level is part of a statistical, quantum-mechanical method of describing the probability of an electron occupying a specific energy level in a solid-state material. This is based on something called Fermi-Dirac statistics, which are too complex to cover in this course. Basically, this statistical function gives the probability that an available energy state will be occupied by an electron at a given temperature (note that no energy states are available within the band gap). In this model, the Fermi level is the energy at which the probability of an energy state being occupied is . For our simplified use in this class, the Fermi level will simply act as an indicator as to whether or not electrons are in the conduction band or holes are in the valence band. In an insulator, the Fermi energy is in the valance band and in a conductor the Fermi energy is in the conduction band. In a semiconductor, the Fermi energy is between the valance and conduction bands and it can act as either an insulator or a conductor. As a result, a pure semiconductor (called intrinsic semiconductor) at has no free carriers (thus acting as an insulator), but if given just a little energy, either through thermal excitation or through an applied electric field, the outer electrons can be freed (thus acting as a conductor). Figure 1.4 shows Fermi-Dirac distribution and the Fermi level as a function of temperature. Notice that as T (temperature) increases, the distribution spreads and it becomes possible to get electrons at energies well above the Fermi level. Figure 1.5 shows three examples of the effects of having the Fermi level at different points within the band gap. Notice that as the Fermi level gets closer to the conduction band, we find a higher density of electrons in the conduction band, and when the Fermi level gets closer to the valence band, we find more holes in the valence band.

Prof. Andrew Mason

Handout 1

EE 562 Intro to Solid State Physics Notes Determining Number of Free Carriers

Fall 00

Intrinsic Carrier Concentration Silicon (Si), the primary element in sand, is the mo st common semiconductor. When the silicon atoms come together they form a crystal lattice structure, which is a very regular cube-like arrangement of atoms that gives Si some of the useful electrical properties that is has. Si is a group IV element and has 4 electrons in its outer shell that can be freed and shared with other atoms in the crystal lattice. When an electron (which is a negative charge carrier) is freed from the atom, it leaves behind a hole , or the absence of an electron (which acts as a positive charge carrier). Free carriers are generated when electrons have gained enough energy to escape their bonds to the atom and move from the valence band to the conduction band . This process is called electron-hole pair generation. Electron-hole pairs can be created by any mechanism which delivers sufficient energy to an electron, including absorbing energy from light (as in a photo diode) and thermal excitation (absorbing heat energy). Under normal circumstances, heat energy is the largest contributor to the creation of free carriers. Thus, the number of available free carriers is a strong function of temperature. A material is defined as intrinsic when it consists purely of one element and no outside force (like light energy) affects the number of free carrier other than heat energy. In intrinsic Si, the heat energy available at room temperature generates approximately 1.5x1010 carriers per cm3 of each type (holes and electrons) . The number of free carriers doubles for approximately every 11C increase in temperature. This number represents a very important constant (at room temperature), and we define ni = 1.5x10 cm where ni denotes the carrier concentration in intrinsic silicon at room temperature (constant for a given temperature). Dopants in Semiconductors To create active circuit devices like diodes and transistors, other elements are typically added to the semiconductor material to alter its electrical characteristics. This process is called doping , and a material that has been doped is called extrinsic. If elements with a different number of valence electrons are added to the silicon lattice, these elements are called dopants or impurities and they increase the number of free charge carriers available to conduct current. Typically, silicon is doped with elements having either 5 or 3 valence electrons (group 5 of group 3 atoms). At room temperature, there is sufficient thermal energy to ionize the impurity atoms and introduce additional free electrons or free holes depending on the type of impurity1 . In this manner, doping has a strong effect on the electrical properties of silicon. Atoms with 5 valence electrons are said to donate their extra electrons to the silicon crystal, thus the impurity is called a donor. Commonly used donor elements are phosphorus, P, and arsenic, As. These impurities are called ntype since they introduce negatively charged carriers. When n-type dopants give up there extra electrons they become ionized, and charge neutrality is maintained since the number of extra electrons is equal to the number of ionized atoms. In a similar manner, impurities with 3 valence electrons introduce a positive charge carrier, or hole, and are called acceptors. A common acceptor in silicon device fabrication is boron, B. When silicon is doped with acceptors, the material is said to be p-type because positively charged carriers (holes) have been introduced. When p-type dopants accept free electrons they become ionized, and charge neutrality is maintained since the number of holes generated is equal to the number of ionized atoms.
10 -3

n-type Donor +

p-type Acceptor

P
group V element

P+
ion

electron

B
group III element

B+
ion

+
hole

free carrier

free carrier

Recall that atoms with four valence electrons have very tightly bound outer shells. In atoms with five electrons, the fifth electron is not as tightly bound and will easily escape the bond at room temperature to become a free carrier (leaving behind a positively charged ion). Likewise, atoms with only three valence electrons will be more stable if they ionize and take on an extra electron. Thus a group five atom provides one free carrier and a group three atom provides one hole.

Prof. Andrew Mason

Handout 1

EE 562 Intro to Solid State Physics Notes

Fall 00

At equilibrium, with no external influences such as light sources or applied voltages, the concentration of electrons, n0, and the concentration of holes, p0, are related by no po = ni where ni is the intrinsic carrier concentration. In an intrinsic material, no = po = ni since every free electron leave 2 behind a hole. In an extrinsic (doped) material, no po but no po = ni is still true. Based on charge neutrality, for a sample doped with ND donor atoms per cm-3 and NA acceptor atoms per cm-3 we can write no + NA = po + ND which shows that the sum of the electron concentration plus the ionized acceptor atoms is equal to the sum of the hole concentration plus the ionized donor atoms. The equation assumes that all donors and acceptors are fully ionized, which is generally true at or above room temperature. Given the impurity concentration, the above equations can be solved simultaneously to determine electron and hole concentrations. In electronic devices, we typically add only one type of impurity within a given area to form either n-type or p-type regions. In n-type regions there are typically only donor impurities and the donor concentration is mu ch greater than the intrinsic carrier concentration, NA=0 and ND >>ni. Under these conditions we can write nn = ND where nn is the free electron concentration in the n-type material and ND is the donor concentration (number of added impurity atoms/cm3 ). Since there are many extra electrons in n-type material due to donor impurities, the number of holes will be much less than in intrinsic silicon and is given by, 2 pn = ni / ND where pn is the hole concentration in an n-type material and ni is the intrinsic carrier concentration in silicon.
2

Similarly, in p-type regions we can generally assume that ND =0 and NA>>ni. In p-type regions, the concentration of positive carriers (holes), pp, will be approximately equal to the acceptor concentration, NA. pp = NA and the number of negative carriers in the p-type material, np, is given by 2 np = ni / NA Notice the use of notation, where negative charged carriers are n, positive charged carriers are p, and the subscripts denote the material, either n-type or p-type. This notation will be used throughout our discussion of pn junctions and bipolar transistors. The above relationships are only valid when ND or NA is >> ni , which will always be the case in our problems related to integrated circuit design. EXAMPLE A Si sample at room temperature is doped with 1011 As atoms/cm3 . What are the equilibrium electron and hole concentrations at 300 K? SOLUTION Since the NA is zero we can write, 2 no po = ni no + NA = po + ND no ND no ni = 0
2 2

Solving this quadratic equations results in n0 = 1.02x10 [cm ] and thus, 2 20 11 p0 = ni / n0 = 2.25x10 / 1.02x10 p0 = 2.2x10 [cm ] Notice that, since ND >n i , the results would be very similar if we assumed n n =ND =1011 cm-3 , although there would be a slight error since ND >is not much greater than n i . Prof. Andrew Mason Handout 1 5
9 -3 11 -3

EE 562 Intro to Solid State Physics Notes

Fall 00

HERE Impurities and Fermi Level Adding impurities to pure silicon will affect number of free carriers in the material, thus it will affect the statistics that determine the Fermi energy level. The semiconductor shown in Figure 1.5(a) represents a pure intrinsic material with a Fermi level in the middle of the band gap. In an n-type material, the presence of donor elements will cause a change in the distribution statistics and the Fermi level to rise closer to the conduction band as a result. Similarly, in p-type silicon the acceptors will cause the Fermi level to lower closer to the valance band. Figure 1.5 (b) and (c) illustrate this effect. The following equations demonstrate the relationship between the dopant concentration and the Fermi level. Although we will not cover the derivation of these equations, they are very important relationships. The subscript 0 means the material is at equilibrium, and it could be either an n or a p depending on the type of material being studied. n0 = Nc e
-(Ec-Ef)/kT -(Ef -Ev)/kT

p0 = Nv e where Ef is the Fermi level, Ec is the energy at the conduction band, and Ev is the energy at the valence band. k is Boltzmans constant (k = 8.62x10-5 eV/K = 1.38x10-23 J/K), and kT at room temperature is 0.0259 eV. Nc is the effective density of states in the conduction band, and Nv is the effective density of states in the valence band. We will consider both of these values to be constant at room temperature. 19 -3 Nc = 2.8x10 cm Nv = 1.04x10 These equations can also be written as n0 = ni e
(Ef -Ei)/kT (Ei-Ef)/kT 19

cm

-3

p0 = ni e where Ei is the intrinsic Fermi level (before doping) which can be assumed to be exactly in the middle of the band gap, thus Ec-Ei = Eg/2. This relationship is very useful because it directly shows that as Ef increases from Ei there are more electrons in the conduction band, and as Ef decreases from Ei there are more holes in the valence band. Although these relationship will be adequate for the problems we will cover in this class, they are actually only valid when Ec-Ef (or Ef-Ev) is >> than kT. Because of the direct relationship between dopant concentration and Ef, this restriction amounts to saying that the equations are valid as long as the dopant concentration is not very high. Generally speaking, these equations are valid for an impurity concentration below 1018 , which will almost always be the case in integrated circuits. EXAMPLE A Si sample is doped with 1017 As atoms/cm3 . What is the equilibrium hole concentration p 0 at 300 K? Where is Ef relative to Ei? SOLUTION Since ND >> ni, we can approximate n 0 = ND and 2 20 17 p0 = ni / n0 = 2.25x10 / 10 p0 = 2.25x10 [cm ] Since n 0 = n i e (Ef-Ei)/kT we can write Ef Ei = kT ln(n0/ni) Ef Ei = 0.0259 ln(10
17 3 -3

/ 1.5x10 )

10

Ef Ei = 0.407 [eV] Thus we see the Fermi level is above the intrinsic level suggesting the material is n-type. This is further supported by the fact that the electron concentration is much higher than the hole concentration. Finally, the dopant was As, which is an n-type dopant, so all the answers make sense intuitively.

Prof. Andrew Mason

Handout 1

EE 562 Intro to Solid State Physics Notes

Fall 00

Conductivity, Drift, and Mobility Charge carriers in a solid are in constant motion. However, this random scattering produces no net motion of electrons within the material, and therefore no net current flow. In the presence of an electric filed E, electrons experience a force -q E from the field. This force will case a net drift that can generate a current density J which is described by J x = qnnEx (n-type) J x = qppEx (p-type) where q is the electron charge (q=1.6x10 C), n and p are the carrier concentrations in n- and p-type material respectively, is the carrier mobility, and the x subscript denotes the direction vector for the electric field. Notice that the current density and the electric field are in the same direction. The mobility, , is different for n- and p-type carriers, and it represents the average velocity of the carrier per unit of electric field = <vx > / Ex Thus the units of mobility are (cm/s)/(V/cm) = cm /V-s. This unit can be related by conductivity, ( -cm)-1 , by
2 -19

= qnn and similarly = qpp for p-type material. Generally, one type of carrier will dominate, but the general expression for the current density is J x = q (nn + pp) Ex = Ex Mobility is a parameter determined by a number of material properties related to band structure, and is strongly affected by impurity concentration, and temperature. Because of difference in the conduction and valence band properties, there is a significant difference in the mobility for n- and p-type materials. Although mobility is not truly a constant, it is nearly constant for the types of circuits we will cover in this class. Unless otherwise stated in a homework problem or on a test, for room temperature mobility we will use n = 1100 cm /V-s p = 500 cm /V-s Mobility is strongly affected by temperature, and this will be covered in more detail near the end of the semester when we discuss temperature stability in circuits and temperature insensitive voltage references. For now it is worth noting that mobility is lower at low temperatures as well as at high temperatures and has its maximum value in the middle of the range. Mobility is also affected by impurity concentration, and it will decrease as the number of dopants increases. The resistance of a material to flow of current, R, is given by R = L / wt where L is the length of the material in the direction of current flow (i.e. the direction of the electric field), w and t are the cross-sectional dimensions (wt = A, where A is the area of the cross section), and is the resistivity ( -cm). Resistivity is indirectly proportional to the conductance =1/ EXAMPLE If the carrier mobilities for Germanium, Ge, are n = 3900 cm2 /V-s, and p = 1900 cm2 /V-s, what is the resistivity of intrinsic Ge at 300 K. Assume the intrinsic carrier concentration for Ge is 2.5x1013 . SOLUTION Since the material is intrinsic, n = p = n i = 2.5x1013 i = q (n + p) ni = 1.6x10
-2 -1 -19 2 2

(5800)(2.5x10 )
-1

13

i = 2.32x10 [ -cm)

i = i = 43 [ -cm] Summary of Constants

Prof. Andrew Mason

Handout 1