1.1 Doped semiconductors  (Page 3/3)

How can we get a sample of semiconductor with a lot of holes in it? What if, instead ofphosphorus, we dope our silicon sample with a group III element, say boron? This is shown in [link] . Now we have some missing orbitals, or places whereelectrons could go if they were around. This modifies our energy picture as follows in [link] . Now we see a set of new levels introduced by the boron atoms. They arelocated within the band gap, just a little way above the top of the almost full, or valence band. Electrons in the valence bandcan be thermally excited up into these new allowed levels, creating empty states, or holes, in the valence band. Theexcited electrons are stuck at the boron atom sites called acceptors , since they "accept" an electron from the valence band, and hence act as fixednegative charges, localized there. A semiconductor which is doped predominantly with acceptors is called p-type , and most of the electrical conduction takes place through the motion of holes. A semiconductor which isdoped with donors is called n-type , and conduction takes place mainly through the motion of electrons.

In n-type material, we can assume that all of the phosphorous atoms, or donors , are fully ionized when they are present in the silicon structure. Since the number of donors isusually much greater than the native, or intrinsic electron concentration, (≈ 10 ¹⁰ cm ^-3 ), if N _d is the density of donors in the material, then n , the electron concentration, ≈ N _d . If an electron deficient material such as boron is present, then the material is called p-type silicon, and the hole concentration is just ≈ N _a the concentration of acceptors , since these atoms "accept" electrons from the valence band.

If both donors and acceptors are in the material, then whichever one has the higher concentration wins out. This is called compensation. If there are more donors thanacceptors then the material is n-type and n ≈ N _d - N _a . If there are more acceptors than donors then the material is p-type and p ≈ N _a - N _d . It should be noted that in most compensated material, one type of impurity usually has a much greater(several order of magnitude) concentration than the other, andso the subtraction process described above usually does not change things very much, e.g., 10 ¹⁸ - 10 ¹⁶ ≈ 10 ¹⁸ .

One other fact which you might find useful is that, again,because of quantum mechanics, it turns out that the product of the electron and holeconcentration in a material must remain a constant. In silicon at room temperature:

Thus, if we have an n-type sample of silicon doped with 10 ¹⁷ donors per cubic centimeter, then n , the electron concentration is just p , the hole concentration, is10 ²⁰ /10 ¹⁷ = 10 ³ cm ^-3 . The carriers which dominate a material are called majority carriers , which would be the electrons in the above example. The other carriers are called minority carriers (the holes in the example) and while 10 ³ might not seem like much compared to 10 ¹⁷ the presence of minority carriers is still quite important and can not be ignored. Note that if the material isundoped, then it must be that n = p and n = p = 10 ¹⁰ .

The picture of "cups" of different allowed energy levels is useful for gaining a pictorial understanding of whatis going on in a semiconductor, but becomes somewhat awkward when you want to start looking at devices which are made up ofboth n and p type silicon. Thus, we will introduce one more way of describing what is going on in our material. The pictureshown in [link] is called a band diagram. A band diagram is just a representation of the energy as a function of position with a semiconductor device. In aband diagram, positive energy for electrons is upward, while for holes, positive energy is downwards. That is, if an electron moves upward, its potential energy increases just as a with a normal mass in a gravitational field. Also, just as a mass will "fall down" if given a chance, an electron will move down a slope shown in a band diagram. On the other hand, holes gain energy by moving downward and so they have a tendancy to "float" upward if given the chance - much like a bubble in a liquid. The line labeled E _v in [link] shows the edge of the conduction band, or the bottom of the lowestunoccupied allowed band, while E _v is the top edge of the valence, or highest occupied band. The band gap, E _g for the material is obviously E _c - E _v . The dotted line labeled E _f is called the Fermi level and it tells us something about the chemical equilibrium energy of the material, and also something about thetype and number of carriers in the material. More on this later. Note that there is no zero energy level on a diagram such as this. We often useeither the Fermi level or one or other of the band edges as a reference level on lieu of knowing exactly where "zero energy" is located.

The distance (in energy) between the Fermi level and either E _c and E _v gives us information concerning the density of electrons and holes in that region of thesemiconductor material. The details, once again, will have to be begged off on grounds of mathematical complexity. It turns out that you can say:

Both N _c and N _v are constants that depend on the material you are talking about, but are typically on the order of 10 ¹⁹ cm ^-3 . The expression in the denominator of the exponential is just Boltzman's constant (8.63 x 10 ^-5 eV/K), k , times the temperature T of the material (in absolute temperature or Kelvin). At room temperature kT = ¹ / ₄₀ of an electron volt. Look carefully at the numerators in the exponential. Note first that there is a minus sign infront, which means the bigger the number in the exponent, the fewer carriers we have. Thus, the top expression says that ifwe have n-type material, then E _f must not be too far away from the conduction band, while if we have p-type material, then the Fermi level, E _f must be down close to the valence band. The closer E _f gets to E _c the more electrons we have. The closer E _f gets to E _v , the more holes we have. [link] therefore must be for a sample of n-type material. Note also that if we know how heavily a sample is doped (i.e., we knowwhat N _d is) and from the fact that n ≈ N _d we can use [link] to find out how far away the Fermi level is from the conduction band, [link] .

To help further in our ability to picture what isgoing on, we will often add to this band diagram, some small signed circles to indicate the presence of mobile electrons andholes in the material. Note that the electrons are spread out in energy. From our "cups" picture we know they like to stay inthe lower energy states if possible, but some will be distributed into the higher levels as well. What is distortedhere is the scale. The band-gap for silicon is 1.1 eV, while the actual spread of the electrons wouldprobably only be a few tenths of an eV, not nearly as much as is shown in [link] . Lets look at a sample of p-type material, just for comparison. Note that for holes, increasingenergy goes down not up, so their distribution is inverted from that of the electrons. You cankind of think of holes as bubbles in a glass of soda or beer, they want to float to the top if they can. Note also for both nand p-type material there are also a few "minority" carriers, or carriers of the opposite type, which arise from thermalgeneration across the band-gap.