# KATIHAL FİZİĞİ MNT-510

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1 KATIHAL FİZİĞİ MNT-510 YARIİLETKENLER Kaynaklar: Katıhal Fiziği, Prof. Dr. Mustafa Dikici, Seçkin Yayıncılık Katıhal Fiziği, Şakir Aydoğan, Nobel Yayıncılık, Physics for Computer Science Students: With Emphasis on Atomic and Semiconductor Physics, Narciso Garcia, Arthur Damask, Steven Schwarz, Springer Lecture Notes, Prof. A. Niknejad, University of California, Berkley

2 The highest energy band completely filled with electrons (at T = 0 K) is called the Valence Band The next band is called the Conduction Band The energy difference between the bottom of the Conduction and the top of the Valence bands is called the Band Gap

3 Conduction Electron Conduction is easy to imagine: electrons (in the conduction band) move almost like free particles Hole Conduction is due to positively charged particles in the valence band

4 Intrinsic Semiconductors Consider nominally pure semiconductor at T = 0 K There is no electrons in the conduction band At T > 0 K a small fraction of electrons is thermally excited into the conduction band, leaving the same number of holes in the valence band

5 Intrinsic Semiconductors at T >0 K Electrons and holes contribute to the current when a voltage is applied

6 Carrier Concentrations at T >0 K Let s take E V = 0, then E C = E G The number of electrons equals the number of holes, n e = n h The Fermi level lies in the middle of the band gap n e = n h increase rapidly with temperature

7 Carrier Concentrations E G of selected semiconductors Si: 1.1eV Ge: 0.7eV GaAs: 1.4eV ZnSe: 2.7eV Carrier effective masses for selected semiconductors GaAs: m e* = 0.067m 0 ; m h* = 0.45m 0 Si: m e * = 0.26m 0 ; m h * = 0.49m 0 Ge: m e * = 0.04m 0 ; m h * = 0.28m 0 ZnSe: m e * = 0.21m 0 ; m h * = 0.74m 0 Carrier concentration falls with 1/T, i.e. increase with T

8 Doping Semiconductors can be easily doped Doping is the incorporation of [substitutional] impurities into a semiconductor according to our requirements In other words, impurities are introduced in a controlled manner Impurities change the conductivity of the material so that it can be fabricated into a device

9 Extrinsic Semiconductors Electrical Properties of Semiconductors can be altered drastically by adding minute amounts of suitable impurities to the pure crystals Impurities: Atoms of the elements different from those forming solid Interstitial: foreign atoms squeezed between regular sites crystal sites Substitutional: foreign atoms occupying the sites of host atoms

10 Donors We use Silicon (Si) as an example Substitute one Si (Group IV) atom with a Group V atom (e.g. As or P) Si atoms have four valence electrons that participate in covalent bonding When a Group V atom replaces a Si atom, it will use four of its electrons to form the covalent bonding What happens with the remaining electron?

11 Donors The remaining electron will not be very tightly bound, and can be easily ionized at T > 0K Ionized electron is free to conduct In term of the band structure, this electron is now in the conduction band Such Group V impurities are called Donors, since they donate electrons into the Conduction Band Semiconductors doped by donors are called n-type semiconductors

12 Donors: Energy Levels The Band Structure View Such impurities create an energy level within the band gap, close to the conduction band A donor is similar to a hydrogen atom A positive charge with a single electron within its potential Such impurities are called hydrogenic donors They create so-called shallow levels - the levels that are very close to the conduction band, so the energy required to ionize the atom is small and a sizable fraction of donor atoms will be ionized at room temperature

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15 This crystal has been doped with a pentavalent impurity. The free electrons in n type silicon support the flow of current.

16 Acceptors Use Silicon (Si) as an example Substitute one Group III atom (e.g. Al or In) with a Si (Group IV) atom Si atoms have four valence electrons that participate in the covalent bonding When a Group III atom replaces a Si atom, it cannot complete a tetravalent bond scheme An electronic vacancy hole is formed when an electron from the valence band is grabbed by the atom so that the core is negatively charged, the hole created is then attracted the negative core At T = 0 K this hole stays with atom localized hole At T > 0 K, electron from the neighboring Si atom can jump into this hole the hole can then migrate and contribute to the current

17 Acceptors At T > 0 K, electron from the neighboring Si atom can jump into this hole the hole starts to migrate, contributing to the current We can say that this impurity atom accepted an electron, so we call them Acceptors Acceptors accept electrons, but donate free holes

18 Acceptors By incorporating the electron into the impurity atom we can represent this (T = 0 K) as a negative charge in the core with a positive charge (hole) outside the core attracted by its [Coulomb] potential At T > 0 K this hole can be ionized Such semiconductors are called p-type semiconductors since they contribute positive charge carriers

19 Acceptor: Energy Levels From the Band Structure View Such impurities create energy levels within the band gap, close to the valence band They are similar to negative hydrogen atoms Such impurities are called hydrogenic acceptors They create shallow levels - levels that are very close to the valence band, so the energy required to ionize the atom (accept the electron that fills the hole and creates another hole further from the substituted atom) is small

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21 This crystal has been doped with a trivalent impurity. The holes in p type silicon contribute to the current. Note that the hole current direction is opposite to electron current so the electrical current is in the same direction

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23 Carrier Concentrations in Extrinsic Semiconductors The carrier densities in extrinsic semiconductors can be very high It depends on doping levels ([net] dopant concentration) and ionization energy of the dopants Often both types of impurities are present If the total concentration of donors (N D ) is larger than the total concentration of acceptors (N A ) have an n-type semiconductor In the opposite case have a p-type semiconductor

24 Charge Neutrality Equation To calculate the charge concentration, the charge neutrality condition is used, since the net charge in a uniformly doped semiconductor is zero Otherwise, there will be a net flow of charge from one point to another resulting in current flow p is the concentration of holes in the valence band n is the electron concentration N D is the ionized donor concentration N A - is the ionized acceptor concentration

25 Resistivity of Semiconductors The carrier concentration and thus the conductivity is dominated by its essentially exponential dependence on temperature For intrinsic semiconductors For impurity semiconductors E F is first between the impurity level and the band edge and then approaches E g /2 after most of the impurities are ionized

26 Optical Properties If semiconductor or insulator does not have many impurity levels in the band gap, photons with energies smaller than the band gap energy can t be absorbed There are no quantum states with energies in the band gap This explains why many insulators or wide band gap semiconductors are transparent to visible light, whereas narrow band semiconductors (Si, GaAs) are not

27 Optical Properties Some applications Emission: light emitting diode (LED) and Laser Diode (LD) Absorption: Filtering Sunglasses Si filters: transmission of infra red light with simultaneous blocking of visible light

28 Optical Properties If there are many impurity levels the photons with energies smaller than the band gap energy can be absorbed, by exciting electrons or holes from these energy levels into the conduction or valence band, respectively Example: Colored Diamonds

29 Photoconductivity Charge carriers (electrons or holes or both) created in the corresponding bands by absorbed light can also participate in current flow, and thus should increase the current for a given applied voltage, i.e., the conductivity increases This effect is called Photoconductivity

30 Photoconductivity Important Applications (Garcia 26.6) Night vision systems imaging IR radiation Solar cells Radiation detectors Photoelectric cells (e.g., used for automatic doors) Xerography CCD ( Digital Cameras )

31 PN Junctions: Overview The most important device is a junction between a p-type region and an n-type region When the junction is first formed, due to the concentration gradient, mobile charges transfer near junction Electrons leave n-type region and holes leave p-type region These mobile carriers become minority carriers in new region (can t penetrate far due to recombination) Due to charge transfer, a voltage difference occurs between regions This creates a field at the junction that causes drift currents to oppose the diffusion current In thermal equilibrium, drift current and diffusion must balance p-type N A N D n-type V

32 Diffusion Diffusion occurs when there exists a concentration gradient In the figure below, imagine that we fill the left chamber with a gas at temperate T If we suddenly remove the divider, what happens? The gas will fill the entire volume of the new chamber. How does this occur?

33 Diffusion (cont) The net motion of gas molecules to the right chamber was due to the concentration gradient If each particle moves on average left or right then eventually half will be in the right chamber If the molecules were charged (or electrons), then there would be a net current flow The diffusion current flows from high concentration to low concentration:

34 Total Current and Boundary Conditions When both drift and diffusion are present, the total current is given by the sum: In resistors, the carrier is approximately uniform and the second term is nearly zero For currents flowing uniformly through an interface (no charge accumulation), the field is discontinous

35 Carrier Concentration and Potential In thermal equilibrium, there are no external fields and we thus expect the electron and hole current densities to be zero:

36 Carrier Concentration and Potential (2) We have an equation relating the potential to the carrier concentration If we integrate the above equation we have We define the potential reference to be intrinsic Si:

37 Carrier Concentration Versus Potential The carrier concentration is thus a function of potential Check that for zero potential, we have intrinsic carrier concentration (reference). If we do a similar calculation for holes, we arrive at a similar equation Note that the law of mass action is upheld

38 The Doping Changes Potential Due to the log nature of the potential, the potential changes linearly for exponential increase in doping: Quick calculation aid: For a p-type concentration of cm -3, the potential is -360 mv N-type materials have a positive potential with respect to intrinsic Si

39 PN Junction Currents Consider the PN junction in thermal equilibrium Again, the currents have to be zero, so we have

40 PN Junction Fields p-type n-type N A N D Transition Region

41 Total Charge in Transition Region To solve for the electric fields, we need to write down the charge density in the transition region: In the p-side of the junction, there are very few electrons and only acceptors: Since the hole concentration is decreasing on the p-side, the net charge is negative:

42 Charge on N-Side Analogous to the p-side, the charge on the n-side is given by: The net charge here is positive since: Transition Region

43 Exact Solution for Fields Given the above approximations, we now have an expression for the charge density We also have the following result from electrostatics Notice that the potential appears on both sides of the equation difficult problem to solve A much simpler way to solve the problem

44 Depletion Approximation Let s assume that the transition region is completely depleted of free carriers (only immobile dopants exist) Then the charge density is given by The solution for electric field is now easy Field zero outside transition region

45 Depletion Approximation (2) Since charge density is a constant If we start from the n-side we get the following result Field zero outside transition region

46 Plot of Fields In Depletion Region p-type N A Depletion Region n-type N D E-Field zero outside of depletion region Note the asymmetrical depletion widths

47 Continuity of E-Field Across Junction Recall that E-Field diverges on charge. For a sheet charge at the interface, the E-field could be discontinuous In our case, the depletion region is only populated by a background density of fixed charges so the E-Field is continuous What does this imply? Total fixed charge in n-region equals fixed charge in p-region! Somewhat obvious result.

48 Potential Across Junction From our earlier calculation we know that the potential in the n-region is higher than p-region The potential has to smoothly transition form high to low in crossing the junction Physically, the potential difference is due to the charge transfer that occurs due to the concentration gradient Let s integrate the field to get the potential:

49 Potential Across Junction We arrive at potential on p-side (parabolic) Do integral on n-side Potential must be continuous at interface (field finite at interface)

50 Solve for Depletion Lengths We have two equations and two unknowns. We are finally in a position to solve for the depletion depths (1) (2)

51 Sanity Check Does the above equation make sense? Let s say we dope one side very highly. Then physically we expect the depletion region width for the heavily doped side to approach zero: Entire depletion width dropped across p-region

52 Total Depletion Width The sum of the depletion widths is the space charge region This region is essentially depleted of all mobile charge Due to high electric field, carriers move across region at velocity saturated speed

53 Have we invented a battery? Can we harness the PN junction and turn it into a battery?? Numerical example:

54 Contact Potential The contact between a PN junction creates a potential difference Likewise, the contact between two dissimilar metals creates a potential difference (proportional to the difference between the work functions) When a metal semiconductor junction is formed, a contact potential forms as well If we short a PN junction, the sum of the voltages around the loop must be zero: n p

55 Diode under Thermal Equilibrium Minority Carrier Close to Thermal Junction Generation p-type n-type Carrier with energy Recombination below barrier height Diffusion small since few carriers have enough energy to penetrate barrier Drift current is small since minority carriers are few and far between: Only minority carriers generated within a diffusion length can contribute current Important Point: Minority drift current independent of barrier! Diffusion current strong (exponential) function of barrier

56 Reverse Bias Reverse Bias causes an increases barrier to diffusion Diffusion current is reduced exponentially p-type n-type Drift current does not change Net result: Small reverse current

57 Forward Bias Forward bias causes an exponential increase in the number of carriers with sufficient energy to penetrate barrier Diffusion current increases exponentially p-type n-type Drift current does not change Net result: Large forward current

58 Diode I-V Curve Diode IV relation is an exponential function This exponential is due to the Boltzmann distribution of carriers versus energy For reverse bias the current saturations to the drift current due to minority carriers

59 Minority Carriers at Junction Edges Minority carrier concentration at boundaries of depletion region increase as barrier lowers the function is (minority) hole conc. on n-side of barrier (majority) hole conc. on p-side of barrier (Boltzmann s Law)

60 Law of the Junction Minority carrier concentrations at the edges of the depletion region are given by: Note 1: N A and N D are the majority carrier concentrations on the other side of the junction Note 2: we can reduce these equations further by substituting V D = 0 V (thermal equilibrium) Note 3: assumption that p n << N D and n p << N A

61 Minority Carrier Concentration Minority Carrier Diffusion Length The minority carrier concentration in the bulk region for forward bias is a decaying exponential due to recombination

62 Steady-State Concentrations Assume that none of the diffusing holes and electrons recombine get straight lines This also happens if the minority carrier diffusion lengths are much larger than W n,p

63 Diode Current Densities

64 Diode Circuits Rectifier (AC to DC conversion) Average value circuit Peak detector (AM demodulator) DC restorer Voltage doubler / quadrupler /

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