ME 432 Fundamentals of Modern Photovoltaics. Discussion 15: Semiconductor Carrier Sta?s?cs 3 October 2018

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1 ME 432 Fundamentals of Modern Photovoltaics Discussion 15: Semiconductor Carrier Sta?s?cs 3 October 2018

2 Fundamental concepts underlying PV conversion input solar spectrum light absorp?on carrier excita?on & thermaliza?on charge transport charge separa?on charge collec?on output You Are Here Courtesy: Yosuke Kanai, University of North Carolina

3 A)er Light Absorp2on & Thermaliza2on: energy We are le) with electrons at the conduc2on band edge, and holes in the valence band edge. These are free carriers that contribute to conduc2on Effec2vely, there are two possible outcomes from here: x

4 A)er Light Absorp2on & Thermaliza2on: Outcome 1: Electrons and holes are physically separated in space, and extracted into an external circuit to do useful work energy direc?on of electron flow direc?on of hole flow x The electron and hole are extracted into opposite terminals of an external circuit, and used to do electrical work This is the desired outcome for a solar cell

5 A)er Light Absorp2on & Thermaliza2on: Outcome 2: Electrons and holes recombine before they can be physically separated and collected energy hv hv Recombina?on takes place in the interior of the solar cell The stored energy is lost, as a photon of light is emiqed and the electron drops back to the valence band Undesirable for a solar cell (whereas this is the desired outcome for an LED) x

6 Typical Conduc2vi2es of Materials Of all the experimentally measurable proper?es of materials, the electrical conduc?vity s is believed to exhibit the greatest varia?on there is a factor of in the difference in the conduc?vity of the best metals rela?ve to the strongest insulators. Example: Room Temperature Conduc?vi?es Copper (Metal) Silicon (Semiconductor) Quartz (Insulator) 6 (10 7 ) (Wm) -1 2 (10-14 ) (Wm) -1 2 (10-3 ) (Wm) -1

7 Conduc2on in Semiconductors Key concepts: 1. Conduc?vity in intrinsic semiconductors depends on the semiconductor band gap and the temperature 2. Conduc?vity in semiconductors can also be controlled by doping (the inten?onal introduc?on of impuri?es)

8 Learning Objec2ves: Conduc2on in Semiconductors 1. Describe conceptually how conduc?vity in intrinsic semiconductors depends on the band gap and the temperature. 2. Describe what the effec?ve mass, available density of states, Fermi energy, and the Fermi-Dirac distribu?on are. 3. Describe mathema?cally how conduc?vity in intrinsic semiconductors depends on the band gap and temperature 4. Describe conceptually how doping is used to increase the conduc?vity of a semiconductor, and what type of dopants result in n-type and p-type doping. 5. Describe mathema?cally how the dopant concentra?on relates to the change in the Fermi energy. Some Suggested Readings: Green, Chapter 2 (posted online) Luque & Hegedus, Chapter 3 (available for download through UIUC online library) Chapters 3 & 4

9 Learning Objec2ves: Conduc2on in Semiconductors 1. Describe conceptually how conduc?vity in intrinsic semiconductors depends on the band gap and the temperature. 2. Describe what the effec?ve mass, available density of states, Fermi energy, and the Fermi-Dirac distribu?on are. 3. Describe mathema?cally how conduc?vity in intrinsic semiconductors depends on the band gap and temperature 4. Describe conceptually how doping is used to increase the conduc?vity of a semiconductor, and what type of dopants result in n-type and p-type doping. 5. Describe mathema?cally how the dopant concentra?on relates to the change in the Fermi energy. Some Suggested Readings: Green, Chapter 2 (posted online) Luque & Hegedus, Chapter 3 (available for download through UIUC online library) Chapters 3 & 4

10 Two-Level Parking Garage Analogy CB VB filled valence band, empty conduc>on band no free carriers, and no conduc>vity

11 Two-Level Parking Garage Analogy CB VB holes in valence band, electrons in conduc>on band movement is possible both in VB & CB, so conduc>on is possible

12 Conduc2vity Band Diagram (E vs. x) Density of States Energy hqp://pvcdrom.pveduca?on.org/ Density of States

13 Conduc2vity: Dependence on Temperature At absolute zero, the electrons fill the lowest available energy levels. Valence band is completely filled, and there is no conduc2vity (perfect insulator). Band Diagram (E vs. x) Density of States Covalentlybonded electrons Energy Density of States

14 Ques?on: If we raise the temperature, what changes? The band gap? The density of states? The occupa?on of the available states by electrons?

15 Conduc2vity: Dependence on Temperature At T > 0 K, some carriers are thermally excited across the bandgap. Band Diagram (E vs. x) Density of States Energy Density of States

16 Conduc2vity: Dependence on Temperature At T > 0 K, some carriers are thermally excited across the bandgap. Band Diagram (E vs. x) Covalentlybonded excited Thermally electrons Density of States Energy Density of States

17 Conduc2vity: Dependence on Temperature At T > 0 K, some carriers are thermally excited across the bandgap, crea2ng electrons in the CB and holes in the VB that can contribute to conduc2on. Band Diagram (E vs. x) n = concentra?on of electrons in the conduc?on band (typical units: # electrons/cm 3 ) Covalentlybonded Intrinsic Carriers electrons Density of States Energy Note: For an intrinsic semiconductor, n = p = n i p = concentra?on of holes in the valence band (typical units: # holes/cm 3 ) Density of States

18 Temperature Dependence of Intrinsic Carrier Concentra2on n i = intrinsic carrier density n i = intrinsic carrier density or intrinsic carrier concentra?on. For an intrinsic semiconductor, n=p=n i. Units: #/cm 3. Note that in silicon, there are approximately 7(10 23 ) electrons/cm 3 total, so we are generally talking about a very very small percentage of the electrons being excited from the VB to the CB that are available to conduct.

19 Temperature Dependence of Intrinsic Carrier Concentra2on Arrhenius Equation, generic form: [ ] n i = N o exp E A / k b T

20 Learning Objec2ves: Conduc2on in Semiconductors 1. Describe conceptually how conduc?vity in intrinsic semiconductors depends on the band gap and the temperature. 2. Describe what the effec?ve mass, available density of states, Fermi energy, and the Fermi-Dirac distribu?on are. 3. Describe mathema?cally how conduc?vity in intrinsic semiconductors depends on the band gap and temperature 4. Describe conceptually how doping is used to increase the conduc?vity of a semiconductor, and what type of dopants result in n-type and p-type doping. 5. Describe mathema?cally how the dopant concentra?on relates to the change in the Fermi energy. Some Suggested Readings: Green, Chapter 2 (posted online) Luque & Hegedus, Chapter 3 (available for download through UIUC online library) Chapters 3 & 4

21 New Concept #1: Fermi energy E F Think of the Fermi energy as the average energy of all the carriers (holes in the valence band and electrons in the conduc?on band) Band Diagram (E vs. x) n = concentra?on of electrons in the conduc?on band (typical units: # electrons/cm 3 ) p = concentra?on of holes in the valence band (typical units: # holes/cm 3 ) E F - If there are as many holes in the VB as there are electrons in the CB as in for an instrinsic semiconductor, then the Fermi energy is right in the middle of the gap - If we shij the popula?on distribu?on of holes and electrons, then the Fermi level will shij: - More holes => Fermi level below mid-gap - More electrons => Fermi level above mid gap

22 New Concept #2: Effec2ve Mass Consider a free electron, and an externally applied electric field: e external electric field E F = m e a = q E = d p dt E = 1 2 m ( ev 2 = m v e ) 2 2m e = p2 2m e Force on electron due to external electric field mass of electron?me rate of change of momentum Rela2onship between electron s energy E and its momentum p

23 New Concept #2: Effec2ve Mass Consider in a semiconductor a free carrier electron (in a CB), and an externally applied electric field. The charges inside the semiconductor crystal interfere with the mo?on of the electron e external electric field E F = m e * a = q E = d k dt ( * v) 2 E = 1 2 m * ev 2 = m e = k 2 * 2m e 2m e * Force on electron due to external electric field effec2ve mass of electron?me rate of change of momentum Rela2onship between electron s energy E and its momentum k No>ce the parabolic rela>onship between E and k

24 New Concept #2: Effec2ve Mass Because of their shared crystal structures, many semiconductors have qualita?vely similar electronic band structures and parabolic extrema E L G X k ( E E c = k ) 2 * 2m e parabolic bands approxima?on But: rela?ve heights of the minima and maxima and their curvatures depend on the the semiconductor itself

25 New Concept #2: Effec2ve Mass E = 1 2m e * ( k) 2 E larger curvature, lighter electrons smaller curvature, heavier electrons k No?ce that the effec?ve mass of an electron in a par?cular band is exactly the inverse of the curvature of the band (d 2 E/dk 2 ) -1! # " d 2 E dk 2 $ & % 1 = m e * Thus, the larger the curvature of the band, the lighter the carrier is (and vice-versa)

26 New Concept #3: Available Density of States The available density of states D avail (E) describes the number of states that are available per unit volume in the conduc?on and valence bands, as a func?on of energy E light electrons E E k D avail (E) heavy holes The available density of states also depends on the curvatures, and hence the effec?ve mass. D avail ( * ) 3 2 h 3 ( E) = 8 2π m e ( E E C ) 1 2

27 New Concept #4: Fermi-Dirac Distribu2on The Fermi-Dirac distribu?on f(e) is a mathema?cal func?on that describes the probability (as a func?on of temperature) that an available electron energy level of energy E is filled. (The deriva?on comes from sta?s?cal mechanics, we ll not derive it here but accept it as a given.) f (E) = 1 " 1+ exp E E F $ # kt % ' & where : - E F is the Fermi energy - k is the Boltzmann Constant : 8.617(10-5 ) ev/k - T is the temperature (K) - kt represents the thermal energy available for electrons at temperature T (kt = ev at room temp) As we increase the temperature, the sharp discon?nuity becomes more and more smeared out 1 f(e) T = 0 K T = 300 K T = 1500 K T = 3000 K 0 E=E F E

28 Learning Objec2ves: Conduc2on in Semiconductors 1. Describe conceptually how conduc?vity in intrinsic semiconductors depends on the band gap and the temperature. 2. Describe what the effec?ve mass, available density of states, Fermi energy, and the Fermi-Dirac distribu?on are. 3. Describe mathema?cally how conduc?vity in intrinsic semiconductors depends on the band gap and temperature 4. Describe conceptually how doping is used to increase the conduc?vity of a semiconductor, and what type of dopants result in n-type and p-type doping. 5. Describe mathema?cally how the dopant concentra?on relates to the change in the Fermi energy. Some Suggested Readings: Green, Chapter 2 (posted online) Luque & Hegedus, Chapter 3 (available for download through UIUC online library) Chapters 3 & 4

29 Quan2fying conduc2vity of a semiconductor as a func2on of its band gap and the temperature.

30 New Concept #4: Fermi-Dirac Distribu2on We can use the Fermi-Dirac distribu?on to quan?fy the occupied density of states as a func?on of temperature using the following: D avail (E) f(e,t) = D occ (E,T) available density of states Fermi-Dirac = distribu?on occupied density of states Then the intrinsic carrier concentra?on n i can be obtained by integra?ng over all energies: n i (T ) = all E D occ (E,T )de

31 Intrinsic Conduc2vity: Mathema2cal Formalism electron density in the conduc?on band: n(t ) = all E in CB D occ (E,T )de = f ( E,T )D avail (E)dE E C = 2 2πm * 3 # ekt & 2 # % ( exp E E C F % $ h 2 ' $ kt # n = N c exp E E & C F % ( $ kt ' & ( ' hole density in the valence band: p(t)= all E in VB E V (D empty (E,T))dE = (1 f ( E,T ))D avail (E)dE = 2 2πm * 3 kt 2 h h 2 exp E E F V kt p = N V exp E E F V kt N C = effec?ve density of states in the conduc?on band N V = effec?ve density of states in the valence band See Green, sec?on 2.8 for detailed deriva?on.

32 Conduc2vity: Dependence on Temperature At a zero temperature, zero intrinsic carriers and zero conduc2vity. Probability Distribu2on Func2on f(e) Available Density of States Conduc?on Band Occupied Density of States Conduc?on Band Energy x Energy = Energy Valence Band Valence Band Probability of Occupancy Density of States Density of States

33 Conduc2vity: Dependence on Temperature At a finite temperature, finite conduc2vity (current can flow). Probability Distribu2on Func2on Available Density of States Conduc?on Band Occupied Density of States Conduc?on Band Energy T > 0 x Energy = Energy Valence Band Valence Band Probability of Occupancy Density of States Density of States

34 Ques?on: To reduce noise in a Si CCD camera, should you increase or decrease temperature?

35 Lower Temperature = Lower Intrinsic Carrier Concentra2on CCD inside a LN dewar hqp://msowww.anu.edu.au/observing/detectors/wfi.php hqp://

36 Ques?on: Transistors made from which semiconductor material experience greater electronic noise at room temperature: Germanium or Silicon?

37 Intrinsic Conduc2vity: Dependence on Bandgap At a finite temperature, finite conduc2vity (current can flow). Probability Distribu2on Func2on Density of States Energy T > 0 Energy Silicon Bandgap ~ 1.12 ev Probability of Occupancy Density of States

38 Intrinsic Conduc2vity: Dependence on Bandgap At a finite temperature, finite conduc2vity (current can flow). Probability Distribu2on Func2on Density of States Energy T > 0 Energy Germanium Bandgap ~ 0.67eV Probability of Occupancy Density of States

39 Intrinsic Conduc2vity: Mathema2cal Formalism electron density in the conduc?on band: n(t ) = all E in CB D occ (E,T )de = f ( E,T )D avail (E)dE E C = 2 2πm * 3 # ekt & 2 # % ( exp E E C F % $ h 2 ' $ kt # n = N c exp E E & C F % ( $ kt ' & ( ' hole density in the valence band: p(t)= all E in VB E V (D empty (E,T))dE = (1 f ( E,T ))D avail (E)dE = 2 2πm * 3 kt 2 h h 2 exp E E F V kt p = N V exp E E F V kt N C = effec?ve density of states in the conduc?on band N V = effec?ve density of states in the valence band See Green, sec?on 2.8 for detailed deriva?on.

40 Intrinsic Conduc2vity: Mathema2cal Formalism Recall that for an intrinsic semiconductor, n = p = n i " n(t ) = N c exp E C E F % $ ' # kt & " p(t ) = N V exp E E % F V $ ' # kt & 2 np = n i " = N c N v exp E C E $ V # kt " = N c N v exp E % G $ ' # kt & % ' & expression for intrinsic carrier concentra?on n i (T): n i ( ) 1 2 exp E G $ = N c N v " # 2kT % ' &

41 Intrinsic Conduc2vity: Mathema2cal Formalism Recall that for an intrinsic semiconductor, n = p = n i " n(t ) = N c exp E C E F % $ ' # kt & " p(t ) = N V exp E E % F V $ ' # kt & n = p N c exp E E C F kt = N exp E E F V v kt expression for intrinsic Fermi level E F (T)=E i (T): E i ( T ) = E C + E V 2 + kt 2 ln! N V # " N C $ & %

42 Intrinsic Conduc2vity: Dependence on Bandgap hqp://

43 Temperature Dependence of Intrinsic Carrier Concentra2on Increasing band gap Arrhenius Equation, generic form: n i = N o exp[ E A / k b T ]

44 Learning Objec2ves: Conduc2on in Semiconductors 1. Describe conceptually how conduc?vity in intrinsic semiconductors depends on the band gap and the temperature. 2. Describe what the effec?ve mass, available density of states, Fermi energy, and the Fermi-Dirac distribu?on are. 3. Describe mathema?cally how conduc?vity in intrinsic semiconductors depends on the band gap and temperature 4. Describe conceptually how doping is used to increase the conduc?vity of a semiconductor, and what type of dopants result in n-type and p-type doping. 5. Describe mathema?cally how the dopant concentra?on relates to the change in the Fermi energy. Some Suggested Readings: Green, Chapter 2 (posted online) Luque & Hegedus, Chapter 3 (available for download through UIUC online library) Chapters 3 & 4

45 Inten2onal Modifica2on of Conduc2vity by Doping

46 Effect of Doping on Carrier Concentra2ons Note the room temperature resis>vity of pure (undoped) Si: (ohm-cm) This degree of control (many orders of magnitude!) over the fundamental electrical proper2es of the material, is what makes semiconductors so versa2le.

47 Dopant Atoms Periodic Table hqp://pvcdrom.pveduca?on.org/

48 Example: n-type doping of silicon Periodic Table We want to dope with an element that is not too different from Si, so as not to perturb the band structure too much Consider doping with phosphorus, which has one extra electron, but is isocoric with silicon The extra electron introduced by phosphorus is very weakly bound to the phosphorus atom (typical binding energies are around mev) It takes very liqle energy to free the electron from the P core, pusng the electron in the conduc?on band and ionizing the P atom. How we draw this on the valence/conduc?on band diagrams: ground state (i.e. at T=0 K) 10 mev ionized state (i.e. at room temp) hqp://pvcdrom.pveduca?on.org/

49 Example: n-type doping of silicon ground state (i.e. at T=0 K) 10 mev ionized state (i.e. at room temp) The P atom is called a donor because it donates electrons to the conduc?on band, and hence increases the free electron concentra?on and conduc?vity. We can think of n-type doped silicon as having a number of electrons that are free to move around in the conduc?on band, and posi?vely charged immobile ion cores (nuclei of the dopants).

50 Example: p-type doping of silicon Periodic Table We want to dope with an element that is not too different from Si, so as not to perturb the band structure too much Consider doping with boron, which has one fewer valence electron than silicon The missing electron can be thought of as a hole that is very weakly bound to the boron atom (again, typical binding energies are around mev) It takes very liqle energy to free the hole from the B core, pusng the hole in the valence band and ionizing the B atom. How we draw this on the valence/conduc?on band diagrams: ground state (i.e. at T=0 K) ionized state (i.e. at room temp) 10 mev hqp://pvcdrom.pveduca?on.org/

51 Example: p-type doping of silicon ground state (i.e. at T=0 K) 10 mev ionized state (i.e. at room temp) The B atom is called an acceptor because it accepts an electron from the valence band, and hence increases the free hole concentra?on and conduc?vity. We can think of p-type doped silicon as having a number of holes that are free to move around in the conduc?on band, and nega?vely charged immobile ion cores (nuclei of the dopants).

52 Ques2on: When we dope a semiconductor, typically what percent of the atoms are doped? Typical doping concentra?ons are around dopant atoms per cm 3 of semiconductor Silicon, for example, has around 5x10 22 atoms per cm 3 Therefore, we typically introduce 1 dopant atom per atoms.

53 Let s play a game donor or acceptor? Periodic Table Semiconductor Dopant donor or acceptor? Si Al Acceptor Si Ge Neither Si As Donor Si Se Donor (double) GaAs Se Donor GaAs Ge Acceptor and Donor ( amphoteric ) GaAs As Could be a double donor

54 Let s play a game donor or acceptor? Periodic Table Semiconductor Dopant donor or acceptor? Si Al Acceptor Si Ge Neither Si As Donor Si Se Donor (double) GaAs Se Donor GaAs Ge Acceptor and Donor ( amphoteric ) GaAs As Could be a double donor

55 Impurity levels for many elements in silicon are well known Courtesy: Mark Winkler, IBM Research * We know orders of magnitude more about silicon than any other semiconductor, and possibly any other material

56 Learning Objec2ves: Conduc2on in Semiconductors 1. Describe conceptually how conduc?vity in intrinsic semiconductors depends on the semiconductor band gap and the temperature. 2. Describe what the Fermi energy and the Fermi-Dirac distribu?on are. 3. Describe conceptually how doping is used to increase the conduc?vity of a semiconductor, and what type of dopants result in n-type and p-type doping. 4. Describe mathema2cally how the dopant concentra2on relates to the change in the Fermi energy. Some Suggested Readings: Green, Chapter 2 (posted online) Luque & Hegedus, Chapter 3 (available for download through UIUC online library) Chapters 3 & 4

57 Quan2fying Doping in a Semiconductor Doping is the incorpora?on of defects into a material to inten?onally modify the conduc?vity When we dope a semiconductor, we increase either n or p. It is no longer true that n=p=n i as was true for the intrinsic case. The Fermi energy E F becomes shijed away from the middle of the gap and towards either the valence or the conduc?on band. Density of States intrinsic semiconductor Density of States extrinsic semiconductor n-type doped: n >> p p-type doped: p >> n Energy E F Energy E F Energy E F Density of States Density of States Density of States

58 Quan2fying Doping in a Semiconductor The degree of shij of the Fermi energy E F reflects the degree of doping via the following rela?ons, which are generally true for any semiconductor (whether intrinsic or extrinsic) " n = N c exp E C E $ F # kt " p = N v exp E E F V $ # kt Band Diagram (E vs. x) % ' & % ' & E C where : - E F is the Fermi energy - E V and E C are the VBM and CBM energies - k is the Boltzmann Constant : 8.617(10-5 ) ev/k - T is the temperature (K) - N C and N V are (respec?vely) the effec>ve density of states in the conduc?on and valence bands and can be measured experimentally for a given semiconductor (units: cm -3 ) Regardless of the doping, the following rela?onship always holds: E F E V " np = N c N V exp E % G $ ' # kt &

59 Quan2fying Doping in a Semiconductor Regardless of the doping, the following rela?onship always holds: " np = N c N V exp E G $ # kt % ' & Thus, for an intrinsic (undoped) semiconductor where n=p=n i, we have: n = p = n i = ( N c N V ) 1 2 " exp E G $ # 2kT % ' & Arrhenius Equation, generic form: n i = N o exp[ E A / k b T ]

60 Quan2fying Doping in a Semiconductor Example: Let s say I dope a semiconductor with a concentra?on N D of donor atoms. What is the corresponding change in the Fermi energy, assuming that all of the donors are ionized? Answer: We use the following rela?onship, sesng n=n D : " n = N c exp E C E $ F # kt % ' = N D & We also use the fact that the Fermi energy is in the middle of the gap (at E=E i ) for the intrinsic semiconductor: Taking the ra?o, we have: Rewri?ng: N D n i = " n = N c exp E C E $ i # kt " N c exp E C E F % $ ' # kt & " N c exp E E = exp E E % F i $ ' " % C i # kt & $ ' # kt &! E F = E i + kt ln N $ D # & " % n i % ' = n i &

61 Quan2fying Doping in a Semiconductor Example: Let s say I dope a semiconductor with a concentra?on N A of acceptor atoms. What is the corresponding change in the Fermi energy, assuming that all of the acceptors are ionized? Answer: We use the following rela?onship, sesng p=n A : " p = N V exp E E F V $ # kt % ' = N A & We also use the fact that the Fermi energy is in the middle of the gap (at E=E i ) for the intrinsic semiconductor: Taking the ra?o, we have: Rewri?ng: N A n i = " p = N V exp E E i V $ # kt " N V exp E E % F V $ ' # kt & " N c exp E E = exp E E % i F $ ' " % i V # kt & $ ' # kt & " E F = E i kt ln N % A $ ' # & n i % ' = n i &

62 Varia2on of Fermi Energy with Doping Level donor doping E F acceptor doping dopant concentra?on (# dopants/cm 3 )

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66 Summary Carrier Sta?s?cs n, p = electron in CB, hole in VB concentra?on n i = intrinsic carrier concentra?on E F = Fermi energy f(e) = Fermi-Dirac distribu?on E i = intrinsic energy, Fermi energy of intrinsic semiconductor = (E C +E V )/2 N c, N v = effec?ve density of states at the conduc?on, valence band edges N D, N A = donor, acceptor doping concentra?on " np = N c N V exp E % G $ ' # kt & " n = N c exp E E % C F $ ' # kt & " p = N v exp E F E V % $ ' # kt & For Any Semiconductor:! E F = E i + kt ln N $ D # & " E F = E i kt ln N n i % A $ ' # & n i For Intrinsic Semiconductors: n = p = n i = ( N c N V ) 1 2 " exp E G $ # 2kT % ' &