Lecture 4 - Carrier generation and recombination. February 12, 2007

Transcription

1 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-1 Contents: Lecture 4 - Carrier generation and recombination 1. G&R mechanisms February 12, Thermal equilibrium: principle of detailed balance 3. G&R rates in thermal equilibrium 4. G&R rates outside thermal equilibrium Reading assignment: del Alamo, Ch

2 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-2 Key questions What are the physical mechanisms that result in the generation and recombination of electrons and holes? Which one of these are most relevant for Si at around temperature? What are the key dependencies of the most important mechanisms? If there are several simultaneous but independent mechanisms for generation and recombination, how exactly does one define thermal equilibrium? What happens to the balance between generation and recombination when carrier concentrations are perturbed from thermal equilibrium values?

3 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture Generation and recombination mechanisms a) Band-to-band G&R, by means of: phonons (thermal G&R) photons (optical G&R) Ec Ev hυ > Eg hυ heat heat thermal thermal optical radiative generation recombination absorption recombination thermal G&R: very unlikely in Si, need too many phonons simultaneously (about 20) optical G&R: unlikely in Si, indirect bandgap material, need a phonon to conserve momentum

4 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-4 b) Auger generation and recombination, involving a third carrier Ec Ev hot-electron hot-electron hot-hole hot-hole assisted assisted assisted assisted Auger Auger Auger Auger generation recombination generation recombination Auger generation: energy provided by hot carrier Auger recombination: energy given to third carrier; needs lots of carriers; important only in heavily-doped semiconductors

5 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-5 c) Trap-assisted generation and recombination, relying on electronic states in middle of gap ( deep levels or traps ) that arise from: crystalline defects impurities Ec Et Ev trap-assisted thermal generation trap-assisted thermal recombination Trap-assisted G/R is: dominant in Si engineerable: can introduce deep levels to Si to enhance it

6 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-6 d) Other generation mechanisms Impact ionization: Auger generation event triggered by electricfield-heated carrier Zener tunneling or field ionization: direct tunneling of electron from VB to CB in presence of strong electric field Ec Ec Ev Ev impact ionization Zener tunneling Energetic particles, such as α-particles (bad for DRAMs) Energetic electrons incident from outside: electron microscope characterization techniques

7 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture Thermal equilibrium: principle of detailed balance Define: G i generation rate by process i [cm 3 s 1 ] R i recombination rate by process i [cm 3 s 1 ] G total generation rate [cm 3 s 1 ] R total recombination rate [cm 3 s 1 ] In thermal equilibrium: R o =ΣR oi = G o =ΣG oi Actually, detailed balance is also required: R oi = G oi for all i In the presence of several paths for G & R, each has to balance out in detail [Principle of Detailed Balance]. [see example in notes illustrating impossibility of TE whithout detailed balance]

8 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture G&R rates in thermal equilibrium a) Band-to-band G&R Will not consider thermal G&R as it is negligible. Optical G&R At finite T, semiconductor is immersed in bath of blackbody radiation optical generation. Only a small number of bonds are broken at any one time depends only on T : G G o,rad = g rad (T ) A recombination process demands one electron and one hole depends of n o p o : R R o,rad = r rad (T ) n o p o In TE, detailed balance implies: 2 g rad = r rad n o p o = r rad n i

9 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-9 b) Auger G&R Involving hot electrons: The more electrons there are, the more likely it is to have hot ones capable of Auger generation: G o,eeh = g eeh (T )n o A recombination event demands two electrons and one hole: R o,eeh = r eeh n o 2 p o In TE, detailed balance implies: g eeh = r eeh n o p o Involving hot holes: similar but substitute n o for p o and eeh by ehh above.

10 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-10 c) Trap-assisted thermal G&R: Shockley-Read-Hall model Consider a trap at E t = E i in concentration N t. Trap occupation probability: 1 n i f(e t )= f(e i )= = 1+exp E i E F n i + p o kt Concentration of traps occupied by an electron: n i n to = N t f(e i )= N t n i + p o Concentration of empty traps: N t n to = N t N t n i + p o = N t n i + p o n i p o Trap occupation depends on doping: n-type: p o n i n to N t, most traps are full p-type: p o n i n to N t, most traps are empty E t E F E t E F n-type p-type

11 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-11 Four basic processes: E t electron electron hole hole capture emission capture emission Rates of four subprocesses in TE: electron capture: r o,ec = c e n o (N t n to ) electron emission: r o,ee = e e n to hole capture: r o,hc = c h p o n to hole emission: r o,he = e h (N t n to )

12 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-12 In thermal equilibrium, detailed balance demands: r o,ec = r o,ee r o,hc = r o,he Then, relationships that tie up capture and emission coefficients: N t n to e e = c e n o n to = c e n i n to e h = c h p o N t n to = c h n i Capture coefficients can be calculated from first principles, but most commonly they are measured. Also define: τ eo = τ ho = 1 N t c e 1 N t c h τ eo and τ ho are characteristic of the nature of the trap and its concentration. They have units of s.

13 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-13 All together, rates of communication of trap with CB and VB: r o,ec = r o,ee = 2 1 n i τ eo n i + p o r o,hc = r o,he = 1 n i p o τ ho n i + p o Rates depend on trap nature and doping level.

14 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-14 Simplify for n-type semiconductor: n i r o,ec = r o,ee τeo r o,hc = r o,he = p o τ ho If τ eo not very different from τ ho, r o,ec = r o,ee r o,hc = r o,he The rate at which trap communicates with CB much higher than VB. E t E F lots of electrons in CB and trap r o,ec = r o,ee high few holes in VB and trap r o,hc = r o,he small Reverse situation for p-type semiconductor.

15 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture G&R rates outside equilibrium In thermal equilibrium: n = n o p = p o G oi = R oi G o = R o Outside thermal equilibrium (with carrier concentrations disturbed from thermal equilibrium values): n = n o p = p o G i = R i G = R n o n=no G o = R o G = R p o p=po thermal equilibrium outside thermal equilibrium

16 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-16 If G = R, carrier concentrations change in time. Useful to define net recombination rate, U: U = R G Reflects imbalance between internal G&R mechanisms: if R>G U>0, net recombination prevails if R<G U<0, net generation prevails if R = G U = 0, thermal equilibrium If there are several mechanisms acting simultaneously, define: U i = R i G i and U =ΣU i What happens to the G&R rates of the various mechanisms outside thermal equilibrium?

17 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-17 a) Band-to-band optical G&R n o n>no G o,rad = R o,rad G rad = G o,rad R rad > R o,rad p o p>po thermal equilibrium with excess carriers optical generation rate unchanged since number of available bonds unchanged: G rad = g rad = r rad n o p o optical recombination rate affected if electron and hole concentrations have changed: define net recombination rate: R rad = r rad np U rad = R rad G rad = r rad (np n o p o ) if np > n o p o, U rad > 0, net recombination prevails if np < n o p o, U rad < 0, net generation prevails note: we have assumed that g rad and r rad are unchanged from equilibrium

18 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-18 b) Auger G&R Involving hot electrons: n o n>no G o,eeh = R o,eeh G eeh > G o,eeh R eeh > R o,eeh p o p>po thermal equilibrium with excess carriers G eeh = g eeh n R eeh = r eeh n2 p If relationship between g eeh and r eeh unchanged from TE: U eeh = R eeh G eeh = r eeh n(np n o p o ) Involving hot holes, similarly: Total Auger: U ehh = r ehh p(np n o p o ) U Auger =(r eeh n + r ehh p)(np n o p o )

19 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 4-19 Key conclusions Dominant generation/recombination mechanisms in Si: trapassisted and Auger. In TE, G and R processes must be balanced in detail. Auger R rate in TE is proportional to the square of the majority carrier concentration and is linear on the minority carrier concentration. Trap-assisted G/R rates in TE depend on the nature of the trap, its concentration, the doping type and the doping level. In n-type semiconductor, midgap trap communicates preferentially with conduction band. In p-type semiconductor, midgap trap communicates preferentially with valence band.