ECE 255: Fall 2019 Purdue University Energy Band Diagrams Mark Lundstrom School of ECE Purdue University West Lafayette, IN USA Lundstrom: Fall 2019 1 Goal for this lecture R ID

+ VR + VDD VD - Lundstrom: Fall 2019 I R ( VR ) =VR R I D ( VD ) =? To set the stage for understanding how PN junction diodes work. i.e. what is ID(VD)?

2 Outline 1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing energy band diagrams Lundstrom: Fall 2019 3 Energy band diagrams An energy band diagram is a plot of the bottom of the conduction band and the top of the valence band vs. position.

Energy band diagrams are a powerful tool for understanding semiconductor devices because they provide qualitative solutions to the semiconductor equations. Lundstrom: Fall 2019 4 Energy band diagrams https://www.pbs.org/wgbh/americanexperience/features/silicon-timeline-silicon/ 5 Band bending in an MOS structure SiO2 oxide

What happens when we apply a voltage to the gate? nB =ni2 NA EC ( x ) =ECB V =0 +VG EVB semiconductor metal gate pB =NA

x Lundstrom: Fall 2019 6 Electrostatic potential vs. position V ( x) +VG V =0 x Lundstrom: Fall 2019 7

Voltage and electron potential energy E =- qV - +V A positive potential lowers the energy of an electron. Lundstrom: Fall 2019 8 Electrostatic potential causes band bending PE =EC ( x) =ECB - qV ( x) E

nB =ni2 NA ECB VG > 0 EC ( x) V =0 EVB pB =NA EV ( x) x Lundstrom: Fall 2019

9 Electric field E EC ( x) =ECB - qV ( x) nB =ni2 NA ECB V =0 VG > 0 EVB dEC ( x) dV ( x) =- q

=q dx dx pB =NA x The electric field is proportional to the slope of EC 10 Electric field 1 dEC ( x) = q dx

Fe 0 x The electric field is proportional to the slope of EC 11 Electron concentration E n( x) e- EC kBT

eqV( x) kBT nB =ni2 NA ECB qV ( x) EIB V =0 EVB n( x) =nBeqV( x)

kBT pB =NA x Lundstrom: Fall 2019 12 Electron concentration n( x) =nBeqV( x) log10 n(x) n( x=0 ) =( ni2 NA ) eqV( x=0 ) kBT

kBT nB =ni2 NA x Lundstrom: Fall 2019 13 Hole concentration p( x) e- qV( x) E kBT nB =ni2 NA

ECB qV ( x) VG > 0 V =0 EVB p( x) =pBe- qV( x) kBT pB =NA x Lundstrom: Fall 2019

14 Electron and hole concentrations (log y-axis) log10 n0 (x), p0 ( x) ni2 eqV( x=0 ) NA NA e- qV( x=0) NA kBT depletion region nB =ni2 NA

kBT W Lundstrom: Fall 2019 x 15 Electron and hole concentrations (linear y-axis) p0 (x) n0 (x), p0 ( x) depletion region n0 (x)

Lundstrom: Fall 2019 x 16 Reading band diagrams A band diagram Reading the band diagram V ( x) - EC ( x) E EC EV

x Lundstrom: Fall 2019 dEC ( x) dx n( x) e- EC ( x) p( x) eEV ( x) kBT kBT eqV( x) e- qV( x) kBT

kBT 17 Outline 1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing equilibrium energy band diagrams Lundstrom: Fall 2019 18 Draw the E-band diagram

N P electron concentration high electron concentration low (hole concentration low) (hole concentration high) Lundstrom: Fall 2019 19 Step 1: Draw a reference line

Eref =EF Lundstrom: Fall 2019 20 Step 2: Draw EC(x) E EC ( x) Eref =EF EC is low where the electron concentration is high, and EC is high where the electron concentration is low. x

Lundstrom: Fall 2019 21 Step 3: Draw EV(x) E EC ( x) EG Eref =EF EV ( x) x Lundstrom: Fall 2019 22

Read the E-band diagram: V(x) E EC ( x) =ECB - qV ( x) EC ( x) Eref =EF EV ( x) x Lundstrom: Fall 2019 23 Read the E-band diagram: V(x) E

V ( x) x Lundstrom: Fall 2019 24 Read the E-band diagram: electric field E EC ( x) =ECB - qV ( x) dEC dV =q dx dx

EC ( x) Eref =EF EV ( x) x Lundstrom: Fall 2019 25 Read the E-band diagram: electric field E ( x) x Lundstrom: Fall 2019

26 Read the E-band diagram: n(x) and p(x) E n0 ( x) e- EC ( x) kBT EC ( x) Eref =EF EV ( x) p0 ( x) e+EV ( x)

kBT x Lundstrom: Fall 2019 27 Read the E-band diagram: n(x) and p(x) log10 n( x) ,log10 p( x) pp =NA nn =ND np =ni2 NA pn =ni2 ND

N - xn xp Lundstrom: Fall 2019 x P 28 Summary Energy band diagrams are a powerful tool for understanding the operation of semiconductor devices.

To draw an E-band diagram: 1) Draw a horizontal line as an energy reference 2) Draw EC(x) low where n(x) is large and EC(x) high where n(x) is small. 3) Draw EV(x) = EC(x) - EG Lundstrom: Fall 2019 29 Summary To read an E-band diagram: 1) Find the electrostatic potential vs. position by turning turn EC(x) upside down. 2) Find the electric field vs. position by taking the slope of EC(x). 3) Find the carrier density vs. position by beginning where it is known, and then exponentially increasing or decreasing

it according to the local EC(x). 30 Energy band diagrams 1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing energy band diagrams Lundstrom: Fall 2019 31