Physics & Technology of Nano-CMOS Devices Dae Mann Kim Korea Institute for Advanced Study

NANO-CMOS OVERVIEW CMOS technology providing VLSI basis for IT - technology driver for microelectronics & also for nanoelectronics - scaling speed progressing at greater speed than anticipated e.g. 6 nm channel length PMOS device in # of transistors in a chip doubling in 1.5 – 2 years (Moor’s law) - 10 GHz speed in about 10 years Strength - level of integration - cost, manufacturability, reliability, scalability Issues - power, fab. cost, variation in device parameters, parasitic effect, interconnect Nanodevices - utilizing physical processes operative in nanostructures - enhance and not replace CMOS technology

PART I FUNDAMENTALS & p-n JUNCTION DIODE PART II MOSFET PART III MOSFET SCALING PART IV NANO-CMOS & CHALLENGES

E g =1.12 eV in Si ECEC EVEV EiEi Conduction band Valence band Semiconductor Acceptor impurity Donor impurity EAEA EDED Semiconductor Energy Band & Impurity Energy Levels

- physical quantities are time-invariant - every process is balanced by its inverse process - single Fermi energy level (E F ) quantifies both electron (n) & hole (p) concentration - E F is flat Thermal Equilibrium - physical quantities are time-invariant - every process is balanced by its inverse process - E F is flat - single Fermi energy level (E F ) quantifies both electron (n) & hole (p) concentration - n p=n i 2 – law of mass action ; n i is the intrinsic carrier concentration

Equilibrium Statistics -Electrons / holes exhibiting the duality (wave-like & particle-like nature) reside in bands: ECEC EVEV EiEi Conduction band E g =1.12 eV in Si Valence band

where g n (E)dE is the number of electronic states between E and E+dE with the 3-D density of states given by

f E EFEF T=0 K T1T1 T2T2 T3T3 T 1 <T 2 <T 3 and is the Fermi-Dirac distribution function denoting the electron occupation probability with Fermi energy level, E F,

The integration yields with denoting the effective density of states at conduction band bottom, and the Fermi 1/2 –integral. For non-degenerate case where E C - E F  2kT

Similarly, For non-degenerate case where E F - E V  2kT with

E F in extrinsic semiconductor in equilibrium Donor atoms are incorporated as The donor energy level E D lies a few kT below E C ; the degeneracy factor g in Si is 2. Similarly, for acceptor atoms The acceptor energy level E A lies a few kT above E V ; the degeneracy factor g in Si is 4.

E F is determined by charge neutrality condition with N D, N A as parameters: T (K) E F -E i [eV] Conduction band Valence band N A =10 12 N D =

Intrinsic & extrinsic (n- & p-type) semiconductor - E Fi + E Fi where, n i is the intrinsic carrier concentration, and E Fi (=E i ) the intrinsic Fermi energy level determined by n = p, i.e., EiEi and - E Fi + E Fi

ECEC EVEV EiEi q  Fp E Fp ECEC EVEV EiEi q  Fn E Fn Fermi potential (  Fn ) in n-typeFermi potential (  Fp ) in p-type

In non-degenerate semiconductor, n p = n i 2 in thermal equilibrium (law of mass action).

Carrier Concentration N-type Intrinsic P-type f(E)N(E)n(E), p(E) n p E E E N(E) E f(E) E n(E), p(E) p n E N(E) E f(E) E n(E), p(E) n p E

Charge Transport; drift & diffusion Recall charge conservation / continuity equation; xx+dx Jn(x)Jn(x)J n (x+dx) J n consists of drift & diffusion components: drift velocity electron mobility electron diffusion coefficient Thus Continuity equation

Similarly, for holes In 3-D

Transport Coefficient Conceptually, v d is driven by E between collisions: mean collision time with electron mobility specified by Likewise

Non-uniform n / p gives rise to diffusion flux: l x The net # of electrons crossing the x-plane of cross sectional area A is given in terms of the mean free path l n as n(x)n(x) Now, electron flux

The mean free path ( l n ) is associated with the thermal velocity ( v T ) by where the thermal velocity ( v T ) is given by equipartition theorem: Thus, Similarly, i.e., one obtains Einstein relation in equilibrium

E F in equilibrium a) Single S/C system Recall in 1-D that E C, E V, E i all represent the electronic potential energy: electrostatic potential Since In equilibrium, J n = 0 and

b) Two S/C systems in equilibrium contact S/C L S/C R F LR (E) F RL (E) Now, in view of Pauli exclusion principle Transfer matrix occupied electron states vacant electron states Thus and No net flux leads to f L (E)=f R (E), i.e. Therefore, Fermi level lines up, E FL =E FR. In equilibrium E F is flat and lines up.

Non-equilibrium and quasi-Fermi levels A system, when under bias or illumination, is driven away from equilibrium to non-equilibrium conditions. In non-equilibrium, quantifying n, p requires respective quasi-Fermi level (imref), E Fn & E Fp The role of E Fn, E Fp identical to that of E F in equilibrium;

Why imrefs? In intrinsic S/C under irradiation where the photogenerated component is given via recombination time  and generation rate g ( ∝ light intensity). When g  n >>n i, g  n >>n i, E F should be above E i for quantifying n & below E i for p. Impossible to meet the condition with single E F. ECEC EVEV EiEi E Fn E Fp Splitting of two imrefs ∝ light intensity.

Shockley Hall Read Recombination: equilibrium vs. non-equilibrium Recall in equilibrium If np > n i 2 : charge injection If np < n i 2 : charge extraction In the presence of single intermediate trap level, E t, In equilibrium, # of electrons captured is trap density EtEt ECEC EVEV e capturee emissionh capture h emission

Away from equilibrium, ec rate is empty trap sites thermal velocity electron capture cross-section occupation probability in non-equilibrium Likewise emission probability Similarly, for holes

Similarly e n, e p can be extracted from equilibrium condition, i.e. r ee = r ec with

Steady state (S.S.) vs. equilibrium In S/C uniformly irradiated with denoting e/h generation rate via band to band excitation At S.S., i.e. the net recombination of e/h is same In equilibrium, a process is balanced by its inverse process, The S.S. condition is automatically satisfied.

The S.S. condition reads as Hence, the S.S. distribution function or the probability of electron being trapped at E t is given by Compare this with equilibrium distribution function,

Next, the recombination rate is given by For simplicity, take  n =  p =  and let 1/  =  v T N t, obtaining In equilibrium, n p=n i 2, U =0 If n p>n i 2 (charge injection), U >0: net recombination If n p<n i 2 (charge extraction), U <0: generation

Minority carrier lifetime & surface recombination In n-type S/C, n n >>p n, n i  p being the minority carrier lifetime. Near interface, effective thickness v R : recombination velocity

V I FORWARD REVERSE BREAKDOWN Definition: p-n junction diode is a rectifier and constitutes a key element in semiconductor transistors. p-n JUNCTION DIODE Keywords: forward & reverse bias and current, breakdown. pn V I

p- & n-type semiconductors brought in equilibrium contact to form p-n junction. pn V V=V F >0 : Forward bias V=V R <0 : Reverse bias Operation: Under bias, the junction is pushed away from equilibrium and current flows to restore the equilibrium.

Equilibrium junction band bending p- & n-type S/C are brought into equilibrium contact via exchange of e/h. In equilibrium, E F should line up and be flat, leading to energy band bending. qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E F n E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i qq Fp qq Fn E C E V E Fp E i E C E E Fn E i qq Fp qq Fn E C E V E Fp E i E C E V E Fn E i p-type n-type q  bi = q (  Fn +  Fp ) No net flows of electrons and holes, J ndrift = J ndiff, J pdrift =J pdiff

Electrostatics in depletion approximation The band bending is supported by the space charge developed in depletion width. p  -x p xnxn  m   bi 0 Q x qN d qN a n q  bi E x EFEF ECEC EVEV Specifically, e/h spills over from n/p to p/n regions, leaving behind uncompensated N D + /N A -, viz. space charge, .

Junction band bending under bias Under forward bias, V>0 -- p- and n-bulk are preserved as in equilibrium These two necessitate introduction of E Fn, E Fp & bias is accommodated via splitting of E Fn & E Fp, qV=E Fn - E Fp. -- p- side is lowered by qV relative to n-side, reducing band bending. Equivalently, n-side is raised by qV relative to p-side  x x x  (  bi -V) Q pn V > 0 W ECEC EVEV E q(  bi -V) E FN E FP -x p xnxn

Under reverse bias, V<0 -- p- and n-bulk are preserved as in equilibrium These two necessitate introduction of E Fn, E Fp & bias is accommodated via splitting of E Fn & E Fp, q|V|=E Fn -E Fp. -- p- side is raised by q|V| relative to the n-side, increasing band bending. Equivalently, n-side is lowered by qV relative to p-side Key words: N D, N A, maximum field ( E max ), built-in or contact potential (  bi ), depletion width (W ) ECEC EVEV Q   E q(  bi +|V|) E FN E FP (  bi +|V|) pn -x p xnxn W

ECEC EVEV E q(  bi +|V|) E Fn E Fp -x p xnxn pn V =V R < 0 q|V| Carrier concentration under non-equilibrium (non-degenerate case) ECEC EVEV E q(  bi -V) E Fn E Fp pn V =V F > 0 -x p xnxn qV Note that in W : Forward bias (V>0)Charge injection Reverse bias (V<0)Charge extraction inducing recombination or generation, i.e. current flows.

Ideal current-voltage characteristics: Shockley equation Shockley assumptions: The abrupt depletion-layer approximation The built-in potential & applied voltages are supported by a dipole layer with abrupt boundaries, and outside the depletion layer S/C is assumed neutral. The Boltzmann approximation Low injection level The injected minority carrier densities are much smaller than the majority carrier densities in equilibrium; No recombination/generation currents exist in the depletion layer Therefore, the electron and hole currents are constant through the depletion layer. The total current can be approximated as It gives simplified boundary conditions

The continuity equation and boundary conditions in n-region, for example, reads as where L p is the hole diffusion length in n-region given by The general solution is given by Likewise, one can write

The respective diffusion currents are given by and total current is given by where is the saturation current.

pn V = V F > 0 -xp-xp xnxn x x n, p np(x)np(x) n p0 p n0 J pn(x)pn(x) JnJn JpJp JFJF LnLn LpLp xnxn -xp-xp x x n p0 p n0 pn(x)pn(x) np(x)np(x) |J| JnJn JpJp JRJR p n V = -V R < 0 n, p Note (i) the condition that I is constant throughout is met by respective majority drift component (ii) the current is dominated by minority diffusion current

qV/k B T J/J S FORWARD REVERSE |J/J S | q |V| /k B T FORWARD REVERSE Ideal I-V characteristics: linear & semilog plot

Non-ideal current Recombination / generation current There is recombination (V>0) or generation (V<0) throughout the depletion and the quasi-neutral region near the junction edges For V>0For V<0 Generation of e/h in depletion occurs primarily via intermediate trap sites inducing alternating e/h emissions

High level injection The quasi-neutral approximation is no longer valid, and electric field exists outside the depletion region, leading to minority drift current In this case, the effective voltage drop through the depletion region is smaller than V, and the current expression becomes exp(qV/mkT) Series resistance It further reduces the junction voltage drop and the current is better approximated by exp(qV/mkT) Thus, in general, with fitting parameter m between 1~2.

p-n junction used as LED or diode laser

lasing due to feedback

p-n junction used as photodiode