Nanoelectronics (unvollständig)

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Nanoelectronics

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Wichtige Begriffe:

dispersion Verteilung lattice Kristallgitter impurities Fremdstoffe to scatter streuen

Abk¨urzungen:

CVD chemical vapour deposition CNT carbon nanotube DoS Density of States

HOPG Highly Ordered Pyrolytic Graphite PMMA Polymethylmethacrylat (Acrylglas) STM Scanning tunneling microscope

1. Moores Law – scaling

1. Transistormaterial: Germanium

Transistor scaling 22nm between drain and source of a MOSFET scaling cant continue indefinitely Against Moores Law: the rising costs of fabrication, the limits of lithography, and the size of the transistor. Advan- tages of scaling: smaller, cheaper, and faster and to consume less power.

2. Quantum mechanics

Klassische Bewegungsgleichung:m^d2r(t) dt2 =F Classical wave equation:

1 c²

∂²Ψ

∂t² − n X

i=1

∂²Ψ

∂x²_i = 0 c=λf

ω= 2πf k=2π

λ Waves

behave as particles. Electrons and photons are both, particles and waves.

Electron Orbits:mvr=n~ nλ= 2πr Bohr atom model: E_n≈ −13.6Z²

n²eV Z: count of protons De Broglie Wavelength:h=pλ p=~k h= 2π~ Uncertainty principle: ∆x·∆px≥~

2 ∆E·∆t≥^~ 2 transistor dimension:L_crit≈q ^h

2m∗Eb

≈4 nm

withm^∗≈0.19m0 E_b= 0.5eV

2.1. Schroedinger Equation

−~² 2m

∇²+V(r, t)

!

Ψ(r, t) = i~

∂

∂tΨ(r, t) (1) Potential energyV(r, t)∈R (for HydrogenatomV(r) =^−e_r²) HamiltonianHˆ=

−~2

2m∇²+V(r, t)

ProbabilitydensityP(r, t) = Ψ(r, t)·Ψ^∗(r, t) =|Ψ²| Normalized:´

|Ψ(r)|²dr= 1

2.1.1 time-independent Schroedinger equation ifV(r, t)is time-independent:

Ψ(r, t) = Ψ(r) exp_iEt

~

⇒ HΨ(r) =ˆ EΨ(r)

1D Confinement (infinite Quantum Well):

Ψn(x) =q ₂

Lxsin^√_2mE

~ x En=_2m^~ k²_n=^~2_2mL^π2n2 2D ConfinementΨ(x, y) =q ₄

LxLysin(k_xx)·sin(k_yy) E_n=_2m^~ (k_x²k²_y)

δ-D Confinement wirh

i=x, y, ..., δ Ψ(r) =q _2δ

QLi δ Q i=1

sin(k_i·i) En=_2m^~ (k_x²k²_y)

Analytical solutions are only possible for the infinite quantum well

2.2. Quantenphysik

E_{P h}=f·h=~·ω=^hc_λ λ·p=h p=~k=^hk_λ ~=_2π^h k=^2π_λ

2.3. Phonons

are quasiparticles to describe modes of vibrations of elastic structures of interacting particles. there are acoustic and optical phonons.

3. Semiconductors

3.1. bandstructure

FermienergieF_E: H¨ochste Energie eines Elektrons beiT = 0KIsola- tor: große Bandl¨uckeE_G >3eV Halbleiter: kleine Bandl¨ucke1eV <

E_G<3eV kann durch thermische Energie ¨uberwunden werden Materials in columns:

IV: Si,Ge, III-V(GaAs, InP, GaN(BluRay), InSB),II-VI(CdSe, CsTe) IV- VI(PbS,PbSe)

Silicon in crystal structure: 5 per Cube

Chemical band structure: energylevels of diffrent atoms moving close together

At finite temperature some electrons can move around. n ∝ exp(T b_gap)

At300K:n= 1.5×10¹0cm³

doping with donors(P,As) or acceptors(B,In) to lower the energy for emission or capture an electron

atoms:10²3per cm³, dopants:10¹5per cm³ E_kin=_2m^p2 = ^~_2m^{2k2 d2E}

dp2 =_m¹ Effektiv mass:_m∗¹

ef f

=¹

~·^d2E dk2 Resistorequation:R_{M at}=ρ_{M atwt}^l conductivity:σ=¹_ρ =qµ_nn_i resistivity:ρ

uncrtainity for electron:∆x≥^0.5·10_∆v⁻⁴ vsatfor Si:2·10^7cm_s

I_DS= ¹₂µCoxW

L(V_DD−V_T)² P=V_DD² C_oxf_max

4. Transistors

I_D= ¹₂µ_nC_ox^{0 W}_L ·(V_GS−V_th)² µ_n≈250·10^−4m_{V s}²,µ_p≈100·10^−4m_{V s}² P_cap=α₀₁f C_oxV_DD²

fmax= ^Isat VDD Cox

4.1. scaling with factor

S <1 reduce areaA=W·L A⁰=A·S² increase speedτ= ^L_v τ⁰=τ·S reduce powerP=V Iτ P⁰=P·S³

Transistorscaling in nm: 90(2003), 65(2005), 45(2007), 32(2009), 22(2011)

4.1.1 Problem of scaling 1. Tunneling across the oxide 2. Need for new lithographic techniques 3. Parasitic effects due to inteconnects 4. Melting interconnects due to voids 5. High field and breakdown effects

⇒new materials, processes and technologies needed!

High-K Material (high dielectricε) as Gate isolator: ⇒1,6·C_G, 0.01·I_leak

Example Intels 45nm MOSFET: High-K with silicon gaten: Problems: un- even

Form: Normalgate, Dualgate, Trigate

Best: Surrounding Gate: CNT – high-K – metal-gate

4.2. Silicon Nanowire

Fabrication: growth on a gold(Au) particle

4.3. GaN - Transistors

Why GaN?

• Wide bandgaps of GaN and AlGaN (high breakdown volt.)

• high drift velocity(hf)

• strong piezoelectric effekt

• High temperature operation

HEMT-Transistor: Two substrate materials, doped and undoped

⇒electrons move on a 2D-Sheet Cut-Off-Frquencyf_T= ^vsat

2πLg:gm= 1(no amplification anymore) Oxide-CapacitanceCox= ^ε_tox^0εr

T-Gate: smooth electric field in the channel.

4.4. Quantum Wire

Ideal: just one subband in two dimensions But for good conductan- ce(mobility, drift velocity) one need20nmFabrication Methods: Stressor, Etching, Ion implantation, Vicinal Growth

Split Gate Transistor: 1D tunnel in the gate between source and drain.

electron wave transistor:

5. Graphene

2D Network of 3D Carbon Atoms.

Stacked Layers of Graphene form Graphite.

applications: seperation membranes, capacitors

a

₁

a

₂

y x

b d

b≈0.14 nm d= 2b≈2.8 nm a₁= ³₂bx+

√ 3 2 by=

√ 3 2 ax+¹₂ay a₂= ³₂bx−

√ 3 2 by=

√ 3 2 ax−¹₂ay a₁

= a₂

=a=√

3b≈0.246 nm kxk=

y

= 3b

5.1. Properties

thinnest material sheet imagineable extremly strong (5 times stronger than steel)

semimetall: better conduction than metal, can switched ON and OFF very light, good head conductor size of one cell: edged≈0.14nm, edge2edgea=√

3d

5.2. production

•Exfoliated Graphene: peeling HOPG with foil. Good for scienece not for manufacturing

•Epitaxial growth: silicon carbide(SiC) is heated (>1100^◦C) to re- duce it to graphene.

5.3. Carbon-Nano-Tubes CNT

Propertys: diameter:d≈10 nm

Application: wires, transitors, sensors, Molecular tweezers Single Walled and Multiwalled: SWCNT: single layer of graphite(graphene) rolled up as cylinder

Kind of curls: zig-zag(n,0), armchair(n, n), quiral(n, m) chiral vector (tube circumfence):C_h=na₁+ma₂

translation vectorT= [(2m+n)a₁−(2n+m)a₂]/gcd(n, m) Tube-Diameter: d_T = kchk

π = ^a_πp

n²+nm+m² with a= 0.246nm

Kind of Nanotube:

(metalic if(n−m)/3∈N0 semiconductor else

Bandenergy in dependency ofk:

E(k) =ε₀±t s

1 + 4 cos √

3akx 2

cos_aky

2

+ 4 cos²_aky 2

Periodic Boundary Conditions:C^>_h·k= 2πn

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