EEM214 Elektronik I: Ders 2 – Taşıyıcı Modelleme (Carrier Modelling)

Bu yazı, Başkent Üniversitesi EEM214 Elektronik I dersinin Lecture 2: Carrier Modelling sunumunun içeriğidir. Sunum dersin öğretim elemanı Yrd. Doç. Dr. Derya Yılmaz'ın ders slaytları arasında yer alıyor; son düzenleme tarihi Şubat 2013. Konu, diyot denkleminin arkasındaki yarıiletken fiziği: akımı belirleyen taşıyıcıların sayısı ve dağılımı. Slaytlar İngilizce olduğu için metinler de İngilizce aktarıldı, formüller düz metin olarak yazıldı; bant diyagramları sayfa görselleriyle verildi. Sunumun orijinal dosyası yazının sonunda indirilebilir.

Semiconductor physics

How do we go from a p-n junction picture to the diode equation?

i_D = I_S·(e^(v_D/ηV_T) − 1)

What determines current?

• Number of charge carriers – state and carrier distribution
• Speed of charge carriers – mobility, drift, diffusion

Crystalline silicon

• 3D lattice: each Si atom bonded to 4 neighbors (tetrahedral coordination)
• Density: 5×10²² atoms/cm³
• Distance between atoms: 0.235 nm
• Q: If Si transistor gate is 15 nm wide, how many atoms wide is the gate?

Review of electrostatics

• Charge – basic unit: Coulomb; q (charge of an electron) = 1.6×10⁻¹⁹ C
• Current – 1 A = 1 C / 1 sec
• Electric field: force per unit charge, |ε on Q₂| = Q₁ / (4π·ε₀·r²)
• Energy = Work: force × distance – basic unit: Joules
• Potential = Work per unit charge – basic unit: 1 Volt = 1 J / 1 C

Moving the negative charge Q₂ away from the positive charge Q₁ means increasing energy, lower potential.

Energy levels

For the hydrogen atom:

E_H = −13.6/n² eV, n = 1, 2, 3, …

1 eV = 1 electron-volt = 1.6×10⁻¹⁹ joules. There are actually sublevels within these levels.

Rules for electrons

• Tend to settle to lowest levels.
• Need extra energy (light or thermal energy) to move to higher levels.
• Can move to lower or equal energy levels if not filled. Give up energy when moving to lower levels.

Periodic table and the isolated Si atom

The Roman numeral at the top of each column indicates the number of valence electrons. The isolated Si atom (+14) has 2 electrons in n = 1, 8 electrons in n = 2 and 4 electrons in n = 3, where 8 electrons are allowed (4 states empty).

Covalent bonding and energy bands

Each Si atom shares 4 electrons with its 4 neighbors, filling in 4 previously empty n = 3 levels (covalent bond).

As many Si atoms are brought closer together, the energy levels for every atom shift: at the crystal spacing d = 0.235 nm the n = 3 level splits into a filled band (E_bottom … E_v) and an empty band (E_c … E_top), separated by the band gap E_G.

Crystalline Si (N atoms): 4N allowed states in the conduction band (mostly empty, between E_c and E_top) and 4N allowed states in the valence band (mostly filled, between E_bottom and E_v). In the simplified version only E_c and E_v are drawn.

Si in 2D, electrons and holes

In the 2D simplification each line represents a shared electron and each circle represents a Si nucleus and core electrons. Thermal generation: thermal energy breaks a bond, an electron rises to E_c and leaves a hole at E_v.

• Holes "move" when a shared electron moves to fill it in.
• Electrons move faster than holes.

In the energy band diagram: no carriers means an empty E_c and a completely filled E_v; an excited electron in the conduction band leaves a hole in the valence band.

Energy bands for different materials

• Insulator: wide gap, E_G ≈ 8 eV (SiO₂), E_G ≈ 5 eV (Diamond)
• Semiconductor: E_G = 1.42 eV (GaAs), 1.12 eV (Si), 0.66 eV (Ge) at room temperature
• Metal: E_c and E_v overlap, or the bands are partially filled

Carriers and electrical properties

• More carriers = higher conductivity
• Can also increase the number of carriers by doping (introducing impurities)
• Purity of Si used in chips: 99.9999999%! (less than 1 non-Si atom in 10⁹ Si atoms)
• Pure Si: intrinsic semiconductor; doped Si: extrinsic semiconductor

Intrinsic carrier concentration

Thermal excitation creates electron-hole pairs: n = number of free electrons/cm³, p = number of free holes/cm³. Intrinsic carrier concentration at room temperature:

For Si at room temperature, less than 1 bond in 10¹³ is broken (assuming 5×10²² atoms/cm³).

Doping

Group III elements (B, Al, Ga, In) are acceptor dopants, group V elements (N, P, As, Sb) are donor dopants. 1 in 10³ to 1 in 10⁸ dopants are typically introduced to alter Si's electrical properties.

• Donor (electron-increasing dopant): e.g. As, |E_B| = 0.054 eV. N_D = donor dopant density. n ≈ N_D for typical dopings at room temperature.
• Acceptor (hole-increasing dopant): e.g. B, |E_B| = 0.045 eV. N_A = acceptor dopant density. p ≈ N_A for typical dopings at room temperature.

Carrier distribution

In the intrinsic material E_F lies near the middle of the gap; in n-type material it moves toward E_c (more electrons), in p-type toward E_v (more holes).

Fermi level

Carrier density and E_F are related – need one to solve the other:

• n = n_i·e^((E_F − E_i)/kT)
• p = n_i·e^((E_i − E_F)/kT)
• E_i ≈ (E_c + E_v)/2

E_F = Fermi energy/level, k = Boltzmann constant = 8.617×10⁻⁵ eV/K, T = temperature in Kelvin. Q: Show that np = n_i².

Assumes a non-degenerate semiconductor: E_F not too close to E_c or E_v (more than 3kT away).

Computing carrier density in extrinsic semiconductors

• Previous equations apply to both intrinsic and extrinsic semiconductors.
• Charge neutrality relationship (assuming total ionization):

p − n + N_D − N_A = 0, where N_D = number of donors/cm³ and N_A = number of acceptors/cm³.

• Carrier concentrations, assuming non-degenerate and N_D or N_A ≫ n_i: