Concepts of Semiconductors

P- and N- Type Semiconductors

Doping of Semiconductors Pentavalent impurities (5 valence electrons) produce n-type semiconductors by contributing extra electrons. Trivalent impurities (3 valence electrons) produce p-type semiconductors by producing a "hole" or electron deficiency.

Chapter 1 Energy Bands and Carrier Concentration Energy Bands for Solid

Energy Bands Comments

Fermi level

Characterization by Fermi level

Fermi-Dirac Distribution

Distribution functions and comparison Fermi function Bose-Einstein distribution function Maxwell-Boltzmann distribution function

F. D. Distribution vs. Temperature                                                                                                                                                                                                                                                     

Fermi Function

Conduction Electron Population for Semiconductor

Electron Density of States

Concepts of Electron Density The Fermi function gives the probability of occupying an available energy state, but this must be factored by the number of available energy states to determine how many electrons would reach the conduction band.This density of states is the electron density of states, but there are differences in its implications for conductors and semiconductors. For the conductor, the density of states can be considered to start at the bottom of the valence band and fill up to the Fermi level, but since the conduction band and valence band overlap, the Fermi level is in the conduction band so there are plenty of electrons available for conduction. In the case of the semiconductor, the density of states is of the same form, but the density of states for conduction electrons begins at the top of the gap.

Example Calculate the effective densities of states in the conduction and valence bands of germanium, silicon and gallium arsenide at 300 K. Solution The effective density of states in the conduction band of germanium equals: where the effective mass for density of states was used (Appendix 3). Similarly one finds the effective densities for silicon and gallium arsenide and those of the valence band:                                                                                                                                                                                                                                                                                                                                                                                                                            Note that the effective density of states is temperature dependent and can be obtain from:                                                                                                             where Nc(300 K) is the effective density of states at 300 K.

Analyze of Carrier Concentration

Intrinsic carrier density versus temperature

Example Example Calculate the intrinsic carrier density in germanium, silicon and gallium arsenide at 300, 400, 500 and 600 K. Solution The intrinsic carrier density in silicon at 300 K equals:                                                                                                                                                                                                  Similarly one finds the intrinsic carrier density for germanium and gallium arsenide at different temperatures, yielding:                                                                                                                                                                                                   

Energy Level for Impurities in Silicon

Mass action law

PN Junction

Doped semiconductors The generation of free carriers requires not only that impurities are present, but also that the impurities give off electrons to the conduction band in which case they are called donors. If they give off holes to the valence band, they are called acceptors

Fermi energy                                                                          

Fermi energy and the free carrier density Graphical solution of the Fermi energy based on the general analysis. The value of the Fermi energy and the free carrier density is obtained at the intersection of the two curves, which represent the total positive and total negative charge in the semiconductor. Na equals 1016 cm-3 and Nd equals 1014 cm-3

Fermi energy of n-type and p-type silicon Fermi energy of n-type and p-type silicon, EF,n and EF,p, as a function of doping density at 100, 200, 300, 400 and 500 K. Shown are the conduction and valence band edges, Ec and Ev. The midgap energy is set to zero

Electron density vs. temperature Electron density and Fermi energy as a function of temperature in silicon with Nd = 1016 cm-3, Na = 1014 cm-3 and Ec - Ed = Ea - Ev = 50 meV. The activation energy at 70 K equals 27.4 meV

Chapter 2. Carrier Transport Phenomena Two transport mechanisms will be considered. The drift of carriers in an electric field and the diffusion of carriers due to a carrier density gradient will be discussed. Recombination mechanisms and the continuity equations are then combined into the diffusion equation. Finally, we present the drift-diffusion model, which combines all the essential elements discussed in this chapter

Intrinsic Semiconductor Both electrons and holes contribute to current flow in an intrinsic semiconductor

Random motion of carriers

Mobility Combining both relations yields an expression for the average particle velocity: We now consider only the steady state situation in which the particle has already accelerated and has reached a constant average velocity. Under such conditions, the velocity is proportional to the applied electric field and we Define the mobility as the velocity to field ratio: The mobility of a particle in a semiconductor is therefore expected to be large if its mass is small and the time between scattering events is large. The drift current, described by, can then be rewritten as a function of the mobility, yielding: Throughout this derivation, we simply considered the mass, m, of the particle. However in order to incorporate the effect of the periodic potential of the atoms in the semiconductor we must use the effective mass, m*, rather than the free particle mass:

Mobility vs. Scattering Scattering by lattice waves: Scattering by impurities: m µ T -s Germanium Silicon Gallium Arsenide Electron mobility µ T -1.7 µ T -2.4 µ T -1.0 Hole mobility µ T -2.3 µ T -2.2 µ T -2.1

Mobility vs. Doping

Drift Current

2.1.2 Resistivity Current density                                                                                                                                 (13) The conductivity due to electrons and holes is then obtained from:                                                                                  (14) The resistivity is defined as the inverse of the conductivity, namely:                                                                                       (15)

Mobility vs. Temperature

2.1.3 Hall Effect

Hall Effect

Current Mode

Applications of Hall Effect

3D Compass 90 degrees bending to obtain magnetic vector Lead frame design 90 degrees bending to obtain magnetic vector Solid state compass component Prototype 3D packaging

Compass Watches

Magnetoresistance ■ Magnetoresistance is an effect in which the resistance of magnetic film, or a multilayer film, varies as a function of applied magnetic field ■ The magnetoresistance curve is hysteretic, i.e. the value of the resistance depends on whether we are ramping the field up or down. When we measure magnetoresistance we ramp the applied field to a value high enough to achieve magnetization saturation, then gradually scan the applied field all the way to negative saturation, and back. When we plot the resistance as a function of applied field we obtain a curve similar to the "M" in our logo in the upper left corner of this page

磁阻簡介     磁阻現象﹝導體在外加磁場下電阻改變的現象﹞大概在100年前就已經發現了。當一導體在一外在磁場作用下通以一電流，由於移動的電子受勞倫茲力﹝Lorentz force﹞的作用，其電阻值會改變，而且很明顯地，此改變值在電流平行磁場或電流垂直磁場下是不一樣的。當此導體為磁性材料時，電阻的改變會更為明顯，例如塊材的Ni80Fe20，其電阻變化率﹝dR/R﹞最大約為5% ~ 6% 。這和電流與磁場方向有關的電阻變化稱之為異向性磁阻﹝anisotropy magnetoresistance，簡稱AMR﹞。     在1971年， Hunt[1] 首次提出利用AMR原理應用在磁紀錄讀頭的觀念，並且在1985年IBM真正將它應用在磁帶機上，而1991年9月IBM首次將它用在硬碟機的讀頭。AMR讀頭與傳統感應式磁頭的讀取原理完全不同，其結構為厚度約10~30nm 的Ni80Fe20薄膜，dR/R為2% 。利用儲存媒體產生的磁場作用到Ni80Fe20薄膜上產生電阻的變化，來感應位元的磁矩方向。也即，其訊號僅與磁片的磁通量變化有關，而非傳統感應式磁頭以磁通量的時間變化率所感應的電動勢作為輸出訊號。靈敏度﹝dR/dH﹞約為傳統磁頭的3~4倍，並且提高了訊號雜訊比，在當時的應用上極具潛力，是高密度儲存裝置中不可或缺的重要發現。

磁阻電流感測器應用 磁阻電流感測器改進馬達驅動器性能 更高的裝置 磁頭。 變化關係，因此磁阻器不能夠偵測磁場極性 ■ 超強磁阻（GMR）是一種靈敏度比標準磁阻器 更高的裝置 ■ 應用範圍從虛擬實境位置感應器到硬碟的讀寫 磁頭。 ■ 磁阻和超強磁阻元件的磁場和電阻為拋物線的 變化關係，因此磁阻器不能夠偵測磁場極性

巨磁阻效應 1988年，Baibich等人發現在低溫下， [Fe/Cr]  多層薄膜系統隨外磁場的增加，其dR/R可達50%[2]。由於其電阻變化率遠大於鐵磁性材料的AMR，因此稱之為巨磁阻﹝giant magnetoresistance，簡稱GMR﹞。起初認為只有 [Fe/Cr]   多層薄膜系統具有GMR效應，後來發現其普遍存在於[ 鐵磁性 ﹝Fe,Co,Ni﹞ / 非鐵磁 ﹝Cr,Cu,Ag,Au﹞ ] 的多層膜系統﹝每層後約1~3 nm﹞，其中 [Co/Cu] 系統的電阻變化率更可在室溫下高達65% 。簡單的來討論GMR效應的原理，首先討論電子於多層鐵磁性薄膜之中的傳導現象。若多層膜結構為兩層鐵磁性(FM)薄膜之間以非鐵磁性薄膜(NM)隔開，如圖一所示，當非鐵磁性薄膜的厚度控制在某一範圍時，上下兩鐵磁性薄膜層經由中間非鐵磁性層會交互的作用，偶合成反平行的磁化方向。若加入足夠大的外磁場，則會破壞鐵磁性薄膜之間的偶合作用，使之成為平行的磁化方向。當電子進入鐵磁性層時，由於電子在磁性物質中散射的機率與磁化方向有關─稱為差異性自旋散射﹝differential spin scattering﹞，因此若磁化方向與電子自旋方向相同則散射機率較小，也就是電阻較小；反之若是磁化方向和電子自旋方向反平行時則散射機率較大，也就是電阻較大。當磁場等於零時，反平行的磁化方向使得大部分的電子在通過多層膜時，皆會受到散射，因此呈現高電阻態。而當磁場達到飽和場時，即鐵磁性層的磁化方向為平行時，有部分的電子，其自旋方向與磁化方向一致，因此可順利地通過多層膜而不被散射，呈現低電阻態。我們可藉由外加磁場改變多層膜系統中鐵磁性層的磁化方向組態，使之電阻較未加磁場下來的小。雖然 [鐵磁性 / 非鐵磁性] 多層膜系統有很好的磁阻變化表現，但由於多層膜系統成長不易，並且飽和磁場 ﹝使得鐵磁性層的磁化方向為平行排列所需加的磁場﹞ 太大 ﹝約數千個Oe﹞ ，因此工業應用上並不具實用性。然而隨著磁紀錄媒體產業的發展，AMR讀頭的密度很快就到達極限。在下一代高密度硬碟中，我們需要有更高靈敏度的讀頭，也就是高的磁阻變化以及小的飽和場。

Giant MagnetoResistive Head Giant MagnetoResistive Head (GMR Head)-巨磁阻讀寫頭 1980年代，兩位歐洲發明家發現多層次覆蓋累積的金屬片，能產生重要的磁致電阻效應。1997年，IBM根據先前發現之磁阻效應進行研發，開發出更方便、更具經濟效益的GMR(Giant magnetoresistive)讀寫頭。這項技術使硬碟機之轉盤每平方英吋可儲存多達100GB的資料，一轉盤每平方英吋有150,000個以上之軌跡，大大提升了電腦硬碟機之驅動速率。大型GMR的技術讓複雜的多媒體、數位影音資料能在單一電腦中同時應用，堪稱磁碟密度科技的大躍進。

Superconducting Quantum Interference Device Electronic Materials A Superconducting Quantum Interference Device (SQUID) is the most sensitive sensor known to science. There are several ways to build a SQUID, but here we restrict ourselves to the so called DC-SQUID which has two equal Josephson junctions connected in parallel, as indicated in the figure. As already mentioned, each of them carries a current depending on the phase difference it feels. If the phase differences are the same, their currents add up and the total current has a maximum (constructive interference). However, if the phase differences are opposite to each other, the currents cancel each other and the total current is zero (destructive interference). The interference can be controlled by curling the pair wave of the superconductors in between the junctions. This can be done by a small magnetic flux inside the ring. Tiny changes in the flux can already change the interference from constructive to destructive, and this is readily observable by total current flowing through the device. This makes this device the most sensitive sensor known. As an example, the SQUID can be used to detect fields generated by brain signals which are less than a billionth of the earth's field. Such SQUID sensors are used whenever tiny signals have to be measured, e.g. in medicine technology, non destructive testing or basic research. MOEMS

General solutions for simplified differential equations

Current Densities

Example Example Electrons in undoped gallium arsenide have a mobility of 8,800 cm2/V-s. Calculate the average time between collisions. Calculate the distance traveled between two collisions (also called the mean free path). Use an average velocity of 107 cm/s. Solution The collision time, tc, is obtained from:                                                                                                                                                                                                  where the mobility was first converted in MKS units. The mean free path, l, equals:                                                                                                                                                   

2.2 Carrier Diffusion

Diffusion Current

Einstein Relation

Total Current