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第第四四章章 半导体光子学基础半导体光子学基础Chapter 4 Chapter 4 Fundamental for Fundamental for Semiconductor PhotonicsSemiconductor Photonics11 辐射复合和非辐射复合 1.1 辐射复合 1.2 非辐射复合2 半导体中的光发射 2.1 自发辐射 2.2 受激辐射3 光吸收 4.1 几种光吸收 4.2 带间跃迁光吸收 4.3 自由载流子光吸收4 阀值条件5 光增益谱2heterojunctions0.5m250m150mEmitting spotStripe contactDielectric(oxide)Key elementsDirect electrical injection by p-n junction:population inversion of gain mediumInternal optical waveguideMirrors to form an optical cavityThe optical gain is generated by a quantum well or quantum dot system激光的三个基本要素产生激光的物质粒子数反转谐振腔3量子阱激光器的能带图claddingcladdingcurrentwaveguideActive region LzEvnNPeVfEcQuantum wellWide-gap materials confine the light:guide the amplified optical mode.Quantum well confines carriers and provides the optical gain.Forward bias injects carriers into the well to invert the population.4波导中的模式增益Quantum-confined structures are much smaller than the optical wavelength:mode extends beyond the gain region.Modal Gain (G)is defined as the fractional increase in the energy in the whole mode per unit distance.WtotF(z)Quantum wellSlab waveguideGuided modeA(z)Lzxy5IntensityDistance,z,across waveguideCoreWellCladdingBandgap=Energy coupled to wellTotal energyModal gain=(material gain):G=g光学限制因子61.辐射复合和非辐射复合1.1 辐射复合a、带间复合b、浅杂质与带间的复合c、施主-受主复合d、激子复合e、其它辐射复合7辐射复合a、带间复合 半导体材料中导带底的电子同导带顶的空穴复合,其能量 大小为:(4-1)所以有:(4-2)式中和Eg的单位分别为m和eV。一般来说,载流子不完全位于导带底最低处和导带顶最高处,而是导带底和价带顶附近的载流子都会参与这种带间复合,因而这种带间复合的发射光谱具有一定的宽度。b、浅杂质与带间的复合 浅施主价带、导带浅受主间的载流子复合产生的辐射光为边缘发射,其光子能量总比禁带宽度小。8c、施主受主复合 施主能级上的电子同受主能级上的空穴复合产生辐射复合,其光子能量小于Eg,简称对复合。d、激子复合 在某些情况下,晶体中的电子和空穴可以稳定地结合在一起,形成一个中性“准粒子”,能在晶体中作为一个整体存在,这种“准粒子”就叫做激子。e、其它辐射复合 深能级复合、等电子陷阱复合。以等电子杂质替代晶格基质原子,因其原子大小和电负性等性质与基质原子原子不同,造成电子和空穴的束缚态,其作用好象陷阱,故通常称之为等电子陷阱。利用等电子陷阱复合,可以使间接带隙材料的发光效率得到提高。91.2 非辐射复合a、多声子跃迁 晶体中的电子与空穴复合时,可以激发多个声子,从而释放出其能量,由于发光半导体的通常在1eV以上,而一个声子的能量通常为0.06eV。因此,电子空穴复合可以通过杂质、缺陷产生多声子跃迁。多声子跃迁是一个几率很低的多级过程。b、俄歇复合 电子空穴复合时,把多余的能量传输给第三个载流子,使其在导带或价带内部激发,第三个载流子在能带的连续态中的多声子跃迁,并耗散其多余的能量,回至其初始的状态,这种复合过程称之为俄歇复合。因有多声子参与,俄歇复合是非辐射复合。c、表面复合和界面态复合 晶体表面的晶格中断,产生悬链,能够产生高浓度的深的和浅的能级,它们可以充当复合中心。表面复合是通过表面连续的跃迁进行的,因而是非辐射复合。10In real lasers there are current paths in addition to the spontaneous recombination current.The spontaneous current may include contributions for higher states which do not necessarily contribute to the gain.Other current paths:俄歇复合 Auger recombination(an intrinsic process)非辐射复合 Non-radiative recombination in the dots,via defect states浸润层/量子阱中的复合 Recombination in the wetting layer/quantum well漂移或扩散引起的载流子泄露 Carrier leakage by drift and/or diffusion.其他电流通道111VB2342C C C H1 2 3 4EVBCB134C H H LRequires conservation of energy and momentum.Energy is ultimately given up to the lattice.Diagrams show two processes in bulk and quantum well structures.俄歇复合ECBVB2C C C H1 2 3 412The probability of the CCCH process iswhere E4-Ec is the energy of state 4 above the conduction band minimum.The probability of an Auger transition is greatest when this difference is a minimum,subject to conservation requirements.This minimum defines a threshold energy ET which is proportional to Eg Since np,the Auger recombination rate isWhereandsoAuger important in narrower gap materials.俄歇复合(4-3)(4-3)(4-4)(4-4)(4-5)(4-5)(4-6)(4-6)13n0EfeEccJdiffEcc-Efen00n(x)n(E)/areaCarriers thermally activated above the confining barrier are extracted by diffusion down the concentration gradient in the cladding layer.扩散引起的载流子泄露14eJ=.minncontactJh=J/majmajority carriers=pehIf there is an electric field in the cladding layer the rate of extraction of carriers is increased by drift.Rate increases with increasing majority carrier current.Occurs in materials where p-cladding conductivity is low,eg AlGaInP,and wide gap nitrides漂移引起的载流子泄露152.半导体中的光发射自发辐射和受激辐射自发辐射半导体中电子空穴复合发光。自发辐射是随机过程,其波长、相位等特性上彼此互不关联:光谱较宽,光强较弱,相位不一致,没有偏振特性。受激辐射如果半导体中的光子诱发电子从导带向价带的跃迁,与价带顶部的空穴复合,发射出能量、相位等特性与入射光子相同的光子,这一过程称之为受激发射。受激辐射的光谱窄,相位一致,有偏振方向,输出功率大。16量子效率(4-7)(4-7)r为半导体中辐射复合过程的寿命nr为非辐射复合过程的寿命17费米分布 电子数 空穴数 (4-8)(4-9)Fn:导带中电子的准费米能级,Fp:价带中空穴的准费米能级.导带中能量为E处的电子数为:(4-10)相应地,价带中的能量为(E-h)处的空穴数为:(4-11)式中Nc(E)、Nv(E-h)分别为能量E处的电子能级密度和能量(E-h)处空穴能级密度。18发射和吸收的光子数 Bcv和Bvc为受激发射的爱因斯坦系数可以证明:Bcv=Bvc(4-12)(4-12)(4-13)(4-13)19要想获得激光,必须:dNedNa (4-14)将和的表达式代入上式,可得:fc(E)fv(E-h)(4-15)即:(4-16)所以:(4-17)即:(4-18)20设x处的光强为I(x),传输dz之后,因增益引起的光强增加量为:(4-19)式中g为增益系数。同样,因吸收引起的光强减少量为:(4-20)式中为吸收系数。光强为的光沿向传播了距离dz之后,其光强的总变化为:(4-21)21如果工作物质是均匀的,体内的g和处处一致,不随位置而变化,则有:(4-22)对上式积分,则有:(4-23)所以 (4-24)223.光吸收3.1 几种光吸收带间跃迁光吸收自由载流子光吸收激子吸收杂质能级吸收光双子吸收、喇曼散射233.2 带间跃迁光吸收直接带隙半导体中,假定导带底附近的关系呈抛物线关系:(4-254-25)式中mr为折合质量:(4-264-26)从量子力学的一级微扰理论出发,可以推导出带间跃迁的光子吸收系数为:(4-274-27)式中系数为:(4-284-28)上式中n为材料的折射率。24Pm0为在电子由第0态到第态跃迁的矩阵元。(4-29)0和m分别为初态和末态的波函数,*表示复数共轭。在直接带隙材料中,初态和末态的波矢相同:(4-30)此时上式积分不为0,也就是说竖直跃迁的几率最大。假定mr=me/2,(4-28)式可以简化为:(4-31)在这一讨论中,我们只考虑导带形状对光吸收的影响,实际半导体中,既要考虑杂质能级的光吸收,还要同时考虑导带和价带的能带形状对光吸收的影响。在间接带隙半导体中,光吸收过程还会又声子参与,因而需要采用二级微扰理论来处理光子吸收问题。253.3 自由载流子光吸收假定入射光的偏振方向为x轴,则入射光波的电场为:(4-32)在此电场的作用下,晶体中电荷的运动方程为:(4-33)式中m为电荷质量,mr(dx/dt)和m0 x分别为阻尼力项和恢复力项,r和0均为材料本身决定的参量。求解方程(4-33),可知电荷也以圆频率振动,其振幅x为:(4-34)可以看出,X0为复数。一方面,电荷的运动同入射光电场有相位差;另一方面,由于有 r 的存在,入射光会在体内吸收过程中发生损耗。26在半导体材料中,假定其复数折射率为:(4-35)则复数介电常数为:(4-36)从电学课程得知,电极化强度可表达为:对于一维运动来说,(4-37)将(4-34)的x0表达式代入(4-42)式,并与(4-36)式相结合,则的实部和虚部都可以表达为:(4-38)(4-39)27电磁波E在折射率为n的介质中传播时,如果传播方向为z,偏振方向为y,则其电场为:(4-40)式中 (4-41)为k复折射率的虚部,0为光波的自由空间波长。上式表明,吸收系数同折射率n的虚部直接相关,亦即同介电常数的虚部直接相关。由(4-39)式看出,如果入射光的角频率逼近晶体中线性振子的角频率0,则2nk逼近其最大值,此时出现强烈的光吸收.28对于半导体中的自由载流子电子和空穴来说,其振荡角频率0远比光波角频率0小得多,0上式可以简化为:(4-42)(4-43)上述公式的物理意义在于,半导体材料的折射率(实部)的平方同载流子浓度之间的关系存有一负号,也就是说,当半导体中注入高浓度载流子时,其折射率会因其浓度的增加而下降。由于注入载流子会引起激光器中不同区域的浓度差,因而引起折射率差,并由此构成波导层,对于激光器的工作模式与波长频谱特性都会产生影响。29如果晶体中吸收损耗足够小,在复折射率中,与n相比,k可以忽略不计,这相应于re,上式可进一步简化为:(4-49)该式表明,当半导体中具有自由载流子时,就会引起吸收损耗,并且其吸收系数同载流子浓度成正比。对于接近本征的半导体来说,电子浓度和空穴浓度大小接近,此时必须考虑它们两者对光吸收的影响,此时的光学吸收系数可表示为:(4-50)314.阈值条件设z处的强度为I0的光,在谐振腔中经过一个来回反射后,z处的光强为:(4-51)如果来回反射一次后,光强还维持不变,即,这就是阀值条件:(4-52)上式可以改写为:(4-53)该式表明,只有增益足以克服内部损耗和端面损耗时,才能受激发射,这一条件就为阈值条件。32在半导体激光器中,理论和实验都得到注入电流密度J同增益系数之间g的关系为:(4-54)式中为增益因子,为指数。将上两式合并可得阀值电流密度为:(4-55)腔长为无穷长时,(4-56)如果m=1,可求得、的表达式为:(4-57)的表达式表明,增大腔长L或增大端面发射率R1、R2,都可以减小内部吸收系数。相反,如果减小L或R1、R2,都会增加增益因子。不同结构的半导体激光器中,J的变化对Jth的增益起决定性作用。33Lc(RI1).e(G-i)xRI1RI2I2I1mirrormirrorLight outLight out受激发射条件Laser action can be sustained when the round-trip amplification replaces light lost from the laser cavity.Mirrors provide coherent optical feedback.The gain required to achieve this threshold is given by:I internal optical mode lossLc cavity lengthR mirror power reflectivityGth gain required in the lasing optical mode(Modal gain)受激发射行为:循环回路的增益(4-584-58)34LightoutputDrive currentCoherentemissionThreshold forlaser actionAt the threshold current the modal gain matches the losses.Local gain is a property of the gain medium,and depends on the degree of inversion of the material,which is related to the current.Modal gain is a property of the whole waveguide region which incorporates the gain medium.Threshold is a characteristic of the complete laser device.It is determined by the cavity length,mirror reflectivity etc,as well as the gain material.阈值电流5.光增益谱受激发射的光子数和其逆过程光吸收的光子数:(4-59)这两个过程的竞争的结果就会产生光吸收或者光发射,它们之差就为净发射粒子数:通常将光波通过粒子数反转区获得的增益表示为:(4-60)式中g为增益系数,它表示单位长度上光强所获得的增益。36将上式微分,则有:(4-61)单位体积内的光子能量密度为:(4-62)受激发射速率等于受激发射与受激吸收速率之差:(4-63)综合这些分析,最终可推导出增益系数表达为:(4-64)上式中引进了光强限制因子,它是考虑光场扩展出粒子数反转区之外造成光的损失而引进的一个参数。37在半导体材料中,遵守带间跃迁选择定则的增益系数g可以表达为:(4-65)式中red为折合态密度:(4-66)式中N/2为两个自旋方向之一的电子态的增量,Ec和Ev分别为导带中和价带中的能量增量,在此能量范围内有相同的状态数目以保证跃迁在一相同的值下进行。c和v分别为导带态密度和价带态密度。red38The modal gain per unit time is defined as the fractional rate of generation of energy(W)in the mode:where W is integrated over the volume of the mode.We relate dW/dt to the net stimulated emission rate into the mode:The modal gain per unit distance travelled is The material gain can also be calculated from the complex part of the susceptibility of the medium,by consideration of the polarisation in an optical field.This approach gives equivalent results and also provides phase information.(4-67)(4-69)(4-68)增益的定义39Quantum-confined structures are much smaller than the optical wavelength:mode extends beyond the gain region.Definition of gain refers to the energy in the whole mode.WtotF(z)Quantum wellSlab waveguideGuided modeA(z)Lzxy波导中的增益40In a waveguide geometry:A(z)is the vector potential.For harmonically varying fields F(z)is the envelope function of the confined states.fc fv are the occupancy factors of the upper and lower states.M is a matrix elementModal gain depends upon the gain medium and the waveguide geometry,in particular the overlap between the envelope functions and the optical field.This equation can be used for a potential of any shape.(4-70)(4-71)模式增益41模式增益和材料增益G=material gainxconfinement factor=gIf the optical field A(z)is uniform over the envelope functions(Awell)Material gain independent of properties of the waveguide.Calculation of the confinement factor usually assumes a well width can be defined.The local gain cannot be measured directly.P Blood IEEE Journ Quantum Electron 36(2000)354-362(4-72)(4-73)42限制因子IntensityDistanceCoreWellCladdingBandgap=Energy coupled to wellTotal energyAwellwmodIntensityDistanceCoreWellCladdingBandgap=Energy coupled to wellTotal energyAwellUseful to define an effective mode width by(4-74)43有效模式宽度If the optical field A(z)is uniform over the envelope functions(Awell)we can also writeand we can define a effective mode width so that the modal gain can be writtenDoes not depend explicitly on the well width.Can be applied to a potential of any shape provided its overall width is small.(4-75)(4-76)44半导体能带图和载流子分布45典型的增益谱0netGain(G-i)gainPhoton energy(h)Waveguide loss(i)Transparency point(hv=Ef)No material gain below band gap,hence“absorption”is due to the cavity losses alone.Can identify transparency point if losses are independent of photon energy.46Ga0.47In0.53As的增益谱和吸收谱 47不同注入电流下的增益谱48TE和TM模的放大自发发射谱(ASE)和增益谱00.20.40.60.8600630660690720TETMWavelength(nm)Amplified Spontaneous Emission(arb)-20-10010203040650670690710TETMWavelength(nm)G=g-i(cm-1)Compressively strained GaInP well49TE和TM模的增益和电流TMSame scale as TETEyGainSpontaneous emission from ASETM,x10670nm Compressively strained GaInP50增益谱的理论同实验曲线的比较Lines:theory(Weng Chow),points:data(P M Smowton).homogeneous broadening calculated from first principlesBand edge obtained from absorption spectra51习习 题题1.名词解释:带间复合、杂质能级复合、激子复合、表面复合。2.什么叫俄歇复合,俄歇复合速率与哪些因素有关?为什么在InGaAsP等长波长激光器中俄歇复合是影响其Jth、温度稳定性和可靠性的重要因素?3.试推导出激光器工作的阀值条件,并解释其物理意义。4.试推导出粒子数反转的条件,同时证明爱因斯坦系数为什么,并解释它们的物理意义。52习习 题题5.图示增益系数和吸收系数同注入的电流密度的关系,试说明、g随E、T、Jm变化的关系及其物 理 意 义,解 释gmax的变化的原因是什么?53
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