⇦Chinese Physics Letters, 2021, Vol. 38, No. 8, Article code 087103Express Letter Universal Theory and Basic Rules of Strain-Dependent Doping Behaviors in Semiconductors Xiaolan Yan (晏晓岚)1, Pei Li (李培)1, Su-Huai Wei (魏苏淮)1,2*, and Bing Huang (黄兵)1,2* Affiliations 1Beijing Computational Science Research Center, Beijing 100193, China 2Department of Physics, Beijing Normal University, Beijing 100875, China Received 10 June 2021; accepted 15 July 2021; published online 23 July 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11634003, 11991060, and 12088101), and NSAF (Grant No. U1930402).
*Corresponding authors. Email: suhuaiwei@csrc.ac.cn; bing.huang@csrc.ac.cn Citation Text: Yan X L, Li P, Wei S H, and Huang B 2021 Chin. Phys. Lett. 38 087103    Abstract Enhancing the dopability of semiconductors via strain engineering is critical to improving their functionalities, which is, however, largely hindered by the lack of basic rules. In this study, for the first time, we develop a universal theory to understand the total energy changes of point defects (or dopants) with different charge states under strains, which can exhibit either parabolic or superlinear behaviors, determined by the size of defect-induced local volume change ($\Delta V$). In general, $\Delta V$ increases (decreases) when an electron is added (removed) to (from) the defect site. Consequently, in terms of this universal theory, three basic rules can be obtained to further understand or predict the diverse strain-dependent doping behaviors, i.e., defect formation energies, charge-state transition levels, and Fermi pinning levels, in semiconductors. These three basic rules could be generally applied to improve the doping performance or overcome the doping bottlenecks in various semiconductors. DOI:10.1088/0256-307X/38/8/087103 © 2021 Chinese Physics Society Article Text The application of semiconductors in electronic and optoelectronic devices critically depends on their dopability. Generally, there are three important factors that can basically limit the dopability in semiconductors: (i) The desirable defects or dopants (generally denoted as defects hereafter) have limited solubility, i.e., their formation energies ($H_{\rm f}^{{D},{q}}$) are too high.[1,2] (ii) The desirable defects have sufficient solubility, but they are too difficult to be ionized at room temperature, i.e., their charge-state transition levels ($\varepsilon^{q/q'}$) are too deep inside the bandgap.[1,2] (iii) The desirable defects have low $H_{\rm f}^{{D},{q}}$ and shallow $\varepsilon^{q/q'}$, unfortunately, the intrinsic compensating defects can easily form and pin the Fermi-level position ($E_{\rm pin}$) deep inside the bandgap, preventing the further increase of desired free carriers.[1,2] Comparing (i) and (ii), (iii) is the most difficult one to be overcome as it belongs to the intrinsic property of semiconductors. In the past decades, strain engineering is widely adopted to enhance the performances of semiconductors, e.g., optimize the electronic structures,[3–9] improve phase stabilities,[10,11] generate spin currents,[12] control carrier excitations or transports,[13–15] and modulate ion diffusion paths.[16,17] It is unsurprising that strain engineering has also been used to tune the doping performances in semiconductors.[18–30] Meanwhile, the strain effects are unavoidable during the growth or operation of various semiconductors.[23,29,31] However, it is rather puzzled that the strain-induced changes of doping behaviors for different point defects in different semiconductors are dramatically different.[18,20–26,28,30] Unfortunately, a universal theory that can intuitively understand all these diverse doping behaviors in different systems is still lacking, which prevents us to establish the basic rules to overcome the doping bottlenecks in semiconductors. In this Letter, we develop a simple but universal theory for understanding the strain-dependent total energy changes of isolated point defects ($\Delta E_{\rm t}^{{D},{q}}$) under different charge states, which is critically determined by the defect-induced local volume change ($\Delta V$). Depending on the size of $\Delta V$, the $\Delta E_{\rm t}^{{D},{q}}$ of a defect can exhibit either parabolic ($\Delta V \sim$ 0) or monotonic ($dE_{\rm t}^{{D},{q}}/dV \sim -\Delta V$) dependences. Noticeably, the $\Delta V$ is charge-state $q$-dependent, which increases (decreases) for more negatively (positively) charged defects. Based on this universal theory of $\Delta E_{\rm t}^{{D},{q}}$, we can establish three basic rules on understanding the strain-dependent $H_{\rm f}^{{D},{q}}$, $\varepsilon^{q/q'}$, and $E_{\rm pin}$, which may consequently be applied to overcome the above (i)–(iii) doping problems in various semiconductor systems. A Universal Theory on $\Delta {E} _{\rm t}^{{D},{q}}$. For a system without a defect, its total energy $E_{\rm t}^{\rm host}(V)$ as a function of volume $V$, to the lowest order, follows $E_{\rm t}^{\rm host}(V)=\alpha_{0}(V-V_{0})^{2}$, where $V_{0}$ is the equilibrium volume of host lattice and $\alpha_{0}=\frac{1}{2}B_{0}/V_{0}$, with $B_{0}$ being the bulk modulus. Similarly, for a system with a point defect in the $q$ charge state, its total energy $E_{\rm t}^{{D},{q}}(V)$ follows $E_{\rm t}^{{D},{q}}(V) = (\alpha_{0}+\Delta \alpha)[V-(V_{0}+\Delta V)]^{2}$, where $\Delta \alpha$ and $\Delta V$ are the changes of $\alpha_{0}$ and $V_{0}$ induced by the defect, respectively.[22] Ignoring the high order terms, the total energy changes $\Delta E_{\rm t}^{{D},{q}}$ induced by the defect in the charge-state $q$ as a function of $V$ can be derived as $$\begin{alignat}{1} \Delta E_{\rm t}^{D,q}(V)={}&E_{\rm t}^{D,q}(V)-E_{\rm t}^{\rm host}(V)\\ ={}&-2\alpha_{0}\Delta V(V-V_{0})+\Delta \alpha (V-V_{0})^{2}.~~~~ \tag {1} \end{alignat} $$ Obviously, $\Delta E_{\rm t}^{{D},{q}}(V)$ is determined by the two terms that are mainly associated with $\Delta \alpha$ and $\Delta V$ in Eq. (1). The $\Delta \alpha$ is usually negligible, especially for substitutional defects where the chemical and size differences are small.[32] The $\Delta V$ depends on the size difference between the dopant and the host element, therefore, is noticeable in most cases.[21,22,26] In these cases, $\Delta E_{\rm t}^{{D},{q}}(V)$ is largely determined by the first term of Eq. (1), giving rise to a linear dependence on $V$ [Fig. 1(a)]. However, if the defect-induced $\Delta V$ is not significant, the high-order second term in Eq. (1) could become dominant, giving rise to a parabolic change of $\Delta E_{\rm t}^{{D},{q}}(V)$ under strain [Fig. 1(b)].[23–25,30] Importantly, the $\Delta V$ can be rather sensitive to the different charge states of a defect, i.e., the $\Delta V$ of negatively (positively) charged state is usually larger (smaller) than that of the corresponding neutral one for a defect, due to the increased (decreased) electron occupation. Therefore, it is expected that dramatically different $q$-dependent $\Delta E_{\rm t}^{{D},{q}}(V)$ under strains could exist even for the same defect in a semiconductor.
cpl-38-8-087103-fig1.png
Fig. 1. Schematic plotting of the universal theory of $\Delta {V}$-dependent $\Delta {E} _{\rm t}^{{D},{q}} ({V})$. $E_{\rm t}^{{D},{q}}(V)$ and $E_{\rm t}^{\rm host}(V)$ are the total energies as a function of volume $V$ for a host with and without a defect, respectively. $\Delta E_{\rm t}^{{D},{q}}(V)$ is indicated by the black arrows, which are mostly determined by the (a) first and (b) second terms of Eq. (1) for the large and small $\Delta V$ cases, respectively.
The defect formation energy $H_{\rm f}^{{D},{q}}(V, E_{\rm F}$) as a function of $V$ can be written as[2,33–35] $$ H_{\rm f}^{D,q}(V,E_{\rm F})=\Delta E_{\rm t}^{D,q}(V)+\sum {n_{i}\mu_{i}+qE_{\rm F}},~~ \tag {2} $$ where $E_{\rm F}$ is the absolute Fermi energy in the bandgap, $\mu_{i}$ is the chemical potential referenced to the total energy of elements in its lowest energy bulk form. Therefore, $H_{\rm f}^{{D},{q}}(V)$ follows the same trend as $\Delta E_{\rm t}^{{D},{q}}(V)$ as a function of $V$ at a given $E_{\rm F}$. The $\varepsilon ^{q/q'}(V)$ referenced to the valence band maximum (VBM) energy $E_{\rm VBM}(V)$ is the Fermi energy at which the same defect $D$ with different charge states $q$ and $q'$ have the same formation energy, therefore, it can be described as $$\begin{align} \varepsilon^{q/q'}(V) ={}&\Big[\frac{\Delta E_{\rm t}^{D,q}(V)-\Delta E_{\rm t}^{D,q'}(V)}{q'-q}-E_{\rm VBM}(V_{0})\Big]\\ &-\Delta E_{\rm VBM}(V).~~ \tag {3} \end{align} $$ As shown in Eq. (3), the $\varepsilon ^{q/q'}(V)$ includes two terms: the first term represents the absolute value of $\varepsilon ^{q/q'}(V)$ [denoted as a-$\varepsilon ^{q/q'}(V)$] referenced to the VBM energy of the unstrained host, and the second term is the $V$-dependent shift of the VBM energy, $\Delta E_{\rm VBM}(V)$. It should be noticed that in a semiconductor, the $\Delta E_{\rm VBM}(V)$ depends on its absolute volume deformation potentials (AVDP). For most common semiconductors with a dominant bonding VBM state, the AVDP is positive,[36] i.e., the $\Delta E_{\rm VBM}(V)$ will increase (decrease) under tensile (compressive) strains. Therefore, the understanding of $V$-dependent a-$\varepsilon ^{q/q'}(V)$ is the key for understanding the trend of $\varepsilon ^{q/q'}(V)$. Based on the above discussions, in the following, we will derive three basic rules of strain-dependent doping behaviors in semiconductors with numerical verifications. Rule I: Strain-Dependent $H_{\rm f}^{D,q}$. Taking GaN as a typical example, we have employed the first-principles calculations (see the computational methods in the Supplementary Material) to study the uniform strain $\eta$ on the change of $H_{\rm f}^{{D},{q}}$ [$\Delta H_{\rm f}^{{D},{q}}(\eta)$] for different point defects in this system, in order to verify the universal theory we developed. Firstly, we consider N vacancy (V$_{\rm N}$), the dominant intrinsic defect in GaN.[37,38] As shown in Fig. 2(a), our calculation shows that a rather small value of $\Delta V$ ($\Delta V \sim -1.87$ Å$^{3}$/V$_{\rm N}^{0}$) exists for V$_{\rm N}$ under its neutral charge state (V$_{\rm N}^{0}$). According to Fig. 1(b), the $\Delta H_{\rm f}^{{D},{q}}(\eta)$ of V$_{\rm N}^{0}$ will be mostly determined by the second term of Eq. (1) and exhibit a parabolic dependence of $\eta$. Indeed, as shown in Fig. 2(b), the calculated $H_{\rm f}^{{D},{q}}(\eta)$ of V$_{\rm N}^{0}$ tends to decrease under both compressive and tensile $\eta$. The negative sign of $\Delta \alpha \sim -0.06$ suggests that the local bulk modulus is reduced with the formation of V$_{\rm N}$, as is expected. Interestingly, when V$_{\rm N}^{0}$ is converted to its $+$1 state (V$_{\rm N}^{+1}$), its local volume is significantly reduced to $\Delta V \sim -7.3$ Å$^{3}$/V$_{\rm N}^{+1}$ due to reduced charge occupation, giving rise to a large left-shift of its energy curve in Fig. 2(a). Consequently, differing from V$_{\rm N}^{0}$, it is expected that the $\Delta H_{\rm f}^{{D},{q}}(\eta)$ of V$_{\rm N}^{+1}$ (at a given $E_{\rm F}$) is now mainly determined by the first term of Eq. (1) [Fig. 1(a)] and exhibits a linear dependence of $\eta$. Again, as confirmed in Fig. 2(b), the $H_{\rm f}^{{D},{q}}(\eta)$ of V$_{\rm N}^{+1}$ linearly decreases (increases) under the compressive (tensile) $\eta$. In addition, compared to V$_{\rm N}^{+1}$, the linear slope of $\Delta H_{\rm f}^{{D},{q}}(\eta)$ for V$_{\rm N}^{+3}$ is only slightly larger than that for V$_{\rm N}^{+1}$ due to the similar $\Delta V$ in both charge states, a reflection of Coulomb interaction between the defects and its local environment. We emphasize that the similar $q$-dependent $\Delta H_{\rm f}^{{D},{q}}(\eta)$ have also been observed for the intrinsic defects in other semiconductors, e.g., V$_{\rm C}$ in SiC (Fig. S1a), V$_{\rm Ga}$ in GaN (Fig. S1b), and V$_{\rm Zn}$ in ZnTe (Fig. S2a).
cpl-38-8-087103-fig2.png
Fig. 2. Rule I on strain-dependent $H_{\rm f}^{D,q}$. Left panels: schematic plotting of total energies $E_{\rm t}^{{D},{q}}(V)$ as a function of $V$ for (a) N vacancy (V$_{\rm N}$), (c) n-type (O$_{\rm N}$) and p-type (Mg$_{\rm Ga}$), and (e) n-type (S$_{\rm N}$) and p-type (Be$_{\rm Ga}$) in different charge states in GaN, respectively. $E_{\rm t}^{\rm host}(V)$ for GaN host (green lines) are also shown for comparison. Right panels: (b) Change of formation energies $\Delta H_{\rm f}^{{D},{q}}(\eta)$ for V$_{\rm N}$ in different charge states in GaN as a function of strain $\eta$. (d) Similar to (b) but for O$_{\rm N}$ and Mg$_{\rm Ga}$ dopants in different charge states in GaN. (f) Similar to (b) but for S$_{\rm N}$ and Be$_{\rm Ga}$ dopants in different charge states in GaN.
Secondly, we consider the substitutional doping in GaN. In order to understand the size effects of dopants, O and S are selected as n-type dopants, while Mg and Be are selected as p-type ones. For the case of O$_{\rm N}^{0}$, the electronic environment around the anion site induces a positive $\Delta V \sim +5.59$ Å$^{3}$/O$_{\rm N}^{0}$ [Fig. 2(c)]. Therefore, it is expected that the $\Delta H_{\rm f}^{{D},{q}}(\eta)$ of O$_{\rm N}^{0}$ will be dominated by the first term of Eq. (1) and exhibit a linear dependence of $\eta$. Indeed, as shown in Fig. 2(d), the calculated $\Delta H_{\rm f}^{{D},{q}}(\eta)$ of O$_{\rm N}^{0}$ linearly increases (decreases) as a function of compressive (tensile) $\eta$. When O$_{\rm N}^{0}$ is ionized to O$_{\rm N}^{+1}$, its $\Delta V$ largely shrinks to a negative value of $\Delta V \sim -5.75$ Å$^{3}$/O$_{\rm N}^{+1}$, which can make its energy curve in Fig. 2(c) largely left-shift. Consequently, the linear slope of $\Delta H_{\rm f}^{{D},{q}}(\eta)$ for O$_{\rm N}^{+1}$ will be inverted in comparison to that of O$_{\rm N}^{0}$, as also confirmed by our calculations shown in Fig. 2(d). Differing from O$_{\rm N}$, as shown in Fig. 2(e), the larger ionic size of S than N induces a much larger $\Delta V$ after the S$_{\rm N}$ doping ($\Delta V \sim +$17.72 Å$^{3}$/S$_{\rm N}^{0}$), which can give rise to a much larger linear slope of $\Delta H_{\rm f}^{{D},{q}}(\eta)$, as confirmed in Fig. 2(f). When S$_{\rm N}^{0}$ is ionized to S$_{\rm N}^{+1}$, its $\Delta V$ largely shrinks to a (still positive) value of $\sim $+6.13 Å$^{3}$/S$_{\rm N}^{+1}$, with a significant left-shift of its energy curve in Fig. 2(e). As a result, it is expected that the linear slope of $\Delta H_{\rm f}^{{D},{q}}(\eta)$ for S$_{\rm N}^{+1}$ will be reduced in comparison to that of S$_{\rm N}^{0}$, as confirmed by the calculations in Fig. 2(f). Therefore, depending on the initial different $\Delta V$ at neutral charge states, the n-type O$_{\rm N}^{+1}$ and S$_{\rm N}^{+1}$ can surprisingly have an opposite linear dependence of $\Delta H_{\rm f}^{{D},{q}}(\eta)$. However, the $\Delta V$ always decreases when electron is removed from the dopant site (more positively charged), and $H_{\rm f}^{{D},{q}}(\eta)$ follows $dE_{\rm t}^{{D},{q}}(\eta)/dV \sim -\Delta V$. Thirdly, we consider the p-type doping in GaN. As shown in Fig. 2(c), the $\Delta V$ of Mg$_{\rm Ga}^{0}$ is positive ($\Delta V \sim +4.06$ Å$^{3}$/Mg$_{\rm Ga}^{0}$) due to the larger ionic size of Mg than Ga, which can further expand when it is ionized to Mg$_{\rm Ga}^{-1}$ ($\Delta V \sim +8.73$ Å$^{3}$/Mg$_{\rm Ga}^{-1}$). Therefore, it is expected that the Mg$_{\rm Ga}^{-1}$ can have a similar linearity but enlarged slope effect after it is ionized, as indeed confirmed by our calculations in Fig. 2(d). On the other hand, as shown in Fig. 2(e), the $\Delta V$ for Be$_{\rm Ga}^{0}$ ($\Delta V \sim -3.26$ Å$^{3}$/Be$_{\rm Ga}^{0}$) is negative due to the small size of Be and after it is negatively charged, forming Be$_{\rm Ga}^{-1}$, again, the $\Delta V$ increases to $\sim -1.07$ Å$^{3}$/Be$_{\rm Ga}^{-1}$. Because of the rather small (absolute) value of $\Delta V$, it is expected that the $\Delta H_{\rm f}^{{D},{q}}(\eta)$ of Be$_{\rm Ga}$ will be dramatically converted from a linear dependence of $\eta$ to a parabolic dependence of $\eta$ after it is ionized. Indeed, this surprising phenomenon is confirmed by our calculations in Fig. 2(f). Therefore, depending on the initial different $\Delta V$ at the neutral charge states, the p-type Mg$_{\rm Ga}^{-1}$ and Be$_{\rm Ga}^{-1}$ can have either linear or parabolic dependence of $\Delta H_{\rm f}^{{D},{q}}(\eta)$. In general, we emphasize that the diverse trends of $q$-dependent $\Delta H_{\rm f}^{{D},{q}}(\eta)$ for different dopants in different semiconductors, e.g., Cl$_{\rm Te}$ in ZnTe (Fig. S2b), Al$_{\rm Si}$ and N$_{\rm C}$ in SiC (Fig. S3), C$_{\rm N}$ and Ge$_{\rm Ga}$ in GaN (Figs. S4a and S4b), Zn$_{\rm Ga}$, Si$_{\rm P}$, Ge$_{\rm Ga}$, and S$_{\rm P}$ in GaP (Figs. S4c and S4d) and As$_{\rm Si}$ in Si,[21] could be understood in a similar way via tracking the evolution of the $q$-dependent $\Delta V$. Importantly, some related experimental observations, e.g., the solubility of B in Si can be greatly enhanced by a compressive strain,[39,40] can be well understood in a similar way by our above analysis. From the above analysis, we can safely reach the Rule I on $\eta$-dependent $\Delta H_{\rm f}^{{D},{q}}(\eta)$ in semiconductors: $\Delta H_{\rm f}^{{D},{q}}(\eta)$ of point defects under different charge states can have either parabolic ($\Delta V\sim 0$) or linear ($\Delta V\ne 0$) dependence, i.e., $dE _{\rm t}^{{D},{q}}(V)/dV \sim -\Delta V$. Here, the defect-induced $\Delta V$ is strongly $q$-dependent, it increases (decreases) when electron is added (removed) to (from) the dopant site. We would emphasize that this rule is independent of the sizes of supercell calculations (Fig. S5) or the forms of strains (Fig. S6). Rule II: Strain-Dependent ${\varepsilon} ^{q/q'}$. According to Eq. (3), the $\eta$-dependent a-$\varepsilon ^{q/q'}(\eta)$, i.e., without the consideration of $\Delta E_{\rm VBM}(V)$, is solely determined by the $\eta$-dependent $\Delta E_{\rm t}^{{D},{q}}(\eta)$. In general, for a p-type acceptor in a semiconductor [Fig. 3(a)], in order to increase (decrease) the value of a-$\varepsilon ^{0/-1}$, one needs to increase (decrease) its $H_{\rm f}^{D ,-1}$ more than $H_{\rm f}^{D ,0}$, which can be achieved under a negative (positive) $\eta$ in terms of Rule I. Similarly, for an n-type donor [Fig. 3(a)], in order to increase (decrease) the value of a-$\varepsilon ^{0/+1}$, one needs to increase (decrease) its $H_{\rm f}^{D ,0}$ more than $H_{\rm f}^{D ,+1}$, which can also be achieved under a negative (positive) $\eta$. We emphasize that those trends shown in Fig. 3(a) are universal for different point donors or acceptors in different semiconductors, due to the robustness of Rule I on $q$-dependent $\Delta H_{\rm f}^{{D},{q}}(\eta)$ [$\Delta E_{\rm t}^{{D},{q}}(\eta)$]. As shown in Fig. 3(b), taking the p-type Mg$_{\rm Ga}$ and n-type O$_{\rm N}$ in GaN as typical examples, it is unsurprising that the calculated strain-dependent a-$\varepsilon ^{0/-1}(\eta)$ of Mg$_{\rm Ga}$ and a-$\varepsilon ^{0/+1}(\eta)$ of O$_{\rm N}$ in GaN fully confirm our analysis in Fig. 3(a). The a-$\varepsilon ^{0/-1}$ (a-$\varepsilon ^{0/+1}$) moves towards VBM (CBM) under a tensile (compressive) $\eta$. Again, this trend shown in Fig. 3(b) is general for all the isolated point defects due to the generality of Rule I. It is also independent of the $\Delta V$ of defects at their neutral charge states, e.g., see the cases of Be$_{\rm Ga}$ and S$_{\rm N}$ in GaN (Fig. S7), Al$_{\rm Si}$ and N$_{\rm C}$ in SiC (Fig. S8) and Mg$_{Al}$ in AlN (Fig. S9). The above discussions can drive us to Rule II that the negative (positive) $\eta$ is always beneficial for the realization of shallower a-$\varepsilon ^{q/q'}(\eta)$ for the donor (acceptor) in semiconductors.
cpl-38-8-087103-fig3.png
Fig. 3. Rule II on strain-dependent a-${\varepsilon} ^{q/q'}$. (a) Schematic plotting of the change of formation energy $H_{\rm f}^{{D},{q}}(\eta)$ for positively charged donor (blue) and negatively charged acceptor (red) in a semiconductor, with respect to their neutral states (gray), under different $\eta$ as a function of $E_{\rm F}$. Arrows indicate the directions of those changes from without $\eta$ to with $\eta$. (b) The a-$\varepsilon ^{0/-1}(\eta)$ and a-$\varepsilon ^{0/+1}(\eta)$ for Mg$_{\rm Ga}$ and O$_{\rm N}$ in GaN as a function of strain $\eta$, respectively. Here the band edge positions are fixed at the values of $E_{\rm VBM}(V_{0})$ and $E_{\rm CBM}(V_{0})$.
Rule II can explain some related experimental observations, e.g., the ionization energy of B in Ge$_{x}$Si$_{1-x}$ alloy gradually decreases with the increase of Ge composition,[41] attributed to the Ge-induced tensible strain effects. As discussed later, it should be noticed that the shallowness of $\varepsilon ^{q/q'}(\eta)$ relative to the VBM of the strained system also depends on the shift $\Delta E_{\rm VBM}(V)$ as shown in Eq. (3). Rule III: Strain-Dependent $E_{\rm pin}$. Rules I and II demonstrate that the strain can always induce opposite changes of $H_{\rm f}^{{D},{q}}(\eta)$ for donors and acceptors, which can be utilized to tuning the Fermi level pinning $E_{\rm pin}$ positions. The $E_{\rm pin}$ level is determined by the Fermi energy at which the compensating donor and acceptor defects holding opposite charge states have the same energy. It is known that n-type $E_{\rm pin - n}$ and p-type $E_{\rm pin - p}$ positions set up the doping limits of n-type and p-type doping, respectively, in a semiconductor, which is an intrinsic problem of semiconductors that are difficult to overcome.[35] Basically, based on the doping limit rules, a semiconductor with low VBM, e.g. GaN,[35] is difficult to be doped p-type, while a semiconductor with high CBM, e.g. ZnTe,[42] is difficult to be doped n-type. For GaN, Mg (Mg$_{\rm Ga}$) has been widely selected as a p-type dopant,[2,37,43] which has an $\varepsilon ^{0/-1}$ at VBM $+$0.3 eV [Fig. 3(b)]. However, when Fermi level $E_{\rm F}$ is shifted towards VBM after Mg$_{\rm Ga}$ doping, the spontaneous formation of V$_{\rm N}^{+3}$ can compensate the p-type doping induced by Mg$_{\rm Ga}^{-1}$, giving rise to a deep $E_{\rm pin - p}$ position locating at VBM $+$0.7 eV, agreeing with previous calculations.[38] As shown in Fig. 4(a), although the $\varepsilon ^{0/-1}$ of Mg$_{\rm Ga}$ is relatively shallow, its p-type doping performance is strongly downgraded by the formation of V$_{\rm N}^{+3}$. To reduce $E_{\rm pin - p}$ position in GaN, one needs to increase the $H_{\rm f}^{{D},{q}}$ of V$_{\rm N}^{+3}$ but decrease the $H_{\rm f}^{{D},{q}}$ of Mg$_{\rm Ga}^{-1}$, which can be achieved under a tensile-$\eta$-induced synergistic effect. Indeed, as shown in Fig. 4(c), our calculations confirm that the $H_{\rm f}^{q}(\eta)$ of V$_{\rm N}^{+3}$ (Mg$_{\rm Ga}^{-1}$) can gradually increase (decrease) as a function of tensile $\eta$, linearly shifting the absolute $E_{\rm pin - p}(\eta)$ [a-$E_{\rm pin - p}(\eta)$, without consideration of $\Delta E_{\rm VBM}(V)$] towards lower energy positions. For the case of ZnTe, Cl (Cl$_{\rm Te}$) is commonly used as an n-type dopant[44,45] with a calculated $\varepsilon ^{0/+1}$ at CBM $-$0.43 eV, agreeing with the previous calculations.[46] Unfortunately, as shown in Fig. 4(b), the spontaneous formation of V$_{\rm Zn}^{-2}$ can largely pin the $E_{\rm pin - n}$ position deeply inside the bandgap, i.e., $E_{\rm pin - n}$ = CBM $-$0.84 eV, preventing the ionization of Cl$_{\rm Te}$. Similarly, as shown in Fig. 4(d), a compressive-$\eta$-induced synergistic effect may decrease the $H_{\rm f}^{D,q}$ of Cl$_{\rm Te}^{+1}$ but increase the $H_{\rm f}^{{D},{q}}$ of V$_{\rm Zn}^{-2}$, giving rise to the shift of a-$E_{\rm pin - n}$ towards higher energy positions. Based on the above understandings, we can arrive at Rule III on the $\eta$-dependent a-$E_{\rm pin}(\eta)$ in semiconductors: the tensile (compressive) strain is always beneficial for the realization of shallower a-$E_{\rm pin - p}(\eta)$ [a-$E_{\rm pin - n}(\eta)$] in semiconductors. Again, as discussed later, the shallowness of $E_{\rm pin - p}(\eta)$ [$E_{\rm pin - n}(\eta)$] relative to the VBM (CBM) of the strained system also depends on the shift $\Delta E_{\rm VBM}(V)$ [$\Delta E_{\rm CBM}(V)$]. Again, we emphasize that Rule III is robust for different semiconductors due to the robustness of Rules I and II. In addition, Rules I–III are generally valid for different semiconductor systems under the biaxial strain (Fig. S6), because Rule I depends solely on the defect-induced $\Delta V$.
cpl-38-8-087103-fig4.png
Fig. 4. Rule III on strain-dependent a-$E_{\rm pin}$. (a) Formation energy $H_{\rm f}^{{D},{q}}$ of external Mg$_{\rm Ga}$ and intrinsic compensating V$_{\rm N}$ in GaN without and with a $+$2% strain. (b) Formation energy $H_{\rm f}^{{D},{q}}$ of external Cl$_{\rm Te}$ and intrinsic compensating V$_{\rm Zn}$ in ZnTe without and with a $-2$% strain. The a-$E_{\rm pin}$ positions are marked by the vertical lines in (a)–(b). (c) a-$E_{\rm pin-p}(\eta)$ and (d) a-$E_{\rm pin-n}(\eta)$ positions as a function of $\eta$ in GaN and ZnTe, respectively. Here the band edge positions are fixed at the values of $E_{\rm VBM}(V_{0})$ and $E_{\rm CBM}(V_{0})$.
Role of $\Delta {E} _{\rm VBM}(V)/\Delta {E} _{\rm CBM}(V)$. Since Rules II and III are for a-$\varepsilon ^{q/q'}(\eta$) and a-$E_{\rm pin}(\eta)$, in order to obtain the rules of $\varepsilon ^{q/q'}(\eta)$ and $E_{\rm pin}(\eta)$ with respect to the band edge states under strain, the $\Delta E_{\rm VBM}(V)/\Delta E_{\rm CBM}(V)$ needs to be taken into account. As shown in Fig. 5(a), in most common semiconductors with VBM (CBM) as bonding (anti-bonding) states, the $\Delta E_{\rm VBM}(V)$ [$\Delta E_{\rm CBM}(V)$] will almost linearly increase (decrease) as a function of $V$ from compression to tension.[36] Therefore, combining Rules II, III and Fig. 5(a), we can easily reach the conclusion that Rules II and III can also apply for the relative $\varepsilon ^{q/q'}(\eta)$ of acceptors and $E_{\rm pin-p}(\eta)$, because the opposite trend of $\eta$-dependent a-$\varepsilon ^{0/-1}(\eta)$ [a-$E_{\rm pin-p}(\eta)$] and $\Delta E_{\rm VBM}(V)$ can induce a novel synergistic effect to make the $\varepsilon ^{q/q'}(\eta)$ [$E_{\rm pin-p}(\eta)$] of acceptors even shallower under a tensile strain. Indeed, as shown in Figs. 5(b) and 5(c), our calculations on $\varepsilon ^{q/q'}$ and $E_{\rm pin-p}$ in GaN confirm our expectation. The calculated $\varepsilon ^{0/-1}$ [$E_{\rm pin - p}$] of Mg$_{\rm Ga}$ is shifted from VBM $+$0.3 [VBM $+$0.7] eV to VBM $-0.15$ [VBM $+$0.27] eV under $\eta =+$2%, significantly shallower than that of a-$\varepsilon ^{0/-1}(\eta)$ [a-$E_{\rm pin - p}$]. The trend of $\varepsilon ^{q/q'}$ shown in Fig. 5(b) is consistent with the similar experimental observations that the Mg$_{\rm Ga}$ level becomes deeper in Al$_{x}$Ga$_{1-x}$N as the $x$ increases, due to the Al-induced compressive strain and band edge shift.[47] We emphasize that there are no similar rules for $\varepsilon ^{q/q'}(\eta)$ of donors and $E_{\rm pin-n}(\eta)$ in common semiconductors. Because of the $\eta$-dependent a-$\varepsilon ^{0/+1}(\eta)$ [a-$E_{\rm pin-n}(\eta)$] and $\Delta E_{\rm CBM}(V)$ shift in the same direction, the relative shift will depend on their individual changes.
cpl-38-8-087103-fig5.png
Fig. 5. Role of $\Delta {E} _{\rm VBM}({V})/ \Delta {E} _{\rm CBM}({V})$. (a) Schematic plotting of the band edge changes as a function of $V$ for common semiconductors with VBM (CBM) as bonding (anti-bonding) states. (b) The $\varepsilon ^{0/-1}(\eta)$ of Mg$_{\rm Ga}$ and (c) $E_{\rm pin-p}(\eta)$ in GaN as a function of strain $\eta$, with consideration of $\Delta E_{\rm VBM}(V)$.
In summary, for the first time, we have developed a universal theory and consequently established three basic rules for understanding the diverse strain-dependent doping behaviors in semiconductors, which can be applied to tune their $H_{\rm f}^{{D},{q}}$, $\varepsilon ^{q/q'}$, and $E_{\rm pin}$, as successfully confirmed by the first-principles calculations on several exemplary semiconductors. Due to the robustness of our developed universal theory and Rule I, it is reasonable to expect that Rules II and III are also robust in different semiconductors. Therefore, these basic rules can be widely applied to control doping and simultaneously overcome the doping bottlenecks in semiconductors via simple strain engineering. Since the strain effects widely exist in different semiconductor films during the growths or operations, our theory can be adopted as a guideline to induce the “right” strain effects to improve the doping performance of semiconductors. The authors thank L. Kang, L. Hu, and J. F. Wang for helpful discussions. The calculations were performed on Tianhe2-JK at CSRC. References First-principles calculations for point defects in solidsStrained silicon as a new electro-optic materialStrain-engineered artificial atom as a broad-spectrum solar energy funnelA micromachining-based technology for enhancing germanium light emission via tensile strainEffects of strain on the band structure of group-III nitridesFlexo-photovoltaic effectSuperconductivity in Shear Strained SemiconductorsInterplay of Strain and Magnetism in FeSe Monolayers *Density functional theory study of Zn X ( X = O ,   S ,   Se ,   Te ) under uniaxial strainStrain engineering and epitaxial stabilization of halide perovskitesBending strain engineering in quantum spin hall system for controlling spin currentsFirst-principles study of electronic properties of biaxially strained silicon: Effects on charge carrier mobilityStrain-Engineered Surface Transport in Si(001): Complete Isolation of the Surface State via Tensile StrainDynamics and efficient conversion of excitons to trions in non-uniformly strained monolayer WS2Anisotropic Strain Enhanced Hydrogen Solubility in bcc Metals: The Independence on the Sign of StrainStrain-engineered diffusive atomic switching in two-dimensional crystalsLarge enhancement of boron solubility in silicon due to biaxial stressPhysics of strain effects in semiconductors and metal-oxide-semiconductor field-effect transistorsAntimony for n-type metal oxide semiconductor ultrashallow junctions in strained Si: A superior dopant to arsenic?Stress effects on impurity solubility in crystalline materials: A general model and density-functional calculations for dopants in siliconStrain-Enhanced Doping in Semiconductors: Effects of Dopant Size and Charge StateEpitaxial Strain-Induced Chemical Ordering in La 0.5 Sr 0.5 CoO 3−δ Films on SrTiO 3Enhancing magnetic vacancies in semiconductors by strainStrain-controlled oxygen vacancy formation and ordering in CaMnO 3 Strain tuning of native defect populations: The case of Cu 2 ZnSn(S,Se) 4High Mg effective incorporation in Al-rich AlxGa1 - xN by periodic repetition of ultimate V/III ratio conditionsPolaron-enhanced giant strain effect on defect formation: The case of oxygen vacancies in rutile TiO 2 A polarization-induced 2D hole gas in undoped gallium nitride quantum wellsStrain-Engineered Modulation on the Electronic Properties of Phosphorous-Doped ZnOStrained-Layer Epitaxy of Germanium-Silicon AlloysElastic constants of defected and amorphous silicon with the environment-dependent interatomic potentialChemical potential dependence of defect formation energies in GaAs: Application to Ga self-diffusionNative defects and self-compensation in ZnSeOvercoming the doping bottleneck in semiconductorsAb initio all-electron calculation of absolute volume deformation potentials of IV-IV, III-V, and II-VI semiconductors: The chemical trendsComputationally predicted energies and properties of defects in GaNSelf-compensation due to point defects in Mg-doped GaNElectrical and structural characterization of boron-doped Si1−xGex strained layersBulk Silicon Crystals with the High Boron Content, Si 1– x B x : Two Semiconductors Form an Unusual MetalAlloy effects in boron doped Si-rich SiGe bulk crystalsA phenomenological model for systematization and prediction of doping limits in II–VI and I–III–VI2 compoundsHighly P-Typed Mg-Doped GaN Films Grown with GaN Buffer LayersDoping of ZnTe by molecular beam epitaxyPhotoluminescence of Cl-doped ZnTe epitaxial layer grown by atmospheric pressure metalorganic vapor phase epitaxyDonor–donor binding in semiconductors: Engineering shallow donor levels for ZnTePhotoluminescence studies of impurity transitions in Mg-doped AlGaN alloys
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